bifilar and trifilar report

KUWAIT UNIVERSITYCOLLEGE OF ENGINEERING AND PETROLEUM

MECHANICAL AND ENGINEERING DEPARTMENTMechanical Engineering Fundamentals Lab

ME-373

Engineering Formal Laboratory ReportExperiment 5

Bifilar and Trifilar Suspensions(Mass Moment of Inertia Determination)

ByAhmad Abdullah Al-Kandri 206217099Hassan Hussain Saleem 206216926Nayef Nazih Ftouni 206216827

On our honor we pledge that this work of us does not violate the university provisions an academic misconduct By signing below we certify that we understand the university policies on academic misconduct and that when an act of academic misconduct is committed all parties involved in violation

1

Abstract

In this experiment a bifilar and trifilar suspension methods are applied to determine the mass momentum of inertia of a beam platform cylindrical mass and a body with an irregular mass

First with the bifilar method a body was suspended by two parallel cords with the same length and by displacing the system through a small angle and using few equations which will be given further in the report we get the time period and then calculating the mass moment of inertia

Moreover and for the trifilar method a platform was suspended by three cords having the same length and also by getting the time period and the moment of inertia of the platform we calculated the moment of inertia of a cylindrical mass and a body with an irregular mass

2

Table of contents

Abstract helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip (3)Table of contents helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip (4)List of figures and tableshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip (5) Nomenclaturehelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip (6)

Chapter 1

11 Introduction helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip helliphelliphelliphellip(7)12 objectives helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip(8)

Chapter 2

Theoretical background helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip (9)

Chapter 3

Experimental setup and procedure helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip (15) 31 Equipments helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip (15) 32 Procedurehelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip (16)

Chapter 4

Results and discussionhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip (18) 41 Sample calculationshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip (19) 42 Discussion helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip (39) 43 Uncertainty Analysis helliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip (40)

Chapter 5

Conclusionshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip (42)

Referenceshelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip (43)

3

List of figures and tables

Figures

Figure (1) Bifilar suspensionhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip (7)

Figure (2) (a) Bifilar suspension (b) Trifilar suspensionhelliphelliphelliphelliphelliphelliphelliphellip (10)

Figure (3) The relation between log L and log τhelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphelliphellip (38)

4

List of tables

Table 1 Bifilar suspension with cord length equal to 04 (m)25Table 2 Bifilar suspension with cord length equal to 045 (m)26Table 3 Bifilar suspension with cord length equal to 048 (m)27Table 4 Trifilar suspension with cord length equal to 04 (m)35Table 5 Trifilar suspension with cord length equal to 044 (m)36Table 6 Trifilar suspension with cord length equal to 05 (m)38

5

Nomenclature

SymbolsScientific Name

dDistance between the cords (m)

FRestoring force (N)

gAcceleration of gravity (ms2)

IMass-Moment of inertia of the body (m4)

k Radius of gyration

L Length of the cords (m)

m

MR

T

θ

Mass (kg)

Restoring moment (Nm)

Tension in the cords (Nm)

Angle of twist (degrees)

umlθ(ω) Angular velocity (rads)

τ Time period (s)

Chapter 16

Introduction

In this chapter we will introduce to you two methods for determining the mass moment of inertia which are the bifilar and trifilar suspension

11 Bifilar suspension method

The bifilar suspension method is a way to determine the mass momentum of inertia (Fig a) shows a uniform rod of mass M and length L suspended horizontally by two vertical strings The length of each string is l and they are attached symmetrically about the center O a distance R apart If the bat is now twisted horizontally it will undergo SHM We wish to analyze this motion and in particular to find the period

And by calculating the time period we can calculate the mass moment of inertia by using some formulas we will provide later in the report

7

A B

Brsquo

O

R

l

L

Figure 1 Bifilar suspension

BA

Arsquo

φ

θ

12 Trifilar suspension method

The trifilar suspension method is a way to determine the mass moment of inertia for a bodies especially a body with an irregular mass as you can see in figure(c) a platform is suspended with three cords and three cords have the same length and they are spaces around the platform center now if we want to determine the mass of inertia we put the object or the body at the platform in the centroidal axis And by displacing the platform through a small angel we can calculate the mass time period and then using some formulas we will provide later we can determine the mass moment of inertia

13 Objectives

The objectives of this experiment are to introduce the bifilar and trifilar suspension methods as options for determining the mass moment of inertia of a rigid body or group of rigid bodies this should lead to the capability of designing experiments for determining moments of inertia of rigid bodies using either bifilar or trifilar pendulum systems

The other objective is to compare theoretical results obtained from pendulum modeling with experimental results and determining accuracy

8

Chapter 2

Theoretical Background

Inertia Determination

Mass- moment of inertia I about an axis of rotation is theoretically can be determined by

IiquestintM

r2 dm (1)

Where

r Distance between the location of mass element dm and the axis of rotation The integration is performed over the entire mass of the body if the axis of rotation coincides with the body centroidal axis the moment of inertia is given by Ī

The parallel axis theorem is given by

I= Ī + m d2 (2)

Where

I The inertia of the body about the axis of rotation

Ī the inertia of the body about its own centroidal axis

m Mass of the body

d the perpendicular distance from the centroidal axis of the body to the axis of rotation

9

Bifilar suspension

Figure 2 (a) Bifilar suspension (b) Trifilar suspension

In figure1 (a) shows the Bifilar suspension and the body is suspended by two parallel cords of length L at a distance R=d apart the tensions in the cords are respectively

T 1=mgd2

d and T 2=mg

d1

d (4)

d1+d2=d and T 1+T 2=mg(5)

If the system is displaced through a small angle θ about its central axis then angular displacement Φ1and Φ2 will result at the supporting chords If both angles are small then

10

Lϕ1=d1Ɵ and Lϕ2=d2

Ɵ (6)

ϕ1= d1

L Ɵ and ϕ2=

d2

L Ɵ (7)

The horizontal restoring forces at the points of attachment with the plate are

F1=iquest T1 sinΦ1iquest (8)

F2=iquest T2 sin Φ2iquest (9)

And using equation (7) to substitute into (8) and (9) and then get

MR= iquest (10)

Where the minus sign is related to the restoring effect

Using equation (4) the restoring moment is rewritten as

MR= -mg d1d2

L Ɵ (11)

Using Newtonrsquos second law and summing moment about the vertical centroidal axis give equation of motion of the body can be written as

Ī ϴ΄΄ = - mg d1d2

L Ɵ rarr ϴ΄΄+mg

d1d2

Ī LƟ=0 (12)

Where

Ī = Mass-moment of inertia of the body about the centroidal axis

The natural frequency of the system can be directly obtained from as

ω=radicmgd1 d2

Ī L (13)

The period that measured experimentally is related to the natural frequency by

11

τ=2 πω = 2 π radic Ī L

mgd1d2 (14)

Using equation (14) the experimental mass-moment of inertia is given by

Ī =mgd1d2 τ

2

4 π2 L (15)

Equation (15) can be rewritten in terms of the radius of gyration

τ=2πk radic Lgd1 d2

(16)

Wherek = Radius of gyration defined by

Ī = m k 2 (17)

To reduce the error different readings at different cords length L should be taken and a relation between the cord length and the corresponding mass moment of inertia is to be obtained In this regard equation (15) can be written as

L = mgd1d2 τ

2

4 π2 Ī (18)

This can be rewritten as L = k τ 2 (19)

With k defined as

k = mgd1d2

4 π2 Ī (20)

12

Trifilar suspension

The trifilar maybe more useful when the shape is highly irregular and a third suspension is required to give a more stable and consistent oscillation Figure1 (b) outlines the trifilar suspension in detail

Where the platform is suspended by three cords of equal length and is equally spaced about the platform centre And the round platform serves as triangular one The part whose moment of inertia is to be determined is carefully placed on the platform so that its centroidal axis coincides with that of platform

The platform is then made to oscillate and the number of oscillate is counted over a specific period of time

The sum of moment sumM about a vertical axis coinciding with the centroidal axis of the body is

ΣM=minusR (m+mp )g sinemptyminus( Ī+ I p )ϴ΄΄=0 (21)

Where

m mass of body whose moment of inertia is to be determinedm p Mass of platformĪ Moment of inertia of body about its centroidal axisI p Moment of inertia of platform about its centroidal axisR distance form platform center of mass to suspension point Ɵ platform displaced angle120601 cord displaced angleL cord length

Since dealing the small displacement angles the angles can be equated to the angles themselves (in radians) Therefore

120601 = RL

Ɵ

(22)

13

And equation (20) becomes

ϴ΄΄+(m+m p ) R2 g

L ( Ī+ I p )θ=0 (23)

Therefore the period is given by

τ=2π radic L ( Ī + I p )(m+mp ) R2 g

(24)

And the experimental mass moment of inertia is given as

Ī+ I p=(m+m p ) R2 g

L ( τ2π )

2

(25)

To reduce the error during the experiment more than one measurement is required Along this line a relation between the cord length L and the period τ is to be obtained

L = (m+m p )R2 g

4 π2 ( Ī + I p ) τ2 (26)

This can be rewritten as

L = C τ 2 (27)

With C defined as

C = (m+mp )R2 g

4 π2 ( Ī + I p ) (28)

14

Chapter 3

Experimental setup and procedure

This chapter is divided into four parts and each part is illustrated briefly First of all the needed equipments for the Bifilar and Trifilar experiment will be illustrated with some figures for more details

While proceeding in this chapter the procedure involved in the Bifilar and Trifilar experiment will be explained briefly Beginning with adjusting the length of the cords until recording the needed data such as the oscillation time

Finally after the set of procedure is done for the both cases The study of the relationship between the mass-moment of inertia of the body and the time period is simply investigated

31 Equipments

In order to perform the bifilar and trifilar experiment a set of equipments is needed and is illustrated as follows

1 Stop watch2 Ruler3 Electronic scale for weighting parts

For the previous set of equipments some helpful notes should be considered such as being careful while recording the cord length and weighting parts

15

32 Procedure

This section of this chapter is divided into two parts First the procedure related to the bifilar apparatus is illustrated Secondly the procedure for the trifilar apparatus is discussed briefly

321 Bifilar apparatus procedure

In order to perform this case of this experiment sufficiently and achieve the needed results with a great possibility to eliminate the human errors as much as possible a specific set of procedures should be followed and is illustrated as follows

1 Measure the dimension and the mass of the beam

2 Suspend the beam by the cords and adjust their common length L

3 Displace the beam by small angle release it and determine the periodic time of free oscillations by timing 20 oscillations

4 Repeat step 3 for three different values of L

5 Measure the dimensions of the cylindrical masses

6 Place the cylindrical masses of known weight either side of the center of gravity of the beam maintaining a specified distance between the masses

7 Repeat the procedure with an irregularly shaped body that can be fixed on the beam

8 In each test record L time period and the mass of the beam Then use equation (13) and (14) to calculate the mass-moment of inertia and radius of gyration of the beam alone and of the beam with masses placed on it

16

322 Trifilar apparatus procedure

1 Measure the mass and radius of the plate Suspend the plate by the wires and adjust their common length L very accurately

2 Displace the plate with a small angle release it and determine the periodic time of free oscillations by timing 30 oscillations

3 Repeat step 2 for different values of L

4 Find the mass of the body then place it on the plate so heir centers of gravity coincide Repeat steps 2 and 3

5 Apply equation (22) to determine the mass-moment of inertia of the body and radius of gyration in each case Estimate the error involved

17

Chapter 4

Results and discussion

In this chapter some sample calculations are obtained in order to achieve the desired results In the other hand these results which are obtained are discussed briefly

As a first step for both the bifilar and trifilar setups the time period theoretical and experimental mass-moment of inertia of different cases and the percentage error between them and the radius of gyration of the beam alone and of the beam with masses placed on it

18

41 Sample calculations

Bifilar Apparatus

1 Total mass of frame hanging with wires = 1444 kg2 Width of frame = 2525 mm3 Height of frame = 1275 mm4 Length of frame = 503 mm5 Distance between the wires = 15 mm6 Number of holes in hanging rectangular x-section bar = 157 Distance between each hole from center to center = 25 mm8 Diameter of each hole = 1042 mm9 Diameter of each cylinder fixed at the end of rectangular section = 20 mm10 Length of each cylinder fixed at the end of rectangular section = 50 mm11 Length of wires = 400mm 450mm 480mm12 Diameter of cylindrical mass 762513 Height of cylindrical mass = 509 mm14 Mass of each cylindrical mass = 179315 Distance between two masses 150mm 200mm 250mm16 Mass of the irregular shape = 2513 kg

19

For the Bifilar apparatus case the following three cases are to be investigated where the cord length was 400 (mm) as follows

(a) The beam only

Theoretical

I zz=I (beam )minusI ( holes )+ I (iquestcylinders)

I zz=(((112)lowast1444lowast(05032+0025252))minus(2lowast(1444lowast(0005212)minus(000008528lowast(00252))))minus(2lowast(1444lowast(0005212)minus(000008528lowast(0052))))minus(2lowast1444lowast(0005212))minus(000008528lowast(00752)))minus(2lowast(1444lowast(0005212)minus(000008528lowast(012))))minus(2lowast(1444lowast(0005212)minus(000008528lowast(01252))))minus(2lowast(1444lowast(0005212)minus(000008528lowast(0152))))minus(2lowast(1444lowast(0005212)minus(000008528lowast(01752))))+(2lowast(05lowast01233lowast(0012)+01233lowast025252))

= 045721497 (m4)

Experimental

20

For first trail when the length of the cord = 400 mm with having 20 oscillations the calculated mass-moment of inertia of the platform and the time period were found to be

τ = time

cycles = 1791

20 = 08955 (s)

And the mass-moment of inertia of the platform was calculated using equation (13) as follows

I p = (m )d1 d2 gτ 2

4π 2 L=

1444lowast02525lowast02525lowast076752lowast(981)4π 2(04)

= 045501 (m4)

Also the percentage error was found as shown

Error = t h eoriticalminusexperimemtal

t heoriticallowast100=0482

(b) The beam with one cylindrical mass placed on it

21

Theoretical

I=I ( plate )minusI (holes )+ I (iquestcylinders )+ I (cylinder placed)

iquest(((112)lowast1444lowast(05032+0025252))minus(2lowast(1444lowast(0005212)minus(000008528lowast(00252))))minus(2lowast(1444lowast(0005212)minus(000008528lowast(0052))))minus(2lowast1444lowast(0005212))minus(000008528lowast(00752)))minus(2lowast(1444lowast(0005212)minus(000008528lowast(012))))minus(2lowast(1444lowast(0005212)minus(000008528lowast(01252))))minus(2lowast(1444lowast(0005212)minus(000008528lowast(0152))))minus(2lowast(1444lowast(0005212)minus(000008528lowast(01752))))+(2lowast(05lowast01233lowast(0012)+01233lowast025252))+2lowast(05lowast1793lowast00381252)

= 0048327651 (m4)

Experimental

Using equation (13) the following experimental data of the mass-moment of inertia was found as follow

τ = timecycles =

121320

=06065(s)

I = (m )d1 d2 τ

2 g

4π 2 L

iquest (1444+1793 ) (052 )2(02525)(02525)(981)

4 π 2(04)

= 0045155684 (m4 iquest

Error = theoriticalminusexperimemtal

theoritical=65634

( c ) The beam with two cylindrical masses spaced 250 (mm)

Theoretical

22

I = I (plate) ndash I (holes) +I (fixed cylinders) + I (cylindrical masses placed)

=

iquest(((112)lowast1444lowast(05032+0025252))minus(2lowast(1444lowast(0005212)minus(000008528lowast(00252))))minus(2lowast(1444lowast(0005212)minus(000008528lowast(0052))))minus(2lowast1444lowast(0005212))minus(000008528lowast(00752)))minus(2lowast(1444lowast(0005212)minus(000008528lowast(012))))minus(2lowast(1444lowast(0005212)minus(000008528lowast(01252))))minusiquest

= 0104358901 (m4 iquest

Experimental

In this case with the two cylindrical masses placed 250 (mm) the mass-moment of inertia was calculated using equation (13) as follows

τ = timecycles =

150920 = 07545 (s)

I = (m )d1 d2 gτ 2

4π 2 L

= (503 ) (05735 )2 (02525 ) (02525 ) ( 981 )

4 π2 (04 )

= 0104606813 (m4 iquest


theoriticallowast100 = 0294

As the previous cases were discussed briefly on the theoretical and experimental basis the same previous steps were repeated for different cord length as illustrated in the next pages tables

23

24

Table 1 Bifilar suspension with cord length equal to 04 (m)

SystemCord

Length L (m)

No of Oscillations

(cyclesrsquo)

Time of Oscillations

(sec)

Period τ

Average time (s)

Average Period

Plate Moment of inertia Error

Theoretical Experimental

Only beam04 20 1785 08925

1782666667 0891333333 00457214970045196926 1147318216

04 20 1772 0886 0044540993 258194598404 20 1791 08955 0045501282 0481645472

Beam with one mass in the center

04 20 1206 06031209333333 0604666667 0048327651

0044636018 763875978504 20 1213 06065 0045155684 656346126304 20 1209 06045 0044858364 7178679599

Beam with two masses of distance

150 mm

04 20 1185 059251185666667 0592833333 0068498901

0064508686 582522521804 20 1178 0589 0063748809 693455239104 20 1194 0597 0065492285 4389290021


200 mm

04 20 1328 06641324666667 0662333333 0084187651

0081017277 376584191504 20 1332 0666 0081506068 318524499304 20 1314 0657 0079318085 5784180162


250 mm

04 20 1485 074251501 07505 0104358901

0101305821 362651766404 20 1509 07545 0104606813 029447598504 20 1509 07545 0104606813 0294475985

Irregular shape04 20 1635 08175

1634666667 0817333333 00487731420061356248 1494649851

04 20 1625 08125 0060608008 149464985104 20 1644 0822 0062033588 1494649851

25


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only beam045 20 1863 09315

1887666667 0943833333 00457214970043762855 4283854264

045 20 19 095 0045518416 04441707045 20 19 095 0045518416 04441707


045 20 13 0651286666667 0643333333 0048327651

0046102556 4604185043045 20 1282 0641 0044834709 7227626401045 20 1278 0639 0044555366 7805646014


150 mm

045 20 1268 06341264333333 0632166667 0068498901

006565495 4151819235045 20 1272 0636 0066069831 3546145034045 20 1253 06265 0064110788 6406107612


200 mm

045 20 1503 075151504 0752 0084187651

009224587 9571735346045 20 1506 0753 0092614485 10009584045 20 1503 07515 009224587 9571735346


250 mm

045 20 1575 078751580666667 0790333333 0104358901

0101295485 3638795325045 20 1585 07925 010258586 2106058254045 20 1582 0791 010219789 2566897967


1722333333 0861166667 00487731420061414329 1501548924

045 20 1716 0858 0060076597 1501548924045 20 1716 0858 0060076597 1501548924

26


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only beam048 20 1953 09765

1960333333 0980166667 00457214970045087453 1386753778

048 20 1956 0978 0045226077 1083561808048 20 1972 0986 0045968998 0541321699


048 20 1344 06721337666667 0668833333 0048327651

0046196398 4410006579048 20 1328 0664 0045103031 6672411469048 20 1341 06705 0045990395 4836271348


150 mm

048 20 1322 06611303333333 0651666667 0068498901

0066905706 2325869495048 20 1325 06625 0067209707 1882064854048 20 1263 06315 0061067051 1084959004


200 mm

048 20 1463 073151458666667 0729333333 0084187651

0081938668 2671393071048 20 1466 0733 0082275056 2271823432048 20 1447 07235 0080156235 4788607411


250 mm

048 20 1628 08141622 0811 0104358901

0101463315 3439442497048 20 1613 08065 0099602212 5650103475048 20 1625 08125 0101089716 3883211714


1751333333 0875666667 00487731420059651783 1292189689

048 20 1747 08735 0058375127 1292189689048 20 1741 08705 0057974842 1292189689

27

As investigated from the previous tables related to the bifilar suspension it is noticeable that the percentage error related to the theoretical and the experimental results is somehow small and acceptable

28

Trifilar Apparatus

1 Total mass of round plate = 4365 kg2 Diameter of plate = 628mm3 Thickness of plate 334mm4 Length of wire 828mm5 Radius of circle at wire fixing position 272mm6 Inside diameter of hallow cylinder 8362mm7 Outside diameter of hallow cylinder = 200mm8 Height of hallow cylinder = 200mm9 Mass of hallow cylinder = 1386 kg10 Mass of irregular shape = 5656 kg

After recording the data obtained in the lab for this case and the theoretical equation for different shapes the values of (I) the Moment of inertia for different cases are calculated as follows

(a) The platform alone

Theoretical

I zz=25mr2

= 25

(4365 ) (0314 )2 = 048441024 m4

Experimental

Equations (23) (24) and (25) are used to get the theoretical data by substituting the data measured in the lab

For first trail when the length of the cord = 400 mm and we had 30 oscillations And average time for this case was calculated after three trails had been done and it was found to be t average=299 secsec

L = C τ 2

29

C = L τ 2

τ = timecycles

= 29930

= 099667 (sec)

C= 04

(099667)2=iquest0388 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I p = (mp )R2 g

4 π2 C=

4365(0272)2(981)4 π 2(0417)

= 0457430617 (m4)


t heoriticallowast100 iquest5569

30

(b) The platform horizontal cylindrical mass placed on it at the centroid

Theoretical

I cy ondash I cy i=1

12m(3 riquestiquest2+h2)minus 1

12m(3 riquestiquest2+h2)iquestiquest

iquest((3 2)lowast4365lowast02722)+((112)lowast1386lowast((022)+3lowast012))minus((112)lowast1386lowast((022)+3lowast0041812))

= 0513003156 m4

Experimental

Equations (23) (24) and (25) were used to obtain the theoretical data by substitute using the data measured in the lab

For the first trail when the length of the cord = 400 mm and performing d 30 oscillations The average time involved in this case after three repetitions was found to be t average=16533sec

L = C τ 2

C = L τ 2

τ = timecycles =

1653330 = 05511 (sec)

C= 04

(0569)2=iquest1317 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I+ I p = (m p+m )R2 g

4π 2C=0563225526m2

31



And by performing the same procedure for each case for different cord lengths

( c ) The platform with vertical cylindrical mass placed on it at the centroid

Theoretical

I=32mr2+ 1

2mro

2minus12mr i

2

=((32)43650272^2)+((12)138601^2)-((12)138600418^2)

= 0541601867m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

16330 = 054333 (sec)

32

C= 04

(05433)2=iquest13549 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I+ I p = (mp+m )R2 g

4π 2C=0563225526m2


theoriticallowast100=39925

( d ) The platform alone with irregular mass placed on it at the centroid

Experimental

Equations (23) (24) and (25) were used to get the theoretical data by substitute using the data measured in the lab

For first trail when the length of the cord = 400 mm and we had 30 oscillations And we did it three time to get the average time for this case To reduce the personality error and we got t average=2123sec

L = C τ 2

C = L τ 2

τ = timecycles

= 2123

30= 0676 sec

C= 04

(0676)2=iquest0707667 m sec2

C = (m+m p )R2 g

4 π2(I + Ip )

33

I+ I p = (m p+m )R2 g

4π 2C=052239m4

Note

For the irregular shape you canrsquot get the theoretical moment of inertia but you can approximate it for the irregular shape by treating it as group of homogenous solids

So the theoretical moment of inertia approximated to

I=(( 32 )lowast4365lowast02722)+( 2

5 )lowast5656lowast0152=0535314 m4



34

Table 4 Trifilar suspension with cord length equal to 04 (m)

SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate04 30 298 149

2993333333 1496666667 0484410240445393497 805448343

04 30 298 149 0445393497 80544834304 30 302 151 0457430617 5569581401

Plate with horizontal cylindrical mass placed

on it at the centroid

04 30 164 0821653333333 0826666667 0513003156

0563225526 978987534704 30 165 0825 0570115071 111328582804 30 167 0835 0584019806 1384331624

Plate with vertical cylindrical mass placed


04 30 157 0785163 0815 0541601867

05161714 469541704204 30 168 084 0591034996 912720797704 30 164 082 0563225526 399253776

Plate with irregular mass placed on it at

the centroid

04 30 213 10652123333333 1061666667 053531424

052239302 24137636304 30 212 106 0517499435 332791537304 30 212 106 0517499435 3327915373

35


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate044 30 314 157

3156666667 1578333333 0484410240449549951 7196439229

044 30 316 158 045529494 6010463261044 30 317 1585 0458181114 5414651525



044 30 162 0811626666667 0813333333 0513003156

0499610983 2610544028044 30 162 081 0499610983 2610544028044 30 164 082 0512023205 0191022412



044 30 169 0845169 0845 0541601867

0543720061 0391098067044 30 167 0835 0530927096 1970962712044 30 171 0855 0556665324 2781278617

Plate with irregular mass placed on it at the

centroid

044 30 231 115523 115 053531424

0558559487 4342355492044 30 23 115 0553733942 344091388044 30 229 1145 0548929332 2543383077

36


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate05 30 343 1715

339 1695 0484410240472051706 2551253737

05 30 335 1675 0450288593 704395660505 30 339 1695 0461105951 4810857982



05 30 185 09251846666667 0923333333 051300315

6

0573360905 117655706105 30 187 0935 0585824908 141951859305 30 182 091 055491619 8170131796



05 30 186 0931820666667 0910333333 054160186

7

0579576154 701147639405 30 1792 0896 0537973187 06699902505 30 181 0905 0548834963 1335500582


centroid

05 30 238 1192396666667 1198333333 053531424

0521773549 252948448605 30 239 1195 0526167412 170868376705 30 242 121 0539459539 0774367463

37

Linear relation for case (a)

By drawing the relation between log τ and log l we obtained with the following linear graph and it is intersection with the ordinate is equal to the constant C and it slope is equal 2

-002-00100010020030040

005

01

015

02

025

03

035

04

045

log τ

log l

Figure (3) The relation between log L and log τ

The pervious graph indicates that the relation between log L and log τ is linear And it is intercept with y-axis at value equal to = 0388 which is the value of constant C that we had calculated in each case And the linear graph has a slope equal to 2 And thatrsquos what we expected in the theoretical equation

38

42 Discussion

When investigating the previous experimentally and theoretically results that were tabulated it is obvious that the theoretical and experimental results of the mass-moment of inertia for the bifilar and trifilar suspension are somehow close and the error involved is not large

In other words the human errors involved in this experiment such as while recording the length of the cords and weighting the parts are acceptable

In addition as noticed from the tabulated data for the bifilar and trifilar suspensions the relationship between the mass-moment of inertia of the body and the periodic time on log scale is linear

But it should be kept in mind that this linear relation involves some errors In other words this relation is very close to be perfectly linear due to the errors generated while performing this experiment

39

43 Uncertainty Analysis

In this part of this chapter it is important to evaluate the error involved in obtaining the mass-moment of inertia in order to ensure the accuracy related to it These errors occur due to the human errors such as measuring the lengths of the cords and the time of oscillations by the stop watch

This can be achieved by performing the uncertainty analysis as shown below for the first case of the bifilar suspension

I = md1 d2 gτ 2

4 π 2 L

uL=05480

=000104

ut=00051432

=000035

part Ipart L

=iquest (m )d1 d2 gτ 2

(4 π2 L)2

part Ipart t

=iquest (m )d1 d2 gτ

2π 2 L

u I= radic(LIpart Ipart L

uL)2

+( tIpart Ipart t

ut)2

minus iquest+iquestiquest iquest

40

As noticed from the uncertainty analysis the error involved is not large and is acceptable In other words and for an instance the values of the error involved in calculating the viscosities of the given oils indicate that the human error due to the usage of the stop watch was somehow acceptable

41

Chapter 5

Conclusion

The conclusion for this experiment is that the bifilar and trifilar suspension methods for determining the mass-moment of inertia can be considered as successful methods applied for that purpose

Moreover the errors involved in this experiment whether in the case of bifilar or trifilar suspensions are not large which can be considered as a good indicator of how successful this experiment was

Finally when this experiment was performed some recommendations and notices should be taken in mind for the purpose of improving the obtained results such as

Being careful while changing the length of the wires and make sure they are on the same level (exactly having the same length)

Using an electronic scale for weighting parts to increase the accuracy

Using the stop watch properly and carefully while recording the time of oscillation

42

References

1 Mechanical Engineering Fundamentals Laboratory Sheet ldquoLab Ardquo Prepared by ProfAhmed H Elkholy

43

Abstract

In this experiment a bifilar and trifilar suspension methods are applied to determine the mass momentum of inertia of a beam platform cylindrical mass and a body with an irregular mass

First with the bifilar method a body was suspended by two parallel cords with the same length and by displacing the system through a small angle and using few equations which will be given further in the report we get the time period and then calculating the mass moment of inertia

Moreover and for the trifilar method a platform was suspended by three cords having the same length and also by getting the time period and the moment of inertia of the platform we calculated the moment of inertia of a cylindrical mass and a body with an irregular mass

2

Table of contents


Chapter 1


Chapter 2


Chapter 3


Chapter 4


Chapter 5



3


Figures




4

List of tables


5

Nomenclature








m

MR

T

θ

Mass (kg)





τ Time period (s)

Chapter 16

Introduction





7

A B

Brsquo

O

R

l

L


BA

Arsquo

φ

θ



13 Objectives



8

Chapter 2




IiquestintM

r2 dm (1)

Where



I= Ī + m d2 (2)

Where



m Mass of the body


9

Bifilar suspension



T 1=mgd2

d and T 2=mg

d1

d (4)



10


Ɵ (6)

ϕ1= d1

L Ɵ and ϕ2=

d2

L Ɵ (7)





MR= iquest (10)



MR= -mg d1d2

L Ɵ (11)


Ī ϴ΄΄ = - mg d1d2

L Ɵ rarr ϴ΄΄+mg

d1d2

Ī LƟ=0 (12)

Where



ω=radicmgd1 d2

Ī L (13)


11


mgd1d2 (14)


Ī =mgd1d2 τ

2

4 π2 L (15)



(16)


Ī = m k 2 (17)


L = mgd1d2 τ

2

4 π2 Ī (18)


With k defined as

k = mgd1d2

4 π2 Ī (20)

12

Trifilar suspension






Where



120601 = RL

Ɵ

(22)

13


ϴ΄΄+(m+m p ) R2 g

L ( Ī+ I p )θ=0 (23)



(24)



L ( τ2π )

2

(25)


L = (m+m p )R2 g

4 π2 ( Ī + I p ) τ2 (26)


L = C τ 2 (27)

With C defined as

C = (m+mp )R2 g

4 π2 ( Ī + I p ) (28)

14

Chapter 3





31 Equipments




15

32 Procedure












16







17

Chapter 4




18


Bifilar Apparatus


19


(a) The beam only

Theoretical



= 045721497 (m4)

Experimental

20


τ = time

cycles = 1791

20 = 08955 (s)


I p = (m )d1 d2 gτ 2

4π 2 L=


= 045501 (m4)





21

Theoretical



= 0048327651 (m4)

Experimental


τ = timecycles =

121320

=06065(s)

I = (m )d1 d2 τ

2 g

4π 2 L

iquest (1444+1793 ) (052 )2(02525)(02525)(981)

4 π 2(04)

= 0045155684 (m4 iquest


theoritical=65634


Theoretical

22


=


= 0104358901 (m4 iquest

Experimental


τ = timecycles =

150920 = 07545 (s)

I = (m )d1 d2 gτ 2

4π 2 L

= (503 ) (05735 )2 (02525 ) (02525 ) ( 981 )

4 π2 (04 )

= 0104606813 (m4 iquest




23

24


SystemCord

Length L (m)

No of Oscillations

(cyclesrsquo)


(sec)

Period τ

Average time (s)

Average Period



Only beam04 20 1785 08925

1782666667 0891333333 00457214970045196926 1147318216

04 20 1772 0886 0044540993 258194598404 20 1791 08955 0045501282 0481645472


04 20 1206 06031209333333 0604666667 0048327651

0044636018 763875978504 20 1213 06065 0045155684 656346126304 20 1209 06045 0044858364 7178679599


150 mm

04 20 1185 059251185666667 0592833333 0068498901

0064508686 582522521804 20 1178 0589 0063748809 693455239104 20 1194 0597 0065492285 4389290021


200 mm

04 20 1328 06641324666667 0662333333 0084187651

0081017277 376584191504 20 1332 0666 0081506068 318524499304 20 1314 0657 0079318085 5784180162


250 mm

04 20 1485 074251501 07505 0104358901

0101305821 362651766404 20 1509 07545 0104606813 029447598504 20 1509 07545 0104606813 0294475985


1634666667 0817333333 00487731420061356248 1494649851

04 20 1625 08125 0060608008 149464985104 20 1644 0822 0062033588 1494649851

25


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only beam045 20 1863 09315

1887666667 0943833333 00457214970043762855 4283854264

045 20 19 095 0045518416 04441707045 20 19 095 0045518416 04441707


045 20 13 0651286666667 0643333333 0048327651

0046102556 4604185043045 20 1282 0641 0044834709 7227626401045 20 1278 0639 0044555366 7805646014


150 mm

045 20 1268 06341264333333 0632166667 0068498901

006565495 4151819235045 20 1272 0636 0066069831 3546145034045 20 1253 06265 0064110788 6406107612


200 mm

045 20 1503 075151504 0752 0084187651

009224587 9571735346045 20 1506 0753 0092614485 10009584045 20 1503 07515 009224587 9571735346


250 mm

045 20 1575 078751580666667 0790333333 0104358901

0101295485 3638795325045 20 1585 07925 010258586 2106058254045 20 1582 0791 010219789 2566897967


1722333333 0861166667 00487731420061414329 1501548924

045 20 1716 0858 0060076597 1501548924045 20 1716 0858 0060076597 1501548924

26


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only beam048 20 1953 09765

1960333333 0980166667 00457214970045087453 1386753778

048 20 1956 0978 0045226077 1083561808048 20 1972 0986 0045968998 0541321699


048 20 1344 06721337666667 0668833333 0048327651

0046196398 4410006579048 20 1328 0664 0045103031 6672411469048 20 1341 06705 0045990395 4836271348


150 mm

048 20 1322 06611303333333 0651666667 0068498901

0066905706 2325869495048 20 1325 06625 0067209707 1882064854048 20 1263 06315 0061067051 1084959004


200 mm

048 20 1463 073151458666667 0729333333 0084187651

0081938668 2671393071048 20 1466 0733 0082275056 2271823432048 20 1447 07235 0080156235 4788607411


250 mm

048 20 1628 08141622 0811 0104358901

0101463315 3439442497048 20 1613 08065 0099602212 5650103475048 20 1625 08125 0101089716 3883211714


1751333333 0875666667 00487731420059651783 1292189689

048 20 1747 08735 0058375127 1292189689048 20 1741 08705 0057974842 1292189689

27


28

Trifilar Apparatus




Theoretical

I zz=25mr2

= 25

(4365 ) (0314 )2 = 048441024 m4

Experimental



L = C τ 2

29

C = L τ 2

τ = timecycles

= 29930

= 099667 (sec)

C= 04

(099667)2=iquest0388 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I p = (mp )R2 g

4 π2 C=

4365(0272)2(981)4 π 2(0417)

= 0457430617 (m4)



30


Theoretical





= 0513003156 m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

1653330 = 05511 (sec)

C= 04

(0569)2=iquest1317 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I+ I p = (m p+m )R2 g

4π 2C=0563225526m2

31





Theoretical

I=32mr2+ 1

2mro

2minus12mr i

2

=((32)43650272^2)+((12)138601^2)-((12)138600418^2)

= 0541601867m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

16330 = 054333 (sec)

32

C= 04

(05433)2=iquest13549 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )


4π 2C=0563225526m2




Experimental



L = C τ 2

C = L τ 2

τ = timecycles

= 2123

30= 0676 sec

C= 04

(0676)2=iquest0707667 m sec2

C = (m+m p )R2 g

4 π2(I + Ip )

33

I+ I p = (m p+m )R2 g

4π 2C=052239m4

Note



I=(( 32 )lowast4365lowast02722)+( 2

5 )lowast5656lowast0152=0535314 m4



34


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate04 30 298 149

2993333333 1496666667 0484410240445393497 805448343

04 30 298 149 0445393497 80544834304 30 302 151 0457430617 5569581401



04 30 164 0821653333333 0826666667 0513003156

0563225526 978987534704 30 165 0825 0570115071 111328582804 30 167 0835 0584019806 1384331624



04 30 157 0785163 0815 0541601867

05161714 469541704204 30 168 084 0591034996 912720797704 30 164 082 0563225526 399253776


the centroid

04 30 213 10652123333333 1061666667 053531424

052239302 24137636304 30 212 106 0517499435 332791537304 30 212 106 0517499435 3327915373

35


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate044 30 314 157

3156666667 1578333333 0484410240449549951 7196439229

044 30 316 158 045529494 6010463261044 30 317 1585 0458181114 5414651525



044 30 162 0811626666667 0813333333 0513003156

0499610983 2610544028044 30 162 081 0499610983 2610544028044 30 164 082 0512023205 0191022412



044 30 169 0845169 0845 0541601867

0543720061 0391098067044 30 167 0835 0530927096 1970962712044 30 171 0855 0556665324 2781278617


centroid

044 30 231 115523 115 053531424

0558559487 4342355492044 30 23 115 0553733942 344091388044 30 229 1145 0548929332 2543383077

36


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate05 30 343 1715

339 1695 0484410240472051706 2551253737

05 30 335 1675 0450288593 704395660505 30 339 1695 0461105951 4810857982



05 30 185 09251846666667 0923333333 051300315

6

0573360905 117655706105 30 187 0935 0585824908 141951859305 30 182 091 055491619 8170131796



05 30 186 0931820666667 0910333333 054160186

7

0579576154 701147639405 30 1792 0896 0537973187 06699902505 30 181 0905 0548834963 1335500582


centroid

05 30 238 1192396666667 1198333333 053531424

0521773549 252948448605 30 239 1195 0526167412 170868376705 30 242 121 0539459539 0774367463

37



-002-00100010020030040

005

01

015

02

025

03

035

04

045

log τ

log l



38

42 Discussion





39




I = md1 d2 gτ 2

4 π 2 L

uL=05480

=000104

ut=00051432

=000035

part Ipart L


(4 π2 L)2

part Ipart t


2π 2 L


uL)2

+( tIpart Ipart t

ut)2


40


41

Chapter 5

Conclusion







42

References


43

Table of contents


Chapter 1


Chapter 2


Chapter 3


Chapter 4


Chapter 5



3


Figures




4

List of tables


5

Nomenclature








m

MR

T

θ

Mass (kg)





τ Time period (s)

Chapter 16

Introduction





7

A B

Brsquo

O

R

l

L


BA

Arsquo

φ

θ



13 Objectives



8

Chapter 2




IiquestintM

r2 dm (1)

Where



I= Ī + m d2 (2)

Where



m Mass of the body


9

Bifilar suspension



T 1=mgd2

d and T 2=mg

d1

d (4)



10


Ɵ (6)

ϕ1= d1

L Ɵ and ϕ2=

d2

L Ɵ (7)





MR= iquest (10)



MR= -mg d1d2

L Ɵ (11)


Ī ϴ΄΄ = - mg d1d2

L Ɵ rarr ϴ΄΄+mg

d1d2

Ī LƟ=0 (12)

Where



ω=radicmgd1 d2

Ī L (13)


11


mgd1d2 (14)


Ī =mgd1d2 τ

2

4 π2 L (15)



(16)


Ī = m k 2 (17)


L = mgd1d2 τ

2

4 π2 Ī (18)


With k defined as

k = mgd1d2

4 π2 Ī (20)

12

Trifilar suspension






Where



120601 = RL

Ɵ

(22)

13


ϴ΄΄+(m+m p ) R2 g

L ( Ī+ I p )θ=0 (23)



(24)



L ( τ2π )

2

(25)


L = (m+m p )R2 g

4 π2 ( Ī + I p ) τ2 (26)


L = C τ 2 (27)

With C defined as

C = (m+mp )R2 g

4 π2 ( Ī + I p ) (28)

14

Chapter 3





31 Equipments




15

32 Procedure












16







17

Chapter 4




18


Bifilar Apparatus


19


(a) The beam only

Theoretical



= 045721497 (m4)

Experimental

20


τ = time

cycles = 1791

20 = 08955 (s)


I p = (m )d1 d2 gτ 2

4π 2 L=


= 045501 (m4)





21

Theoretical



= 0048327651 (m4)

Experimental


τ = timecycles =

121320

=06065(s)

I = (m )d1 d2 τ

2 g

4π 2 L

iquest (1444+1793 ) (052 )2(02525)(02525)(981)

4 π 2(04)

= 0045155684 (m4 iquest


theoritical=65634


Theoretical

22


=


= 0104358901 (m4 iquest

Experimental


τ = timecycles =

150920 = 07545 (s)

I = (m )d1 d2 gτ 2

4π 2 L

= (503 ) (05735 )2 (02525 ) (02525 ) ( 981 )

4 π2 (04 )

= 0104606813 (m4 iquest




23

24


SystemCord

Length L (m)

No of Oscillations

(cyclesrsquo)


(sec)

Period τ

Average time (s)

Average Period



Only beam04 20 1785 08925

1782666667 0891333333 00457214970045196926 1147318216

04 20 1772 0886 0044540993 258194598404 20 1791 08955 0045501282 0481645472


04 20 1206 06031209333333 0604666667 0048327651

0044636018 763875978504 20 1213 06065 0045155684 656346126304 20 1209 06045 0044858364 7178679599


150 mm

04 20 1185 059251185666667 0592833333 0068498901

0064508686 582522521804 20 1178 0589 0063748809 693455239104 20 1194 0597 0065492285 4389290021


200 mm

04 20 1328 06641324666667 0662333333 0084187651

0081017277 376584191504 20 1332 0666 0081506068 318524499304 20 1314 0657 0079318085 5784180162


250 mm

04 20 1485 074251501 07505 0104358901

0101305821 362651766404 20 1509 07545 0104606813 029447598504 20 1509 07545 0104606813 0294475985


1634666667 0817333333 00487731420061356248 1494649851

04 20 1625 08125 0060608008 149464985104 20 1644 0822 0062033588 1494649851

25


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only beam045 20 1863 09315

1887666667 0943833333 00457214970043762855 4283854264

045 20 19 095 0045518416 04441707045 20 19 095 0045518416 04441707


045 20 13 0651286666667 0643333333 0048327651

0046102556 4604185043045 20 1282 0641 0044834709 7227626401045 20 1278 0639 0044555366 7805646014


150 mm

045 20 1268 06341264333333 0632166667 0068498901

006565495 4151819235045 20 1272 0636 0066069831 3546145034045 20 1253 06265 0064110788 6406107612


200 mm

045 20 1503 075151504 0752 0084187651

009224587 9571735346045 20 1506 0753 0092614485 10009584045 20 1503 07515 009224587 9571735346


250 mm

045 20 1575 078751580666667 0790333333 0104358901

0101295485 3638795325045 20 1585 07925 010258586 2106058254045 20 1582 0791 010219789 2566897967


1722333333 0861166667 00487731420061414329 1501548924

045 20 1716 0858 0060076597 1501548924045 20 1716 0858 0060076597 1501548924

26


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only beam048 20 1953 09765

1960333333 0980166667 00457214970045087453 1386753778

048 20 1956 0978 0045226077 1083561808048 20 1972 0986 0045968998 0541321699


048 20 1344 06721337666667 0668833333 0048327651

0046196398 4410006579048 20 1328 0664 0045103031 6672411469048 20 1341 06705 0045990395 4836271348


150 mm

048 20 1322 06611303333333 0651666667 0068498901

0066905706 2325869495048 20 1325 06625 0067209707 1882064854048 20 1263 06315 0061067051 1084959004


200 mm

048 20 1463 073151458666667 0729333333 0084187651

0081938668 2671393071048 20 1466 0733 0082275056 2271823432048 20 1447 07235 0080156235 4788607411


250 mm

048 20 1628 08141622 0811 0104358901

0101463315 3439442497048 20 1613 08065 0099602212 5650103475048 20 1625 08125 0101089716 3883211714


1751333333 0875666667 00487731420059651783 1292189689

048 20 1747 08735 0058375127 1292189689048 20 1741 08705 0057974842 1292189689

27


28

Trifilar Apparatus




Theoretical

I zz=25mr2

= 25

(4365 ) (0314 )2 = 048441024 m4

Experimental



L = C τ 2

29

C = L τ 2

τ = timecycles

= 29930

= 099667 (sec)

C= 04

(099667)2=iquest0388 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I p = (mp )R2 g

4 π2 C=

4365(0272)2(981)4 π 2(0417)

= 0457430617 (m4)



30


Theoretical





= 0513003156 m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

1653330 = 05511 (sec)

C= 04

(0569)2=iquest1317 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I+ I p = (m p+m )R2 g

4π 2C=0563225526m2

31





Theoretical

I=32mr2+ 1

2mro

2minus12mr i

2

=((32)43650272^2)+((12)138601^2)-((12)138600418^2)

= 0541601867m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

16330 = 054333 (sec)

32

C= 04

(05433)2=iquest13549 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )


4π 2C=0563225526m2




Experimental



L = C τ 2

C = L τ 2

τ = timecycles

= 2123

30= 0676 sec

C= 04

(0676)2=iquest0707667 m sec2

C = (m+m p )R2 g

4 π2(I + Ip )

33

I+ I p = (m p+m )R2 g

4π 2C=052239m4

Note



I=(( 32 )lowast4365lowast02722)+( 2

5 )lowast5656lowast0152=0535314 m4



34


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate04 30 298 149

2993333333 1496666667 0484410240445393497 805448343

04 30 298 149 0445393497 80544834304 30 302 151 0457430617 5569581401



04 30 164 0821653333333 0826666667 0513003156

0563225526 978987534704 30 165 0825 0570115071 111328582804 30 167 0835 0584019806 1384331624



04 30 157 0785163 0815 0541601867

05161714 469541704204 30 168 084 0591034996 912720797704 30 164 082 0563225526 399253776


the centroid

04 30 213 10652123333333 1061666667 053531424

052239302 24137636304 30 212 106 0517499435 332791537304 30 212 106 0517499435 3327915373

35


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate044 30 314 157

3156666667 1578333333 0484410240449549951 7196439229

044 30 316 158 045529494 6010463261044 30 317 1585 0458181114 5414651525



044 30 162 0811626666667 0813333333 0513003156

0499610983 2610544028044 30 162 081 0499610983 2610544028044 30 164 082 0512023205 0191022412



044 30 169 0845169 0845 0541601867

0543720061 0391098067044 30 167 0835 0530927096 1970962712044 30 171 0855 0556665324 2781278617


centroid

044 30 231 115523 115 053531424

0558559487 4342355492044 30 23 115 0553733942 344091388044 30 229 1145 0548929332 2543383077

36


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate05 30 343 1715

339 1695 0484410240472051706 2551253737

05 30 335 1675 0450288593 704395660505 30 339 1695 0461105951 4810857982



05 30 185 09251846666667 0923333333 051300315

6

0573360905 117655706105 30 187 0935 0585824908 141951859305 30 182 091 055491619 8170131796



05 30 186 0931820666667 0910333333 054160186

7

0579576154 701147639405 30 1792 0896 0537973187 06699902505 30 181 0905 0548834963 1335500582


centroid

05 30 238 1192396666667 1198333333 053531424

0521773549 252948448605 30 239 1195 0526167412 170868376705 30 242 121 0539459539 0774367463

37



-002-00100010020030040

005

01

015

02

025

03

035

04

045

log τ

log l



38

42 Discussion





39




I = md1 d2 gτ 2

4 π 2 L

uL=05480

=000104

ut=00051432

=000035

part Ipart L


(4 π2 L)2

part Ipart t


2π 2 L


uL)2

+( tIpart Ipart t

ut)2


40


41

Chapter 5

Conclusion







42

References


43


Figures




4

List of tables


5

Nomenclature








m

MR

T

θ

Mass (kg)





τ Time period (s)

Chapter 16

Introduction





7

A B

Brsquo

O

R

l

L


BA

Arsquo

φ

θ



13 Objectives



8

Chapter 2




IiquestintM

r2 dm (1)

Where



I= Ī + m d2 (2)

Where



m Mass of the body


9

Bifilar suspension



T 1=mgd2

d and T 2=mg

d1

d (4)



10


Ɵ (6)

ϕ1= d1

L Ɵ and ϕ2=

d2

L Ɵ (7)





MR= iquest (10)



MR= -mg d1d2

L Ɵ (11)


Ī ϴ΄΄ = - mg d1d2

L Ɵ rarr ϴ΄΄+mg

d1d2

Ī LƟ=0 (12)

Where



ω=radicmgd1 d2

Ī L (13)


11


mgd1d2 (14)


Ī =mgd1d2 τ

2

4 π2 L (15)



(16)


Ī = m k 2 (17)


L = mgd1d2 τ

2

4 π2 Ī (18)


With k defined as

k = mgd1d2

4 π2 Ī (20)

12

Trifilar suspension






Where



120601 = RL

Ɵ

(22)

13


ϴ΄΄+(m+m p ) R2 g

L ( Ī+ I p )θ=0 (23)



(24)



L ( τ2π )

2

(25)


L = (m+m p )R2 g

4 π2 ( Ī + I p ) τ2 (26)


L = C τ 2 (27)

With C defined as

C = (m+mp )R2 g

4 π2 ( Ī + I p ) (28)

14

Chapter 3





31 Equipments




15

32 Procedure












16







17

Chapter 4




18


Bifilar Apparatus


19


(a) The beam only

Theoretical



= 045721497 (m4)

Experimental

20


τ = time

cycles = 1791

20 = 08955 (s)


I p = (m )d1 d2 gτ 2

4π 2 L=


= 045501 (m4)





21

Theoretical



= 0048327651 (m4)

Experimental


τ = timecycles =

121320

=06065(s)

I = (m )d1 d2 τ

2 g

4π 2 L

iquest (1444+1793 ) (052 )2(02525)(02525)(981)

4 π 2(04)

= 0045155684 (m4 iquest


theoritical=65634


Theoretical

22


=


= 0104358901 (m4 iquest

Experimental


τ = timecycles =

150920 = 07545 (s)

I = (m )d1 d2 gτ 2

4π 2 L

= (503 ) (05735 )2 (02525 ) (02525 ) ( 981 )

4 π2 (04 )

= 0104606813 (m4 iquest




23

24


SystemCord

Length L (m)

No of Oscillations

(cyclesrsquo)


(sec)

Period τ

Average time (s)

Average Period



Only beam04 20 1785 08925

1782666667 0891333333 00457214970045196926 1147318216

04 20 1772 0886 0044540993 258194598404 20 1791 08955 0045501282 0481645472


04 20 1206 06031209333333 0604666667 0048327651

0044636018 763875978504 20 1213 06065 0045155684 656346126304 20 1209 06045 0044858364 7178679599


150 mm

04 20 1185 059251185666667 0592833333 0068498901

0064508686 582522521804 20 1178 0589 0063748809 693455239104 20 1194 0597 0065492285 4389290021


200 mm

04 20 1328 06641324666667 0662333333 0084187651

0081017277 376584191504 20 1332 0666 0081506068 318524499304 20 1314 0657 0079318085 5784180162


250 mm

04 20 1485 074251501 07505 0104358901

0101305821 362651766404 20 1509 07545 0104606813 029447598504 20 1509 07545 0104606813 0294475985


1634666667 0817333333 00487731420061356248 1494649851

04 20 1625 08125 0060608008 149464985104 20 1644 0822 0062033588 1494649851

25


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only beam045 20 1863 09315

1887666667 0943833333 00457214970043762855 4283854264

045 20 19 095 0045518416 04441707045 20 19 095 0045518416 04441707


045 20 13 0651286666667 0643333333 0048327651

0046102556 4604185043045 20 1282 0641 0044834709 7227626401045 20 1278 0639 0044555366 7805646014


150 mm

045 20 1268 06341264333333 0632166667 0068498901

006565495 4151819235045 20 1272 0636 0066069831 3546145034045 20 1253 06265 0064110788 6406107612


200 mm

045 20 1503 075151504 0752 0084187651

009224587 9571735346045 20 1506 0753 0092614485 10009584045 20 1503 07515 009224587 9571735346


250 mm

045 20 1575 078751580666667 0790333333 0104358901

0101295485 3638795325045 20 1585 07925 010258586 2106058254045 20 1582 0791 010219789 2566897967


1722333333 0861166667 00487731420061414329 1501548924

045 20 1716 0858 0060076597 1501548924045 20 1716 0858 0060076597 1501548924

26


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only beam048 20 1953 09765

1960333333 0980166667 00457214970045087453 1386753778

048 20 1956 0978 0045226077 1083561808048 20 1972 0986 0045968998 0541321699


048 20 1344 06721337666667 0668833333 0048327651

0046196398 4410006579048 20 1328 0664 0045103031 6672411469048 20 1341 06705 0045990395 4836271348


150 mm

048 20 1322 06611303333333 0651666667 0068498901

0066905706 2325869495048 20 1325 06625 0067209707 1882064854048 20 1263 06315 0061067051 1084959004


200 mm

048 20 1463 073151458666667 0729333333 0084187651

0081938668 2671393071048 20 1466 0733 0082275056 2271823432048 20 1447 07235 0080156235 4788607411


250 mm

048 20 1628 08141622 0811 0104358901

0101463315 3439442497048 20 1613 08065 0099602212 5650103475048 20 1625 08125 0101089716 3883211714


1751333333 0875666667 00487731420059651783 1292189689

048 20 1747 08735 0058375127 1292189689048 20 1741 08705 0057974842 1292189689

27


28

Trifilar Apparatus




Theoretical

I zz=25mr2

= 25

(4365 ) (0314 )2 = 048441024 m4

Experimental



L = C τ 2

29

C = L τ 2

τ = timecycles

= 29930

= 099667 (sec)

C= 04

(099667)2=iquest0388 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I p = (mp )R2 g

4 π2 C=

4365(0272)2(981)4 π 2(0417)

= 0457430617 (m4)



30


Theoretical





= 0513003156 m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

1653330 = 05511 (sec)

C= 04

(0569)2=iquest1317 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I+ I p = (m p+m )R2 g

4π 2C=0563225526m2

31





Theoretical

I=32mr2+ 1

2mro

2minus12mr i

2

=((32)43650272^2)+((12)138601^2)-((12)138600418^2)

= 0541601867m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

16330 = 054333 (sec)

32

C= 04

(05433)2=iquest13549 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )


4π 2C=0563225526m2




Experimental



L = C τ 2

C = L τ 2

τ = timecycles

= 2123

30= 0676 sec

C= 04

(0676)2=iquest0707667 m sec2

C = (m+m p )R2 g

4 π2(I + Ip )

33

I+ I p = (m p+m )R2 g

4π 2C=052239m4

Note



I=(( 32 )lowast4365lowast02722)+( 2

5 )lowast5656lowast0152=0535314 m4



34


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate04 30 298 149

2993333333 1496666667 0484410240445393497 805448343

04 30 298 149 0445393497 80544834304 30 302 151 0457430617 5569581401



04 30 164 0821653333333 0826666667 0513003156

0563225526 978987534704 30 165 0825 0570115071 111328582804 30 167 0835 0584019806 1384331624



04 30 157 0785163 0815 0541601867

05161714 469541704204 30 168 084 0591034996 912720797704 30 164 082 0563225526 399253776


the centroid

04 30 213 10652123333333 1061666667 053531424

052239302 24137636304 30 212 106 0517499435 332791537304 30 212 106 0517499435 3327915373

35


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate044 30 314 157

3156666667 1578333333 0484410240449549951 7196439229

044 30 316 158 045529494 6010463261044 30 317 1585 0458181114 5414651525



044 30 162 0811626666667 0813333333 0513003156

0499610983 2610544028044 30 162 081 0499610983 2610544028044 30 164 082 0512023205 0191022412



044 30 169 0845169 0845 0541601867

0543720061 0391098067044 30 167 0835 0530927096 1970962712044 30 171 0855 0556665324 2781278617


centroid

044 30 231 115523 115 053531424

0558559487 4342355492044 30 23 115 0553733942 344091388044 30 229 1145 0548929332 2543383077

36


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate05 30 343 1715

339 1695 0484410240472051706 2551253737

05 30 335 1675 0450288593 704395660505 30 339 1695 0461105951 4810857982



05 30 185 09251846666667 0923333333 051300315

6

0573360905 117655706105 30 187 0935 0585824908 141951859305 30 182 091 055491619 8170131796



05 30 186 0931820666667 0910333333 054160186

7

0579576154 701147639405 30 1792 0896 0537973187 06699902505 30 181 0905 0548834963 1335500582


centroid

05 30 238 1192396666667 1198333333 053531424

0521773549 252948448605 30 239 1195 0526167412 170868376705 30 242 121 0539459539 0774367463

37



-002-00100010020030040

005

01

015

02

025

03

035

04

045

log τ

log l



38

42 Discussion





39




I = md1 d2 gτ 2

4 π 2 L

uL=05480

=000104

ut=00051432

=000035

part Ipart L


(4 π2 L)2

part Ipart t


2π 2 L


uL)2

+( tIpart Ipart t

ut)2


40


41

Chapter 5

Conclusion







42

References


43

List of tables


5

Nomenclature








m

MR

T

θ

Mass (kg)





τ Time period (s)

Chapter 16

Introduction





7

A B

Brsquo

O

R

l

L


BA

Arsquo

φ

θ



13 Objectives



8

Chapter 2




IiquestintM

r2 dm (1)

Where



I= Ī + m d2 (2)

Where



m Mass of the body


9

Bifilar suspension



T 1=mgd2

d and T 2=mg

d1

d (4)



10


Ɵ (6)

ϕ1= d1

L Ɵ and ϕ2=

d2

L Ɵ (7)





MR= iquest (10)



MR= -mg d1d2

L Ɵ (11)


Ī ϴ΄΄ = - mg d1d2

L Ɵ rarr ϴ΄΄+mg

d1d2

Ī LƟ=0 (12)

Where



ω=radicmgd1 d2

Ī L (13)


11


mgd1d2 (14)


Ī =mgd1d2 τ

2

4 π2 L (15)



(16)


Ī = m k 2 (17)


L = mgd1d2 τ

2

4 π2 Ī (18)


With k defined as

k = mgd1d2

4 π2 Ī (20)

12

Trifilar suspension






Where



120601 = RL

Ɵ

(22)

13


ϴ΄΄+(m+m p ) R2 g

L ( Ī+ I p )θ=0 (23)



(24)



L ( τ2π )

2

(25)


L = (m+m p )R2 g

4 π2 ( Ī + I p ) τ2 (26)


L = C τ 2 (27)

With C defined as

C = (m+mp )R2 g

4 π2 ( Ī + I p ) (28)

14

Chapter 3





31 Equipments




15

32 Procedure












16







17

Chapter 4




18


Bifilar Apparatus


19


(a) The beam only

Theoretical



= 045721497 (m4)

Experimental

20


τ = time

cycles = 1791

20 = 08955 (s)


I p = (m )d1 d2 gτ 2

4π 2 L=


= 045501 (m4)





21

Theoretical



= 0048327651 (m4)

Experimental


τ = timecycles =

121320

=06065(s)

I = (m )d1 d2 τ

2 g

4π 2 L

iquest (1444+1793 ) (052 )2(02525)(02525)(981)

4 π 2(04)

= 0045155684 (m4 iquest


theoritical=65634


Theoretical

22


=


= 0104358901 (m4 iquest

Experimental


τ = timecycles =

150920 = 07545 (s)

I = (m )d1 d2 gτ 2

4π 2 L

= (503 ) (05735 )2 (02525 ) (02525 ) ( 981 )

4 π2 (04 )

= 0104606813 (m4 iquest




23

24


SystemCord

Length L (m)

No of Oscillations

(cyclesrsquo)


(sec)

Period τ

Average time (s)

Average Period



Only beam04 20 1785 08925

1782666667 0891333333 00457214970045196926 1147318216

04 20 1772 0886 0044540993 258194598404 20 1791 08955 0045501282 0481645472


04 20 1206 06031209333333 0604666667 0048327651

0044636018 763875978504 20 1213 06065 0045155684 656346126304 20 1209 06045 0044858364 7178679599


150 mm

04 20 1185 059251185666667 0592833333 0068498901

0064508686 582522521804 20 1178 0589 0063748809 693455239104 20 1194 0597 0065492285 4389290021


200 mm

04 20 1328 06641324666667 0662333333 0084187651

0081017277 376584191504 20 1332 0666 0081506068 318524499304 20 1314 0657 0079318085 5784180162


250 mm

04 20 1485 074251501 07505 0104358901

0101305821 362651766404 20 1509 07545 0104606813 029447598504 20 1509 07545 0104606813 0294475985


1634666667 0817333333 00487731420061356248 1494649851

04 20 1625 08125 0060608008 149464985104 20 1644 0822 0062033588 1494649851

25


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only beam045 20 1863 09315

1887666667 0943833333 00457214970043762855 4283854264

045 20 19 095 0045518416 04441707045 20 19 095 0045518416 04441707


045 20 13 0651286666667 0643333333 0048327651

0046102556 4604185043045 20 1282 0641 0044834709 7227626401045 20 1278 0639 0044555366 7805646014


150 mm

045 20 1268 06341264333333 0632166667 0068498901

006565495 4151819235045 20 1272 0636 0066069831 3546145034045 20 1253 06265 0064110788 6406107612


200 mm

045 20 1503 075151504 0752 0084187651

009224587 9571735346045 20 1506 0753 0092614485 10009584045 20 1503 07515 009224587 9571735346


250 mm

045 20 1575 078751580666667 0790333333 0104358901

0101295485 3638795325045 20 1585 07925 010258586 2106058254045 20 1582 0791 010219789 2566897967


1722333333 0861166667 00487731420061414329 1501548924

045 20 1716 0858 0060076597 1501548924045 20 1716 0858 0060076597 1501548924

26


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only beam048 20 1953 09765

1960333333 0980166667 00457214970045087453 1386753778

048 20 1956 0978 0045226077 1083561808048 20 1972 0986 0045968998 0541321699


048 20 1344 06721337666667 0668833333 0048327651

0046196398 4410006579048 20 1328 0664 0045103031 6672411469048 20 1341 06705 0045990395 4836271348


150 mm

048 20 1322 06611303333333 0651666667 0068498901

0066905706 2325869495048 20 1325 06625 0067209707 1882064854048 20 1263 06315 0061067051 1084959004


200 mm

048 20 1463 073151458666667 0729333333 0084187651

0081938668 2671393071048 20 1466 0733 0082275056 2271823432048 20 1447 07235 0080156235 4788607411


250 mm

048 20 1628 08141622 0811 0104358901

0101463315 3439442497048 20 1613 08065 0099602212 5650103475048 20 1625 08125 0101089716 3883211714


1751333333 0875666667 00487731420059651783 1292189689

048 20 1747 08735 0058375127 1292189689048 20 1741 08705 0057974842 1292189689

27


28

Trifilar Apparatus




Theoretical

I zz=25mr2

= 25

(4365 ) (0314 )2 = 048441024 m4

Experimental



L = C τ 2

29

C = L τ 2

τ = timecycles

= 29930

= 099667 (sec)

C= 04

(099667)2=iquest0388 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I p = (mp )R2 g

4 π2 C=

4365(0272)2(981)4 π 2(0417)

= 0457430617 (m4)



30


Theoretical





= 0513003156 m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

1653330 = 05511 (sec)

C= 04

(0569)2=iquest1317 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I+ I p = (m p+m )R2 g

4π 2C=0563225526m2

31





Theoretical

I=32mr2+ 1

2mro

2minus12mr i

2

=((32)43650272^2)+((12)138601^2)-((12)138600418^2)

= 0541601867m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

16330 = 054333 (sec)

32

C= 04

(05433)2=iquest13549 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )


4π 2C=0563225526m2




Experimental



L = C τ 2

C = L τ 2

τ = timecycles

= 2123

30= 0676 sec

C= 04

(0676)2=iquest0707667 m sec2

C = (m+m p )R2 g

4 π2(I + Ip )

33

I+ I p = (m p+m )R2 g

4π 2C=052239m4

Note



I=(( 32 )lowast4365lowast02722)+( 2

5 )lowast5656lowast0152=0535314 m4



34


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate04 30 298 149

2993333333 1496666667 0484410240445393497 805448343

04 30 298 149 0445393497 80544834304 30 302 151 0457430617 5569581401



04 30 164 0821653333333 0826666667 0513003156

0563225526 978987534704 30 165 0825 0570115071 111328582804 30 167 0835 0584019806 1384331624



04 30 157 0785163 0815 0541601867

05161714 469541704204 30 168 084 0591034996 912720797704 30 164 082 0563225526 399253776


the centroid

04 30 213 10652123333333 1061666667 053531424

052239302 24137636304 30 212 106 0517499435 332791537304 30 212 106 0517499435 3327915373

35


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate044 30 314 157

3156666667 1578333333 0484410240449549951 7196439229

044 30 316 158 045529494 6010463261044 30 317 1585 0458181114 5414651525



044 30 162 0811626666667 0813333333 0513003156

0499610983 2610544028044 30 162 081 0499610983 2610544028044 30 164 082 0512023205 0191022412



044 30 169 0845169 0845 0541601867

0543720061 0391098067044 30 167 0835 0530927096 1970962712044 30 171 0855 0556665324 2781278617


centroid

044 30 231 115523 115 053531424

0558559487 4342355492044 30 23 115 0553733942 344091388044 30 229 1145 0548929332 2543383077

36


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate05 30 343 1715

339 1695 0484410240472051706 2551253737

05 30 335 1675 0450288593 704395660505 30 339 1695 0461105951 4810857982



05 30 185 09251846666667 0923333333 051300315

6

0573360905 117655706105 30 187 0935 0585824908 141951859305 30 182 091 055491619 8170131796



05 30 186 0931820666667 0910333333 054160186

7

0579576154 701147639405 30 1792 0896 0537973187 06699902505 30 181 0905 0548834963 1335500582


centroid

05 30 238 1192396666667 1198333333 053531424

0521773549 252948448605 30 239 1195 0526167412 170868376705 30 242 121 0539459539 0774367463

37



-002-00100010020030040

005

01

015

02

025

03

035

04

045

log τ

log l



38

42 Discussion





39




I = md1 d2 gτ 2

4 π 2 L

uL=05480

=000104

ut=00051432

=000035

part Ipart L


(4 π2 L)2

part Ipart t


2π 2 L


uL)2

+( tIpart Ipart t

ut)2


40


41

Chapter 5

Conclusion







42

References


43

Nomenclature








m

MR

T

θ

Mass (kg)





τ Time period (s)

Chapter 16

Introduction





7

A B

Brsquo

O

R

l

L


BA

Arsquo

φ

θ



13 Objectives



8

Chapter 2




IiquestintM

r2 dm (1)

Where



I= Ī + m d2 (2)

Where



m Mass of the body


9

Bifilar suspension



T 1=mgd2

d and T 2=mg

d1

d (4)



10


Ɵ (6)

ϕ1= d1

L Ɵ and ϕ2=

d2

L Ɵ (7)





MR= iquest (10)



MR= -mg d1d2

L Ɵ (11)


Ī ϴ΄΄ = - mg d1d2

L Ɵ rarr ϴ΄΄+mg

d1d2

Ī LƟ=0 (12)

Where



ω=radicmgd1 d2

Ī L (13)


11


mgd1d2 (14)


Ī =mgd1d2 τ

2

4 π2 L (15)



(16)


Ī = m k 2 (17)


L = mgd1d2 τ

2

4 π2 Ī (18)


With k defined as

k = mgd1d2

4 π2 Ī (20)

12

Trifilar suspension






Where



120601 = RL

Ɵ

(22)

13


ϴ΄΄+(m+m p ) R2 g

L ( Ī+ I p )θ=0 (23)



(24)



L ( τ2π )

2

(25)


L = (m+m p )R2 g

4 π2 ( Ī + I p ) τ2 (26)


L = C τ 2 (27)

With C defined as

C = (m+mp )R2 g

4 π2 ( Ī + I p ) (28)

14

Chapter 3





31 Equipments




15

32 Procedure












16







17

Chapter 4




18


Bifilar Apparatus


19


(a) The beam only

Theoretical



= 045721497 (m4)

Experimental

20


τ = time

cycles = 1791

20 = 08955 (s)


I p = (m )d1 d2 gτ 2

4π 2 L=


= 045501 (m4)





21

Theoretical



= 0048327651 (m4)

Experimental


τ = timecycles =

121320

=06065(s)

I = (m )d1 d2 τ

2 g

4π 2 L

iquest (1444+1793 ) (052 )2(02525)(02525)(981)

4 π 2(04)

= 0045155684 (m4 iquest


theoritical=65634


Theoretical

22


=


= 0104358901 (m4 iquest

Experimental


τ = timecycles =

150920 = 07545 (s)

I = (m )d1 d2 gτ 2

4π 2 L

= (503 ) (05735 )2 (02525 ) (02525 ) ( 981 )

4 π2 (04 )

= 0104606813 (m4 iquest




23

24


SystemCord

Length L (m)

No of Oscillations

(cyclesrsquo)


(sec)

Period τ

Average time (s)

Average Period



Only beam04 20 1785 08925

1782666667 0891333333 00457214970045196926 1147318216

04 20 1772 0886 0044540993 258194598404 20 1791 08955 0045501282 0481645472


04 20 1206 06031209333333 0604666667 0048327651

0044636018 763875978504 20 1213 06065 0045155684 656346126304 20 1209 06045 0044858364 7178679599


150 mm

04 20 1185 059251185666667 0592833333 0068498901

0064508686 582522521804 20 1178 0589 0063748809 693455239104 20 1194 0597 0065492285 4389290021


200 mm

04 20 1328 06641324666667 0662333333 0084187651

0081017277 376584191504 20 1332 0666 0081506068 318524499304 20 1314 0657 0079318085 5784180162


250 mm

04 20 1485 074251501 07505 0104358901

0101305821 362651766404 20 1509 07545 0104606813 029447598504 20 1509 07545 0104606813 0294475985


1634666667 0817333333 00487731420061356248 1494649851

04 20 1625 08125 0060608008 149464985104 20 1644 0822 0062033588 1494649851

25


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only beam045 20 1863 09315

1887666667 0943833333 00457214970043762855 4283854264

045 20 19 095 0045518416 04441707045 20 19 095 0045518416 04441707


045 20 13 0651286666667 0643333333 0048327651

0046102556 4604185043045 20 1282 0641 0044834709 7227626401045 20 1278 0639 0044555366 7805646014


150 mm

045 20 1268 06341264333333 0632166667 0068498901

006565495 4151819235045 20 1272 0636 0066069831 3546145034045 20 1253 06265 0064110788 6406107612


200 mm

045 20 1503 075151504 0752 0084187651

009224587 9571735346045 20 1506 0753 0092614485 10009584045 20 1503 07515 009224587 9571735346


250 mm

045 20 1575 078751580666667 0790333333 0104358901

0101295485 3638795325045 20 1585 07925 010258586 2106058254045 20 1582 0791 010219789 2566897967


1722333333 0861166667 00487731420061414329 1501548924

045 20 1716 0858 0060076597 1501548924045 20 1716 0858 0060076597 1501548924

26


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only beam048 20 1953 09765

1960333333 0980166667 00457214970045087453 1386753778

048 20 1956 0978 0045226077 1083561808048 20 1972 0986 0045968998 0541321699


048 20 1344 06721337666667 0668833333 0048327651

0046196398 4410006579048 20 1328 0664 0045103031 6672411469048 20 1341 06705 0045990395 4836271348


150 mm

048 20 1322 06611303333333 0651666667 0068498901

0066905706 2325869495048 20 1325 06625 0067209707 1882064854048 20 1263 06315 0061067051 1084959004


200 mm

048 20 1463 073151458666667 0729333333 0084187651

0081938668 2671393071048 20 1466 0733 0082275056 2271823432048 20 1447 07235 0080156235 4788607411


250 mm

048 20 1628 08141622 0811 0104358901

0101463315 3439442497048 20 1613 08065 0099602212 5650103475048 20 1625 08125 0101089716 3883211714


1751333333 0875666667 00487731420059651783 1292189689

048 20 1747 08735 0058375127 1292189689048 20 1741 08705 0057974842 1292189689

27


28

Trifilar Apparatus




Theoretical

I zz=25mr2

= 25

(4365 ) (0314 )2 = 048441024 m4

Experimental



L = C τ 2

29

C = L τ 2

τ = timecycles

= 29930

= 099667 (sec)

C= 04

(099667)2=iquest0388 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I p = (mp )R2 g

4 π2 C=

4365(0272)2(981)4 π 2(0417)

= 0457430617 (m4)



30


Theoretical





= 0513003156 m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

1653330 = 05511 (sec)

C= 04

(0569)2=iquest1317 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I+ I p = (m p+m )R2 g

4π 2C=0563225526m2

31





Theoretical

I=32mr2+ 1

2mro

2minus12mr i

2

=((32)43650272^2)+((12)138601^2)-((12)138600418^2)

= 0541601867m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

16330 = 054333 (sec)

32

C= 04

(05433)2=iquest13549 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )


4π 2C=0563225526m2




Experimental



L = C τ 2

C = L τ 2

τ = timecycles

= 2123

30= 0676 sec

C= 04

(0676)2=iquest0707667 m sec2

C = (m+m p )R2 g

4 π2(I + Ip )

33

I+ I p = (m p+m )R2 g

4π 2C=052239m4

Note



I=(( 32 )lowast4365lowast02722)+( 2

5 )lowast5656lowast0152=0535314 m4



34


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate04 30 298 149

2993333333 1496666667 0484410240445393497 805448343

04 30 298 149 0445393497 80544834304 30 302 151 0457430617 5569581401



04 30 164 0821653333333 0826666667 0513003156

0563225526 978987534704 30 165 0825 0570115071 111328582804 30 167 0835 0584019806 1384331624



04 30 157 0785163 0815 0541601867

05161714 469541704204 30 168 084 0591034996 912720797704 30 164 082 0563225526 399253776


the centroid

04 30 213 10652123333333 1061666667 053531424

052239302 24137636304 30 212 106 0517499435 332791537304 30 212 106 0517499435 3327915373

35


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate044 30 314 157

3156666667 1578333333 0484410240449549951 7196439229

044 30 316 158 045529494 6010463261044 30 317 1585 0458181114 5414651525



044 30 162 0811626666667 0813333333 0513003156

0499610983 2610544028044 30 162 081 0499610983 2610544028044 30 164 082 0512023205 0191022412



044 30 169 0845169 0845 0541601867

0543720061 0391098067044 30 167 0835 0530927096 1970962712044 30 171 0855 0556665324 2781278617


centroid

044 30 231 115523 115 053531424

0558559487 4342355492044 30 23 115 0553733942 344091388044 30 229 1145 0548929332 2543383077

36


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate05 30 343 1715

339 1695 0484410240472051706 2551253737

05 30 335 1675 0450288593 704395660505 30 339 1695 0461105951 4810857982



05 30 185 09251846666667 0923333333 051300315

6

0573360905 117655706105 30 187 0935 0585824908 141951859305 30 182 091 055491619 8170131796



05 30 186 0931820666667 0910333333 054160186

7

0579576154 701147639405 30 1792 0896 0537973187 06699902505 30 181 0905 0548834963 1335500582


centroid

05 30 238 1192396666667 1198333333 053531424

0521773549 252948448605 30 239 1195 0526167412 170868376705 30 242 121 0539459539 0774367463

37



-002-00100010020030040

005

01

015

02

025

03

035

04

045

log τ

log l



38

42 Discussion





39




I = md1 d2 gτ 2

4 π 2 L

uL=05480

=000104

ut=00051432

=000035

part Ipart L


(4 π2 L)2

part Ipart t


2π 2 L


uL)2

+( tIpart Ipart t

ut)2


40


41

Chapter 5

Conclusion







42

References


43

Introduction





7

A B

Brsquo

O

R

l

L


BA

Arsquo

φ

θ



13 Objectives



8

Chapter 2




IiquestintM

r2 dm (1)

Where



I= Ī + m d2 (2)

Where



m Mass of the body


9

Bifilar suspension



T 1=mgd2

d and T 2=mg

d1

d (4)



10


Ɵ (6)

ϕ1= d1

L Ɵ and ϕ2=

d2

L Ɵ (7)





MR= iquest (10)



MR= -mg d1d2

L Ɵ (11)


Ī ϴ΄΄ = - mg d1d2

L Ɵ rarr ϴ΄΄+mg

d1d2

Ī LƟ=0 (12)

Where



ω=radicmgd1 d2

Ī L (13)


11


mgd1d2 (14)


Ī =mgd1d2 τ

2

4 π2 L (15)



(16)


Ī = m k 2 (17)


L = mgd1d2 τ

2

4 π2 Ī (18)


With k defined as

k = mgd1d2

4 π2 Ī (20)

12

Trifilar suspension






Where



120601 = RL

Ɵ

(22)

13


ϴ΄΄+(m+m p ) R2 g

L ( Ī+ I p )θ=0 (23)



(24)



L ( τ2π )

2

(25)


L = (m+m p )R2 g

4 π2 ( Ī + I p ) τ2 (26)


L = C τ 2 (27)

With C defined as

C = (m+mp )R2 g

4 π2 ( Ī + I p ) (28)

14

Chapter 3





31 Equipments




15

32 Procedure












16







17

Chapter 4




18


Bifilar Apparatus


19


(a) The beam only

Theoretical



= 045721497 (m4)

Experimental

20


τ = time

cycles = 1791

20 = 08955 (s)


I p = (m )d1 d2 gτ 2

4π 2 L=


= 045501 (m4)





21

Theoretical



= 0048327651 (m4)

Experimental


τ = timecycles =

121320

=06065(s)

I = (m )d1 d2 τ

2 g

4π 2 L

iquest (1444+1793 ) (052 )2(02525)(02525)(981)

4 π 2(04)

= 0045155684 (m4 iquest


theoritical=65634


Theoretical

22


=


= 0104358901 (m4 iquest

Experimental


τ = timecycles =

150920 = 07545 (s)

I = (m )d1 d2 gτ 2

4π 2 L

= (503 ) (05735 )2 (02525 ) (02525 ) ( 981 )

4 π2 (04 )

= 0104606813 (m4 iquest




23

24


SystemCord

Length L (m)

No of Oscillations

(cyclesrsquo)


(sec)

Period τ

Average time (s)

Average Period



Only beam04 20 1785 08925

1782666667 0891333333 00457214970045196926 1147318216

04 20 1772 0886 0044540993 258194598404 20 1791 08955 0045501282 0481645472


04 20 1206 06031209333333 0604666667 0048327651

0044636018 763875978504 20 1213 06065 0045155684 656346126304 20 1209 06045 0044858364 7178679599


150 mm

04 20 1185 059251185666667 0592833333 0068498901

0064508686 582522521804 20 1178 0589 0063748809 693455239104 20 1194 0597 0065492285 4389290021


200 mm

04 20 1328 06641324666667 0662333333 0084187651

0081017277 376584191504 20 1332 0666 0081506068 318524499304 20 1314 0657 0079318085 5784180162


250 mm

04 20 1485 074251501 07505 0104358901

0101305821 362651766404 20 1509 07545 0104606813 029447598504 20 1509 07545 0104606813 0294475985


1634666667 0817333333 00487731420061356248 1494649851

04 20 1625 08125 0060608008 149464985104 20 1644 0822 0062033588 1494649851

25


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only beam045 20 1863 09315

1887666667 0943833333 00457214970043762855 4283854264

045 20 19 095 0045518416 04441707045 20 19 095 0045518416 04441707


045 20 13 0651286666667 0643333333 0048327651

0046102556 4604185043045 20 1282 0641 0044834709 7227626401045 20 1278 0639 0044555366 7805646014


150 mm

045 20 1268 06341264333333 0632166667 0068498901

006565495 4151819235045 20 1272 0636 0066069831 3546145034045 20 1253 06265 0064110788 6406107612


200 mm

045 20 1503 075151504 0752 0084187651

009224587 9571735346045 20 1506 0753 0092614485 10009584045 20 1503 07515 009224587 9571735346


250 mm

045 20 1575 078751580666667 0790333333 0104358901

0101295485 3638795325045 20 1585 07925 010258586 2106058254045 20 1582 0791 010219789 2566897967


1722333333 0861166667 00487731420061414329 1501548924

045 20 1716 0858 0060076597 1501548924045 20 1716 0858 0060076597 1501548924

26


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only beam048 20 1953 09765

1960333333 0980166667 00457214970045087453 1386753778

048 20 1956 0978 0045226077 1083561808048 20 1972 0986 0045968998 0541321699


048 20 1344 06721337666667 0668833333 0048327651

0046196398 4410006579048 20 1328 0664 0045103031 6672411469048 20 1341 06705 0045990395 4836271348


150 mm

048 20 1322 06611303333333 0651666667 0068498901

0066905706 2325869495048 20 1325 06625 0067209707 1882064854048 20 1263 06315 0061067051 1084959004


200 mm

048 20 1463 073151458666667 0729333333 0084187651

0081938668 2671393071048 20 1466 0733 0082275056 2271823432048 20 1447 07235 0080156235 4788607411


250 mm

048 20 1628 08141622 0811 0104358901

0101463315 3439442497048 20 1613 08065 0099602212 5650103475048 20 1625 08125 0101089716 3883211714


1751333333 0875666667 00487731420059651783 1292189689

048 20 1747 08735 0058375127 1292189689048 20 1741 08705 0057974842 1292189689

27


28

Trifilar Apparatus




Theoretical

I zz=25mr2

= 25

(4365 ) (0314 )2 = 048441024 m4

Experimental



L = C τ 2

29

C = L τ 2

τ = timecycles

= 29930

= 099667 (sec)

C= 04

(099667)2=iquest0388 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I p = (mp )R2 g

4 π2 C=

4365(0272)2(981)4 π 2(0417)

= 0457430617 (m4)



30


Theoretical





= 0513003156 m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

1653330 = 05511 (sec)

C= 04

(0569)2=iquest1317 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I+ I p = (m p+m )R2 g

4π 2C=0563225526m2

31





Theoretical

I=32mr2+ 1

2mro

2minus12mr i

2

=((32)43650272^2)+((12)138601^2)-((12)138600418^2)

= 0541601867m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

16330 = 054333 (sec)

32

C= 04

(05433)2=iquest13549 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )


4π 2C=0563225526m2




Experimental



L = C τ 2

C = L τ 2

τ = timecycles

= 2123

30= 0676 sec

C= 04

(0676)2=iquest0707667 m sec2

C = (m+m p )R2 g

4 π2(I + Ip )

33

I+ I p = (m p+m )R2 g

4π 2C=052239m4

Note



I=(( 32 )lowast4365lowast02722)+( 2

5 )lowast5656lowast0152=0535314 m4



34


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate04 30 298 149

2993333333 1496666667 0484410240445393497 805448343

04 30 298 149 0445393497 80544834304 30 302 151 0457430617 5569581401



04 30 164 0821653333333 0826666667 0513003156

0563225526 978987534704 30 165 0825 0570115071 111328582804 30 167 0835 0584019806 1384331624



04 30 157 0785163 0815 0541601867

05161714 469541704204 30 168 084 0591034996 912720797704 30 164 082 0563225526 399253776


the centroid

04 30 213 10652123333333 1061666667 053531424

052239302 24137636304 30 212 106 0517499435 332791537304 30 212 106 0517499435 3327915373

35


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate044 30 314 157

3156666667 1578333333 0484410240449549951 7196439229

044 30 316 158 045529494 6010463261044 30 317 1585 0458181114 5414651525



044 30 162 0811626666667 0813333333 0513003156

0499610983 2610544028044 30 162 081 0499610983 2610544028044 30 164 082 0512023205 0191022412



044 30 169 0845169 0845 0541601867

0543720061 0391098067044 30 167 0835 0530927096 1970962712044 30 171 0855 0556665324 2781278617


centroid

044 30 231 115523 115 053531424

0558559487 4342355492044 30 23 115 0553733942 344091388044 30 229 1145 0548929332 2543383077

36


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate05 30 343 1715

339 1695 0484410240472051706 2551253737

05 30 335 1675 0450288593 704395660505 30 339 1695 0461105951 4810857982



05 30 185 09251846666667 0923333333 051300315

6

0573360905 117655706105 30 187 0935 0585824908 141951859305 30 182 091 055491619 8170131796



05 30 186 0931820666667 0910333333 054160186

7

0579576154 701147639405 30 1792 0896 0537973187 06699902505 30 181 0905 0548834963 1335500582


centroid

05 30 238 1192396666667 1198333333 053531424

0521773549 252948448605 30 239 1195 0526167412 170868376705 30 242 121 0539459539 0774367463

37



-002-00100010020030040

005

01

015

02

025

03

035

04

045

log τ

log l



38

42 Discussion





39




I = md1 d2 gτ 2

4 π 2 L

uL=05480

=000104

ut=00051432

=000035

part Ipart L


(4 π2 L)2

part Ipart t


2π 2 L


uL)2

+( tIpart Ipart t

ut)2


40


41

Chapter 5

Conclusion







42

References


43



13 Objectives



8

Chapter 2




IiquestintM

r2 dm (1)

Where



I= Ī + m d2 (2)

Where



m Mass of the body


9

Bifilar suspension



T 1=mgd2

d and T 2=mg

d1

d (4)



10


Ɵ (6)

ϕ1= d1

L Ɵ and ϕ2=

d2

L Ɵ (7)





MR= iquest (10)



MR= -mg d1d2

L Ɵ (11)


Ī ϴ΄΄ = - mg d1d2

L Ɵ rarr ϴ΄΄+mg

d1d2

Ī LƟ=0 (12)

Where



ω=radicmgd1 d2

Ī L (13)


11


mgd1d2 (14)


Ī =mgd1d2 τ

2

4 π2 L (15)



(16)


Ī = m k 2 (17)


L = mgd1d2 τ

2

4 π2 Ī (18)


With k defined as

k = mgd1d2

4 π2 Ī (20)

12

Trifilar suspension






Where



120601 = RL

Ɵ

(22)

13


ϴ΄΄+(m+m p ) R2 g

L ( Ī+ I p )θ=0 (23)



(24)



L ( τ2π )

2

(25)


L = (m+m p )R2 g

4 π2 ( Ī + I p ) τ2 (26)


L = C τ 2 (27)

With C defined as

C = (m+mp )R2 g

4 π2 ( Ī + I p ) (28)

14

Chapter 3





31 Equipments




15

32 Procedure












16







17

Chapter 4




18


Bifilar Apparatus


19


(a) The beam only

Theoretical



= 045721497 (m4)

Experimental

20


τ = time

cycles = 1791

20 = 08955 (s)


I p = (m )d1 d2 gτ 2

4π 2 L=


= 045501 (m4)





21

Theoretical



= 0048327651 (m4)

Experimental


τ = timecycles =

121320

=06065(s)

I = (m )d1 d2 τ

2 g

4π 2 L

iquest (1444+1793 ) (052 )2(02525)(02525)(981)

4 π 2(04)

= 0045155684 (m4 iquest


theoritical=65634


Theoretical

22


=


= 0104358901 (m4 iquest

Experimental


τ = timecycles =

150920 = 07545 (s)

I = (m )d1 d2 gτ 2

4π 2 L

= (503 ) (05735 )2 (02525 ) (02525 ) ( 981 )

4 π2 (04 )

= 0104606813 (m4 iquest




23

24


SystemCord

Length L (m)

No of Oscillations

(cyclesrsquo)


(sec)

Period τ

Average time (s)

Average Period



Only beam04 20 1785 08925

1782666667 0891333333 00457214970045196926 1147318216

04 20 1772 0886 0044540993 258194598404 20 1791 08955 0045501282 0481645472


04 20 1206 06031209333333 0604666667 0048327651

0044636018 763875978504 20 1213 06065 0045155684 656346126304 20 1209 06045 0044858364 7178679599


150 mm

04 20 1185 059251185666667 0592833333 0068498901

0064508686 582522521804 20 1178 0589 0063748809 693455239104 20 1194 0597 0065492285 4389290021


200 mm

04 20 1328 06641324666667 0662333333 0084187651

0081017277 376584191504 20 1332 0666 0081506068 318524499304 20 1314 0657 0079318085 5784180162


250 mm

04 20 1485 074251501 07505 0104358901

0101305821 362651766404 20 1509 07545 0104606813 029447598504 20 1509 07545 0104606813 0294475985


1634666667 0817333333 00487731420061356248 1494649851

04 20 1625 08125 0060608008 149464985104 20 1644 0822 0062033588 1494649851

25


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only beam045 20 1863 09315

1887666667 0943833333 00457214970043762855 4283854264

045 20 19 095 0045518416 04441707045 20 19 095 0045518416 04441707


045 20 13 0651286666667 0643333333 0048327651

0046102556 4604185043045 20 1282 0641 0044834709 7227626401045 20 1278 0639 0044555366 7805646014


150 mm

045 20 1268 06341264333333 0632166667 0068498901

006565495 4151819235045 20 1272 0636 0066069831 3546145034045 20 1253 06265 0064110788 6406107612


200 mm

045 20 1503 075151504 0752 0084187651

009224587 9571735346045 20 1506 0753 0092614485 10009584045 20 1503 07515 009224587 9571735346


250 mm

045 20 1575 078751580666667 0790333333 0104358901

0101295485 3638795325045 20 1585 07925 010258586 2106058254045 20 1582 0791 010219789 2566897967


1722333333 0861166667 00487731420061414329 1501548924

045 20 1716 0858 0060076597 1501548924045 20 1716 0858 0060076597 1501548924

26


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only beam048 20 1953 09765

1960333333 0980166667 00457214970045087453 1386753778

048 20 1956 0978 0045226077 1083561808048 20 1972 0986 0045968998 0541321699


048 20 1344 06721337666667 0668833333 0048327651

0046196398 4410006579048 20 1328 0664 0045103031 6672411469048 20 1341 06705 0045990395 4836271348


150 mm

048 20 1322 06611303333333 0651666667 0068498901

0066905706 2325869495048 20 1325 06625 0067209707 1882064854048 20 1263 06315 0061067051 1084959004


200 mm

048 20 1463 073151458666667 0729333333 0084187651

0081938668 2671393071048 20 1466 0733 0082275056 2271823432048 20 1447 07235 0080156235 4788607411


250 mm

048 20 1628 08141622 0811 0104358901

0101463315 3439442497048 20 1613 08065 0099602212 5650103475048 20 1625 08125 0101089716 3883211714


1751333333 0875666667 00487731420059651783 1292189689

048 20 1747 08735 0058375127 1292189689048 20 1741 08705 0057974842 1292189689

27


28

Trifilar Apparatus




Theoretical

I zz=25mr2

= 25

(4365 ) (0314 )2 = 048441024 m4

Experimental



L = C τ 2

29

C = L τ 2

τ = timecycles

= 29930

= 099667 (sec)

C= 04

(099667)2=iquest0388 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I p = (mp )R2 g

4 π2 C=

4365(0272)2(981)4 π 2(0417)

= 0457430617 (m4)



30


Theoretical





= 0513003156 m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

1653330 = 05511 (sec)

C= 04

(0569)2=iquest1317 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I+ I p = (m p+m )R2 g

4π 2C=0563225526m2

31





Theoretical

I=32mr2+ 1

2mro

2minus12mr i

2

=((32)43650272^2)+((12)138601^2)-((12)138600418^2)

= 0541601867m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

16330 = 054333 (sec)

32

C= 04

(05433)2=iquest13549 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )


4π 2C=0563225526m2




Experimental



L = C τ 2

C = L τ 2

τ = timecycles

= 2123

30= 0676 sec

C= 04

(0676)2=iquest0707667 m sec2

C = (m+m p )R2 g

4 π2(I + Ip )

33

I+ I p = (m p+m )R2 g

4π 2C=052239m4

Note



I=(( 32 )lowast4365lowast02722)+( 2

5 )lowast5656lowast0152=0535314 m4



34


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate04 30 298 149

2993333333 1496666667 0484410240445393497 805448343

04 30 298 149 0445393497 80544834304 30 302 151 0457430617 5569581401



04 30 164 0821653333333 0826666667 0513003156

0563225526 978987534704 30 165 0825 0570115071 111328582804 30 167 0835 0584019806 1384331624



04 30 157 0785163 0815 0541601867

05161714 469541704204 30 168 084 0591034996 912720797704 30 164 082 0563225526 399253776


the centroid

04 30 213 10652123333333 1061666667 053531424

052239302 24137636304 30 212 106 0517499435 332791537304 30 212 106 0517499435 3327915373

35


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate044 30 314 157

3156666667 1578333333 0484410240449549951 7196439229

044 30 316 158 045529494 6010463261044 30 317 1585 0458181114 5414651525



044 30 162 0811626666667 0813333333 0513003156

0499610983 2610544028044 30 162 081 0499610983 2610544028044 30 164 082 0512023205 0191022412



044 30 169 0845169 0845 0541601867

0543720061 0391098067044 30 167 0835 0530927096 1970962712044 30 171 0855 0556665324 2781278617


centroid

044 30 231 115523 115 053531424

0558559487 4342355492044 30 23 115 0553733942 344091388044 30 229 1145 0548929332 2543383077

36


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate05 30 343 1715

339 1695 0484410240472051706 2551253737

05 30 335 1675 0450288593 704395660505 30 339 1695 0461105951 4810857982



05 30 185 09251846666667 0923333333 051300315

6

0573360905 117655706105 30 187 0935 0585824908 141951859305 30 182 091 055491619 8170131796



05 30 186 0931820666667 0910333333 054160186

7

0579576154 701147639405 30 1792 0896 0537973187 06699902505 30 181 0905 0548834963 1335500582


centroid

05 30 238 1192396666667 1198333333 053531424

0521773549 252948448605 30 239 1195 0526167412 170868376705 30 242 121 0539459539 0774367463

37



-002-00100010020030040

005

01

015

02

025

03

035

04

045

log τ

log l



38

42 Discussion





39




I = md1 d2 gτ 2

4 π 2 L

uL=05480

=000104

ut=00051432

=000035

part Ipart L


(4 π2 L)2

part Ipart t


2π 2 L


uL)2

+( tIpart Ipart t

ut)2


40


41

Chapter 5

Conclusion







42

References


43

Chapter 2




IiquestintM

r2 dm (1)

Where



I= Ī + m d2 (2)

Where



m Mass of the body


9

Bifilar suspension



T 1=mgd2

d and T 2=mg

d1

d (4)



10


Ɵ (6)

ϕ1= d1

L Ɵ and ϕ2=

d2

L Ɵ (7)





MR= iquest (10)



MR= -mg d1d2

L Ɵ (11)


Ī ϴ΄΄ = - mg d1d2

L Ɵ rarr ϴ΄΄+mg

d1d2

Ī LƟ=0 (12)

Where



ω=radicmgd1 d2

Ī L (13)


11


mgd1d2 (14)


Ī =mgd1d2 τ

2

4 π2 L (15)



(16)


Ī = m k 2 (17)


L = mgd1d2 τ

2

4 π2 Ī (18)


With k defined as

k = mgd1d2

4 π2 Ī (20)

12

Trifilar suspension






Where



120601 = RL

Ɵ

(22)

13


ϴ΄΄+(m+m p ) R2 g

L ( Ī+ I p )θ=0 (23)



(24)



L ( τ2π )

2

(25)


L = (m+m p )R2 g

4 π2 ( Ī + I p ) τ2 (26)


L = C τ 2 (27)

With C defined as

C = (m+mp )R2 g

4 π2 ( Ī + I p ) (28)

14

Chapter 3





31 Equipments




15

32 Procedure












16







17

Chapter 4




18


Bifilar Apparatus


19


(a) The beam only

Theoretical



= 045721497 (m4)

Experimental

20


τ = time

cycles = 1791

20 = 08955 (s)


I p = (m )d1 d2 gτ 2

4π 2 L=


= 045501 (m4)





21

Theoretical



= 0048327651 (m4)

Experimental


τ = timecycles =

121320

=06065(s)

I = (m )d1 d2 τ

2 g

4π 2 L

iquest (1444+1793 ) (052 )2(02525)(02525)(981)

4 π 2(04)

= 0045155684 (m4 iquest


theoritical=65634


Theoretical

22


=


= 0104358901 (m4 iquest

Experimental


τ = timecycles =

150920 = 07545 (s)

I = (m )d1 d2 gτ 2

4π 2 L

= (503 ) (05735 )2 (02525 ) (02525 ) ( 981 )

4 π2 (04 )

= 0104606813 (m4 iquest




23

24


SystemCord

Length L (m)

No of Oscillations

(cyclesrsquo)


(sec)

Period τ

Average time (s)

Average Period



Only beam04 20 1785 08925

1782666667 0891333333 00457214970045196926 1147318216

04 20 1772 0886 0044540993 258194598404 20 1791 08955 0045501282 0481645472


04 20 1206 06031209333333 0604666667 0048327651

0044636018 763875978504 20 1213 06065 0045155684 656346126304 20 1209 06045 0044858364 7178679599


150 mm

04 20 1185 059251185666667 0592833333 0068498901

0064508686 582522521804 20 1178 0589 0063748809 693455239104 20 1194 0597 0065492285 4389290021


200 mm

04 20 1328 06641324666667 0662333333 0084187651

0081017277 376584191504 20 1332 0666 0081506068 318524499304 20 1314 0657 0079318085 5784180162


250 mm

04 20 1485 074251501 07505 0104358901

0101305821 362651766404 20 1509 07545 0104606813 029447598504 20 1509 07545 0104606813 0294475985


1634666667 0817333333 00487731420061356248 1494649851

04 20 1625 08125 0060608008 149464985104 20 1644 0822 0062033588 1494649851

25


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only beam045 20 1863 09315

1887666667 0943833333 00457214970043762855 4283854264

045 20 19 095 0045518416 04441707045 20 19 095 0045518416 04441707


045 20 13 0651286666667 0643333333 0048327651

0046102556 4604185043045 20 1282 0641 0044834709 7227626401045 20 1278 0639 0044555366 7805646014


150 mm

045 20 1268 06341264333333 0632166667 0068498901

006565495 4151819235045 20 1272 0636 0066069831 3546145034045 20 1253 06265 0064110788 6406107612


200 mm

045 20 1503 075151504 0752 0084187651

009224587 9571735346045 20 1506 0753 0092614485 10009584045 20 1503 07515 009224587 9571735346


250 mm

045 20 1575 078751580666667 0790333333 0104358901

0101295485 3638795325045 20 1585 07925 010258586 2106058254045 20 1582 0791 010219789 2566897967


1722333333 0861166667 00487731420061414329 1501548924

045 20 1716 0858 0060076597 1501548924045 20 1716 0858 0060076597 1501548924

26


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only beam048 20 1953 09765

1960333333 0980166667 00457214970045087453 1386753778

048 20 1956 0978 0045226077 1083561808048 20 1972 0986 0045968998 0541321699


048 20 1344 06721337666667 0668833333 0048327651

0046196398 4410006579048 20 1328 0664 0045103031 6672411469048 20 1341 06705 0045990395 4836271348


150 mm

048 20 1322 06611303333333 0651666667 0068498901

0066905706 2325869495048 20 1325 06625 0067209707 1882064854048 20 1263 06315 0061067051 1084959004


200 mm

048 20 1463 073151458666667 0729333333 0084187651

0081938668 2671393071048 20 1466 0733 0082275056 2271823432048 20 1447 07235 0080156235 4788607411


250 mm

048 20 1628 08141622 0811 0104358901

0101463315 3439442497048 20 1613 08065 0099602212 5650103475048 20 1625 08125 0101089716 3883211714


1751333333 0875666667 00487731420059651783 1292189689

048 20 1747 08735 0058375127 1292189689048 20 1741 08705 0057974842 1292189689

27


28

Trifilar Apparatus




Theoretical

I zz=25mr2

= 25

(4365 ) (0314 )2 = 048441024 m4

Experimental



L = C τ 2

29

C = L τ 2

τ = timecycles

= 29930

= 099667 (sec)

C= 04

(099667)2=iquest0388 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I p = (mp )R2 g

4 π2 C=

4365(0272)2(981)4 π 2(0417)

= 0457430617 (m4)



30


Theoretical





= 0513003156 m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

1653330 = 05511 (sec)

C= 04

(0569)2=iquest1317 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I+ I p = (m p+m )R2 g

4π 2C=0563225526m2

31





Theoretical

I=32mr2+ 1

2mro

2minus12mr i

2

=((32)43650272^2)+((12)138601^2)-((12)138600418^2)

= 0541601867m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

16330 = 054333 (sec)

32

C= 04

(05433)2=iquest13549 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )


4π 2C=0563225526m2




Experimental



L = C τ 2

C = L τ 2

τ = timecycles

= 2123

30= 0676 sec

C= 04

(0676)2=iquest0707667 m sec2

C = (m+m p )R2 g

4 π2(I + Ip )

33

I+ I p = (m p+m )R2 g

4π 2C=052239m4

Note



I=(( 32 )lowast4365lowast02722)+( 2

5 )lowast5656lowast0152=0535314 m4



34


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate04 30 298 149

2993333333 1496666667 0484410240445393497 805448343

04 30 298 149 0445393497 80544834304 30 302 151 0457430617 5569581401



04 30 164 0821653333333 0826666667 0513003156

0563225526 978987534704 30 165 0825 0570115071 111328582804 30 167 0835 0584019806 1384331624



04 30 157 0785163 0815 0541601867

05161714 469541704204 30 168 084 0591034996 912720797704 30 164 082 0563225526 399253776


the centroid

04 30 213 10652123333333 1061666667 053531424

052239302 24137636304 30 212 106 0517499435 332791537304 30 212 106 0517499435 3327915373

35


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate044 30 314 157

3156666667 1578333333 0484410240449549951 7196439229

044 30 316 158 045529494 6010463261044 30 317 1585 0458181114 5414651525



044 30 162 0811626666667 0813333333 0513003156

0499610983 2610544028044 30 162 081 0499610983 2610544028044 30 164 082 0512023205 0191022412



044 30 169 0845169 0845 0541601867

0543720061 0391098067044 30 167 0835 0530927096 1970962712044 30 171 0855 0556665324 2781278617


centroid

044 30 231 115523 115 053531424

0558559487 4342355492044 30 23 115 0553733942 344091388044 30 229 1145 0548929332 2543383077

36


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate05 30 343 1715

339 1695 0484410240472051706 2551253737

05 30 335 1675 0450288593 704395660505 30 339 1695 0461105951 4810857982



05 30 185 09251846666667 0923333333 051300315

6

0573360905 117655706105 30 187 0935 0585824908 141951859305 30 182 091 055491619 8170131796



05 30 186 0931820666667 0910333333 054160186

7

0579576154 701147639405 30 1792 0896 0537973187 06699902505 30 181 0905 0548834963 1335500582


centroid

05 30 238 1192396666667 1198333333 053531424

0521773549 252948448605 30 239 1195 0526167412 170868376705 30 242 121 0539459539 0774367463

37



-002-00100010020030040

005

01

015

02

025

03

035

04

045

log τ

log l



38

42 Discussion





39




I = md1 d2 gτ 2

4 π 2 L

uL=05480

=000104

ut=00051432

=000035

part Ipart L


(4 π2 L)2

part Ipart t


2π 2 L


uL)2

+( tIpart Ipart t

ut)2


40


41

Chapter 5

Conclusion







42

References


43

Bifilar suspension



T 1=mgd2

d and T 2=mg

d1

d (4)



10


Ɵ (6)

ϕ1= d1

L Ɵ and ϕ2=

d2

L Ɵ (7)





MR= iquest (10)



MR= -mg d1d2

L Ɵ (11)


Ī ϴ΄΄ = - mg d1d2

L Ɵ rarr ϴ΄΄+mg

d1d2

Ī LƟ=0 (12)

Where



ω=radicmgd1 d2

Ī L (13)


11


mgd1d2 (14)


Ī =mgd1d2 τ

2

4 π2 L (15)



(16)


Ī = m k 2 (17)


L = mgd1d2 τ

2

4 π2 Ī (18)


With k defined as

k = mgd1d2

4 π2 Ī (20)

12

Trifilar suspension






Where



120601 = RL

Ɵ

(22)

13


ϴ΄΄+(m+m p ) R2 g

L ( Ī+ I p )θ=0 (23)



(24)



L ( τ2π )

2

(25)


L = (m+m p )R2 g

4 π2 ( Ī + I p ) τ2 (26)


L = C τ 2 (27)

With C defined as

C = (m+mp )R2 g

4 π2 ( Ī + I p ) (28)

14

Chapter 3





31 Equipments




15

32 Procedure












16







17

Chapter 4




18


Bifilar Apparatus


19


(a) The beam only

Theoretical



= 045721497 (m4)

Experimental

20


τ = time

cycles = 1791

20 = 08955 (s)


I p = (m )d1 d2 gτ 2

4π 2 L=


= 045501 (m4)





21

Theoretical



= 0048327651 (m4)

Experimental


τ = timecycles =

121320

=06065(s)

I = (m )d1 d2 τ

2 g

4π 2 L

iquest (1444+1793 ) (052 )2(02525)(02525)(981)

4 π 2(04)

= 0045155684 (m4 iquest


theoritical=65634


Theoretical

22


=


= 0104358901 (m4 iquest

Experimental


τ = timecycles =

150920 = 07545 (s)

I = (m )d1 d2 gτ 2

4π 2 L

= (503 ) (05735 )2 (02525 ) (02525 ) ( 981 )

4 π2 (04 )

= 0104606813 (m4 iquest




23

24


SystemCord

Length L (m)

No of Oscillations

(cyclesrsquo)


(sec)

Period τ

Average time (s)

Average Period



Only beam04 20 1785 08925

1782666667 0891333333 00457214970045196926 1147318216

04 20 1772 0886 0044540993 258194598404 20 1791 08955 0045501282 0481645472


04 20 1206 06031209333333 0604666667 0048327651

0044636018 763875978504 20 1213 06065 0045155684 656346126304 20 1209 06045 0044858364 7178679599


150 mm

04 20 1185 059251185666667 0592833333 0068498901

0064508686 582522521804 20 1178 0589 0063748809 693455239104 20 1194 0597 0065492285 4389290021


200 mm

04 20 1328 06641324666667 0662333333 0084187651

0081017277 376584191504 20 1332 0666 0081506068 318524499304 20 1314 0657 0079318085 5784180162


250 mm

04 20 1485 074251501 07505 0104358901

0101305821 362651766404 20 1509 07545 0104606813 029447598504 20 1509 07545 0104606813 0294475985


1634666667 0817333333 00487731420061356248 1494649851

04 20 1625 08125 0060608008 149464985104 20 1644 0822 0062033588 1494649851

25


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only beam045 20 1863 09315

1887666667 0943833333 00457214970043762855 4283854264

045 20 19 095 0045518416 04441707045 20 19 095 0045518416 04441707


045 20 13 0651286666667 0643333333 0048327651

0046102556 4604185043045 20 1282 0641 0044834709 7227626401045 20 1278 0639 0044555366 7805646014


150 mm

045 20 1268 06341264333333 0632166667 0068498901

006565495 4151819235045 20 1272 0636 0066069831 3546145034045 20 1253 06265 0064110788 6406107612


200 mm

045 20 1503 075151504 0752 0084187651

009224587 9571735346045 20 1506 0753 0092614485 10009584045 20 1503 07515 009224587 9571735346


250 mm

045 20 1575 078751580666667 0790333333 0104358901

0101295485 3638795325045 20 1585 07925 010258586 2106058254045 20 1582 0791 010219789 2566897967


1722333333 0861166667 00487731420061414329 1501548924

045 20 1716 0858 0060076597 1501548924045 20 1716 0858 0060076597 1501548924

26


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only beam048 20 1953 09765

1960333333 0980166667 00457214970045087453 1386753778

048 20 1956 0978 0045226077 1083561808048 20 1972 0986 0045968998 0541321699


048 20 1344 06721337666667 0668833333 0048327651

0046196398 4410006579048 20 1328 0664 0045103031 6672411469048 20 1341 06705 0045990395 4836271348


150 mm

048 20 1322 06611303333333 0651666667 0068498901

0066905706 2325869495048 20 1325 06625 0067209707 1882064854048 20 1263 06315 0061067051 1084959004


200 mm

048 20 1463 073151458666667 0729333333 0084187651

0081938668 2671393071048 20 1466 0733 0082275056 2271823432048 20 1447 07235 0080156235 4788607411


250 mm

048 20 1628 08141622 0811 0104358901

0101463315 3439442497048 20 1613 08065 0099602212 5650103475048 20 1625 08125 0101089716 3883211714


1751333333 0875666667 00487731420059651783 1292189689

048 20 1747 08735 0058375127 1292189689048 20 1741 08705 0057974842 1292189689

27


28

Trifilar Apparatus




Theoretical

I zz=25mr2

= 25

(4365 ) (0314 )2 = 048441024 m4

Experimental



L = C τ 2

29

C = L τ 2

τ = timecycles

= 29930

= 099667 (sec)

C= 04

(099667)2=iquest0388 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I p = (mp )R2 g

4 π2 C=

4365(0272)2(981)4 π 2(0417)

= 0457430617 (m4)



30


Theoretical





= 0513003156 m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

1653330 = 05511 (sec)

C= 04

(0569)2=iquest1317 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I+ I p = (m p+m )R2 g

4π 2C=0563225526m2

31





Theoretical

I=32mr2+ 1

2mro

2minus12mr i

2

=((32)43650272^2)+((12)138601^2)-((12)138600418^2)

= 0541601867m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

16330 = 054333 (sec)

32

C= 04

(05433)2=iquest13549 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )


4π 2C=0563225526m2




Experimental



L = C τ 2

C = L τ 2

τ = timecycles

= 2123

30= 0676 sec

C= 04

(0676)2=iquest0707667 m sec2

C = (m+m p )R2 g

4 π2(I + Ip )

33

I+ I p = (m p+m )R2 g

4π 2C=052239m4

Note



I=(( 32 )lowast4365lowast02722)+( 2

5 )lowast5656lowast0152=0535314 m4



34


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate04 30 298 149

2993333333 1496666667 0484410240445393497 805448343

04 30 298 149 0445393497 80544834304 30 302 151 0457430617 5569581401



04 30 164 0821653333333 0826666667 0513003156

0563225526 978987534704 30 165 0825 0570115071 111328582804 30 167 0835 0584019806 1384331624



04 30 157 0785163 0815 0541601867

05161714 469541704204 30 168 084 0591034996 912720797704 30 164 082 0563225526 399253776


the centroid

04 30 213 10652123333333 1061666667 053531424

052239302 24137636304 30 212 106 0517499435 332791537304 30 212 106 0517499435 3327915373

35


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate044 30 314 157

3156666667 1578333333 0484410240449549951 7196439229

044 30 316 158 045529494 6010463261044 30 317 1585 0458181114 5414651525



044 30 162 0811626666667 0813333333 0513003156

0499610983 2610544028044 30 162 081 0499610983 2610544028044 30 164 082 0512023205 0191022412



044 30 169 0845169 0845 0541601867

0543720061 0391098067044 30 167 0835 0530927096 1970962712044 30 171 0855 0556665324 2781278617


centroid

044 30 231 115523 115 053531424

0558559487 4342355492044 30 23 115 0553733942 344091388044 30 229 1145 0548929332 2543383077

36


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate05 30 343 1715

339 1695 0484410240472051706 2551253737

05 30 335 1675 0450288593 704395660505 30 339 1695 0461105951 4810857982



05 30 185 09251846666667 0923333333 051300315

6

0573360905 117655706105 30 187 0935 0585824908 141951859305 30 182 091 055491619 8170131796



05 30 186 0931820666667 0910333333 054160186

7

0579576154 701147639405 30 1792 0896 0537973187 06699902505 30 181 0905 0548834963 1335500582


centroid

05 30 238 1192396666667 1198333333 053531424

0521773549 252948448605 30 239 1195 0526167412 170868376705 30 242 121 0539459539 0774367463

37



-002-00100010020030040

005

01

015

02

025

03

035

04

045

log τ

log l



38

42 Discussion





39




I = md1 d2 gτ 2

4 π 2 L

uL=05480

=000104

ut=00051432

=000035

part Ipart L


(4 π2 L)2

part Ipart t


2π 2 L


uL)2

+( tIpart Ipart t

ut)2


40


41

Chapter 5

Conclusion







42

References


43


Ɵ (6)

ϕ1= d1

L Ɵ and ϕ2=

d2

L Ɵ (7)





MR= iquest (10)



MR= -mg d1d2

L Ɵ (11)


Ī ϴ΄΄ = - mg d1d2

L Ɵ rarr ϴ΄΄+mg

d1d2

Ī LƟ=0 (12)

Where



ω=radicmgd1 d2

Ī L (13)


11


mgd1d2 (14)


Ī =mgd1d2 τ

2

4 π2 L (15)



(16)


Ī = m k 2 (17)


L = mgd1d2 τ

2

4 π2 Ī (18)


With k defined as

k = mgd1d2

4 π2 Ī (20)

12

Trifilar suspension






Where



120601 = RL

Ɵ

(22)

13


ϴ΄΄+(m+m p ) R2 g

L ( Ī+ I p )θ=0 (23)



(24)



L ( τ2π )

2

(25)


L = (m+m p )R2 g

4 π2 ( Ī + I p ) τ2 (26)


L = C τ 2 (27)

With C defined as

C = (m+mp )R2 g

4 π2 ( Ī + I p ) (28)

14

Chapter 3





31 Equipments




15

32 Procedure












16







17

Chapter 4




18


Bifilar Apparatus


19


(a) The beam only

Theoretical



= 045721497 (m4)

Experimental

20


τ = time

cycles = 1791

20 = 08955 (s)


I p = (m )d1 d2 gτ 2

4π 2 L=


= 045501 (m4)





21

Theoretical



= 0048327651 (m4)

Experimental


τ = timecycles =

121320

=06065(s)

I = (m )d1 d2 τ

2 g

4π 2 L

iquest (1444+1793 ) (052 )2(02525)(02525)(981)

4 π 2(04)

= 0045155684 (m4 iquest


theoritical=65634


Theoretical

22


=


= 0104358901 (m4 iquest

Experimental


τ = timecycles =

150920 = 07545 (s)

I = (m )d1 d2 gτ 2

4π 2 L

= (503 ) (05735 )2 (02525 ) (02525 ) ( 981 )

4 π2 (04 )

= 0104606813 (m4 iquest




23

24


SystemCord

Length L (m)

No of Oscillations

(cyclesrsquo)


(sec)

Period τ

Average time (s)

Average Period



Only beam04 20 1785 08925

1782666667 0891333333 00457214970045196926 1147318216

04 20 1772 0886 0044540993 258194598404 20 1791 08955 0045501282 0481645472


04 20 1206 06031209333333 0604666667 0048327651

0044636018 763875978504 20 1213 06065 0045155684 656346126304 20 1209 06045 0044858364 7178679599


150 mm

04 20 1185 059251185666667 0592833333 0068498901

0064508686 582522521804 20 1178 0589 0063748809 693455239104 20 1194 0597 0065492285 4389290021


200 mm

04 20 1328 06641324666667 0662333333 0084187651

0081017277 376584191504 20 1332 0666 0081506068 318524499304 20 1314 0657 0079318085 5784180162


250 mm

04 20 1485 074251501 07505 0104358901

0101305821 362651766404 20 1509 07545 0104606813 029447598504 20 1509 07545 0104606813 0294475985


1634666667 0817333333 00487731420061356248 1494649851

04 20 1625 08125 0060608008 149464985104 20 1644 0822 0062033588 1494649851

25


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only beam045 20 1863 09315

1887666667 0943833333 00457214970043762855 4283854264

045 20 19 095 0045518416 04441707045 20 19 095 0045518416 04441707


045 20 13 0651286666667 0643333333 0048327651

0046102556 4604185043045 20 1282 0641 0044834709 7227626401045 20 1278 0639 0044555366 7805646014


150 mm

045 20 1268 06341264333333 0632166667 0068498901

006565495 4151819235045 20 1272 0636 0066069831 3546145034045 20 1253 06265 0064110788 6406107612


200 mm

045 20 1503 075151504 0752 0084187651

009224587 9571735346045 20 1506 0753 0092614485 10009584045 20 1503 07515 009224587 9571735346


250 mm

045 20 1575 078751580666667 0790333333 0104358901

0101295485 3638795325045 20 1585 07925 010258586 2106058254045 20 1582 0791 010219789 2566897967


1722333333 0861166667 00487731420061414329 1501548924

045 20 1716 0858 0060076597 1501548924045 20 1716 0858 0060076597 1501548924

26


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only beam048 20 1953 09765

1960333333 0980166667 00457214970045087453 1386753778

048 20 1956 0978 0045226077 1083561808048 20 1972 0986 0045968998 0541321699


048 20 1344 06721337666667 0668833333 0048327651

0046196398 4410006579048 20 1328 0664 0045103031 6672411469048 20 1341 06705 0045990395 4836271348


150 mm

048 20 1322 06611303333333 0651666667 0068498901

0066905706 2325869495048 20 1325 06625 0067209707 1882064854048 20 1263 06315 0061067051 1084959004


200 mm

048 20 1463 073151458666667 0729333333 0084187651

0081938668 2671393071048 20 1466 0733 0082275056 2271823432048 20 1447 07235 0080156235 4788607411


250 mm

048 20 1628 08141622 0811 0104358901

0101463315 3439442497048 20 1613 08065 0099602212 5650103475048 20 1625 08125 0101089716 3883211714


1751333333 0875666667 00487731420059651783 1292189689

048 20 1747 08735 0058375127 1292189689048 20 1741 08705 0057974842 1292189689

27


28

Trifilar Apparatus




Theoretical

I zz=25mr2

= 25

(4365 ) (0314 )2 = 048441024 m4

Experimental



L = C τ 2

29

C = L τ 2

τ = timecycles

= 29930

= 099667 (sec)

C= 04

(099667)2=iquest0388 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I p = (mp )R2 g

4 π2 C=

4365(0272)2(981)4 π 2(0417)

= 0457430617 (m4)



30


Theoretical





= 0513003156 m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

1653330 = 05511 (sec)

C= 04

(0569)2=iquest1317 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I+ I p = (m p+m )R2 g

4π 2C=0563225526m2

31





Theoretical

I=32mr2+ 1

2mro

2minus12mr i

2

=((32)43650272^2)+((12)138601^2)-((12)138600418^2)

= 0541601867m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

16330 = 054333 (sec)

32

C= 04

(05433)2=iquest13549 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )


4π 2C=0563225526m2




Experimental



L = C τ 2

C = L τ 2

τ = timecycles

= 2123

30= 0676 sec

C= 04

(0676)2=iquest0707667 m sec2

C = (m+m p )R2 g

4 π2(I + Ip )

33

I+ I p = (m p+m )R2 g

4π 2C=052239m4

Note



I=(( 32 )lowast4365lowast02722)+( 2

5 )lowast5656lowast0152=0535314 m4



34


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate04 30 298 149

2993333333 1496666667 0484410240445393497 805448343

04 30 298 149 0445393497 80544834304 30 302 151 0457430617 5569581401



04 30 164 0821653333333 0826666667 0513003156

0563225526 978987534704 30 165 0825 0570115071 111328582804 30 167 0835 0584019806 1384331624



04 30 157 0785163 0815 0541601867

05161714 469541704204 30 168 084 0591034996 912720797704 30 164 082 0563225526 399253776


the centroid

04 30 213 10652123333333 1061666667 053531424

052239302 24137636304 30 212 106 0517499435 332791537304 30 212 106 0517499435 3327915373

35


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate044 30 314 157

3156666667 1578333333 0484410240449549951 7196439229

044 30 316 158 045529494 6010463261044 30 317 1585 0458181114 5414651525



044 30 162 0811626666667 0813333333 0513003156

0499610983 2610544028044 30 162 081 0499610983 2610544028044 30 164 082 0512023205 0191022412



044 30 169 0845169 0845 0541601867

0543720061 0391098067044 30 167 0835 0530927096 1970962712044 30 171 0855 0556665324 2781278617


centroid

044 30 231 115523 115 053531424

0558559487 4342355492044 30 23 115 0553733942 344091388044 30 229 1145 0548929332 2543383077

36


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate05 30 343 1715

339 1695 0484410240472051706 2551253737

05 30 335 1675 0450288593 704395660505 30 339 1695 0461105951 4810857982



05 30 185 09251846666667 0923333333 051300315

6

0573360905 117655706105 30 187 0935 0585824908 141951859305 30 182 091 055491619 8170131796



05 30 186 0931820666667 0910333333 054160186

7

0579576154 701147639405 30 1792 0896 0537973187 06699902505 30 181 0905 0548834963 1335500582


centroid

05 30 238 1192396666667 1198333333 053531424

0521773549 252948448605 30 239 1195 0526167412 170868376705 30 242 121 0539459539 0774367463

37



-002-00100010020030040

005

01

015

02

025

03

035

04

045

log τ

log l



38

42 Discussion





39




I = md1 d2 gτ 2

4 π 2 L

uL=05480

=000104

ut=00051432

=000035

part Ipart L


(4 π2 L)2

part Ipart t


2π 2 L


uL)2

+( tIpart Ipart t

ut)2


40


41

Chapter 5

Conclusion







42

References


43


mgd1d2 (14)


Ī =mgd1d2 τ

2

4 π2 L (15)



(16)


Ī = m k 2 (17)


L = mgd1d2 τ

2

4 π2 Ī (18)


With k defined as

k = mgd1d2

4 π2 Ī (20)

12

Trifilar suspension






Where



120601 = RL

Ɵ

(22)

13


ϴ΄΄+(m+m p ) R2 g

L ( Ī+ I p )θ=0 (23)



(24)



L ( τ2π )

2

(25)


L = (m+m p )R2 g

4 π2 ( Ī + I p ) τ2 (26)


L = C τ 2 (27)

With C defined as

C = (m+mp )R2 g

4 π2 ( Ī + I p ) (28)

14

Chapter 3





31 Equipments




15

32 Procedure












16







17

Chapter 4




18


Bifilar Apparatus


19


(a) The beam only

Theoretical



= 045721497 (m4)

Experimental

20


τ = time

cycles = 1791

20 = 08955 (s)


I p = (m )d1 d2 gτ 2

4π 2 L=


= 045501 (m4)





21

Theoretical



= 0048327651 (m4)

Experimental


τ = timecycles =

121320

=06065(s)

I = (m )d1 d2 τ

2 g

4π 2 L

iquest (1444+1793 ) (052 )2(02525)(02525)(981)

4 π 2(04)

= 0045155684 (m4 iquest


theoritical=65634


Theoretical

22


=


= 0104358901 (m4 iquest

Experimental


τ = timecycles =

150920 = 07545 (s)

I = (m )d1 d2 gτ 2

4π 2 L

= (503 ) (05735 )2 (02525 ) (02525 ) ( 981 )

4 π2 (04 )

= 0104606813 (m4 iquest




23

24


SystemCord

Length L (m)

No of Oscillations

(cyclesrsquo)


(sec)

Period τ

Average time (s)

Average Period



Only beam04 20 1785 08925

1782666667 0891333333 00457214970045196926 1147318216

04 20 1772 0886 0044540993 258194598404 20 1791 08955 0045501282 0481645472


04 20 1206 06031209333333 0604666667 0048327651

0044636018 763875978504 20 1213 06065 0045155684 656346126304 20 1209 06045 0044858364 7178679599


150 mm

04 20 1185 059251185666667 0592833333 0068498901

0064508686 582522521804 20 1178 0589 0063748809 693455239104 20 1194 0597 0065492285 4389290021


200 mm

04 20 1328 06641324666667 0662333333 0084187651

0081017277 376584191504 20 1332 0666 0081506068 318524499304 20 1314 0657 0079318085 5784180162


250 mm

04 20 1485 074251501 07505 0104358901

0101305821 362651766404 20 1509 07545 0104606813 029447598504 20 1509 07545 0104606813 0294475985


1634666667 0817333333 00487731420061356248 1494649851

04 20 1625 08125 0060608008 149464985104 20 1644 0822 0062033588 1494649851

25


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only beam045 20 1863 09315

1887666667 0943833333 00457214970043762855 4283854264

045 20 19 095 0045518416 04441707045 20 19 095 0045518416 04441707


045 20 13 0651286666667 0643333333 0048327651

0046102556 4604185043045 20 1282 0641 0044834709 7227626401045 20 1278 0639 0044555366 7805646014


150 mm

045 20 1268 06341264333333 0632166667 0068498901

006565495 4151819235045 20 1272 0636 0066069831 3546145034045 20 1253 06265 0064110788 6406107612


200 mm

045 20 1503 075151504 0752 0084187651

009224587 9571735346045 20 1506 0753 0092614485 10009584045 20 1503 07515 009224587 9571735346


250 mm

045 20 1575 078751580666667 0790333333 0104358901

0101295485 3638795325045 20 1585 07925 010258586 2106058254045 20 1582 0791 010219789 2566897967


1722333333 0861166667 00487731420061414329 1501548924

045 20 1716 0858 0060076597 1501548924045 20 1716 0858 0060076597 1501548924

26


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only beam048 20 1953 09765

1960333333 0980166667 00457214970045087453 1386753778

048 20 1956 0978 0045226077 1083561808048 20 1972 0986 0045968998 0541321699


048 20 1344 06721337666667 0668833333 0048327651

0046196398 4410006579048 20 1328 0664 0045103031 6672411469048 20 1341 06705 0045990395 4836271348


150 mm

048 20 1322 06611303333333 0651666667 0068498901

0066905706 2325869495048 20 1325 06625 0067209707 1882064854048 20 1263 06315 0061067051 1084959004


200 mm

048 20 1463 073151458666667 0729333333 0084187651

0081938668 2671393071048 20 1466 0733 0082275056 2271823432048 20 1447 07235 0080156235 4788607411


250 mm

048 20 1628 08141622 0811 0104358901

0101463315 3439442497048 20 1613 08065 0099602212 5650103475048 20 1625 08125 0101089716 3883211714


1751333333 0875666667 00487731420059651783 1292189689

048 20 1747 08735 0058375127 1292189689048 20 1741 08705 0057974842 1292189689

27


28

Trifilar Apparatus




Theoretical

I zz=25mr2

= 25

(4365 ) (0314 )2 = 048441024 m4

Experimental



L = C τ 2

29

C = L τ 2

τ = timecycles

= 29930

= 099667 (sec)

C= 04

(099667)2=iquest0388 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I p = (mp )R2 g

4 π2 C=

4365(0272)2(981)4 π 2(0417)

= 0457430617 (m4)



30


Theoretical





= 0513003156 m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

1653330 = 05511 (sec)

C= 04

(0569)2=iquest1317 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I+ I p = (m p+m )R2 g

4π 2C=0563225526m2

31





Theoretical

I=32mr2+ 1

2mro

2minus12mr i

2

=((32)43650272^2)+((12)138601^2)-((12)138600418^2)

= 0541601867m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

16330 = 054333 (sec)

32

C= 04

(05433)2=iquest13549 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )


4π 2C=0563225526m2




Experimental



L = C τ 2

C = L τ 2

τ = timecycles

= 2123

30= 0676 sec

C= 04

(0676)2=iquest0707667 m sec2

C = (m+m p )R2 g

4 π2(I + Ip )

33

I+ I p = (m p+m )R2 g

4π 2C=052239m4

Note



I=(( 32 )lowast4365lowast02722)+( 2

5 )lowast5656lowast0152=0535314 m4



34


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate04 30 298 149

2993333333 1496666667 0484410240445393497 805448343

04 30 298 149 0445393497 80544834304 30 302 151 0457430617 5569581401



04 30 164 0821653333333 0826666667 0513003156

0563225526 978987534704 30 165 0825 0570115071 111328582804 30 167 0835 0584019806 1384331624



04 30 157 0785163 0815 0541601867

05161714 469541704204 30 168 084 0591034996 912720797704 30 164 082 0563225526 399253776


the centroid

04 30 213 10652123333333 1061666667 053531424

052239302 24137636304 30 212 106 0517499435 332791537304 30 212 106 0517499435 3327915373

35


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate044 30 314 157

3156666667 1578333333 0484410240449549951 7196439229

044 30 316 158 045529494 6010463261044 30 317 1585 0458181114 5414651525



044 30 162 0811626666667 0813333333 0513003156

0499610983 2610544028044 30 162 081 0499610983 2610544028044 30 164 082 0512023205 0191022412



044 30 169 0845169 0845 0541601867

0543720061 0391098067044 30 167 0835 0530927096 1970962712044 30 171 0855 0556665324 2781278617


centroid

044 30 231 115523 115 053531424

0558559487 4342355492044 30 23 115 0553733942 344091388044 30 229 1145 0548929332 2543383077

36


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate05 30 343 1715

339 1695 0484410240472051706 2551253737

05 30 335 1675 0450288593 704395660505 30 339 1695 0461105951 4810857982



05 30 185 09251846666667 0923333333 051300315

6

0573360905 117655706105 30 187 0935 0585824908 141951859305 30 182 091 055491619 8170131796



05 30 186 0931820666667 0910333333 054160186

7

0579576154 701147639405 30 1792 0896 0537973187 06699902505 30 181 0905 0548834963 1335500582


centroid

05 30 238 1192396666667 1198333333 053531424

0521773549 252948448605 30 239 1195 0526167412 170868376705 30 242 121 0539459539 0774367463

37



-002-00100010020030040

005

01

015

02

025

03

035

04

045

log τ

log l



38

42 Discussion





39




I = md1 d2 gτ 2

4 π 2 L

uL=05480

=000104

ut=00051432

=000035

part Ipart L


(4 π2 L)2

part Ipart t


2π 2 L


uL)2

+( tIpart Ipart t

ut)2


40


41

Chapter 5

Conclusion







42

References


43

Trifilar suspension






Where



120601 = RL

Ɵ

(22)

13


ϴ΄΄+(m+m p ) R2 g

L ( Ī+ I p )θ=0 (23)



(24)



L ( τ2π )

2

(25)


L = (m+m p )R2 g

4 π2 ( Ī + I p ) τ2 (26)


L = C τ 2 (27)

With C defined as

C = (m+mp )R2 g

4 π2 ( Ī + I p ) (28)

14

Chapter 3





31 Equipments




15

32 Procedure












16







17

Chapter 4




18


Bifilar Apparatus


19


(a) The beam only

Theoretical



= 045721497 (m4)

Experimental

20


τ = time

cycles = 1791

20 = 08955 (s)


I p = (m )d1 d2 gτ 2

4π 2 L=


= 045501 (m4)





21

Theoretical



= 0048327651 (m4)

Experimental


τ = timecycles =

121320

=06065(s)

I = (m )d1 d2 τ

2 g

4π 2 L

iquest (1444+1793 ) (052 )2(02525)(02525)(981)

4 π 2(04)

= 0045155684 (m4 iquest


theoritical=65634


Theoretical

22


=


= 0104358901 (m4 iquest

Experimental


τ = timecycles =

150920 = 07545 (s)

I = (m )d1 d2 gτ 2

4π 2 L

= (503 ) (05735 )2 (02525 ) (02525 ) ( 981 )

4 π2 (04 )

= 0104606813 (m4 iquest




23

24


SystemCord

Length L (m)

No of Oscillations

(cyclesrsquo)


(sec)

Period τ

Average time (s)

Average Period



Only beam04 20 1785 08925

1782666667 0891333333 00457214970045196926 1147318216

04 20 1772 0886 0044540993 258194598404 20 1791 08955 0045501282 0481645472


04 20 1206 06031209333333 0604666667 0048327651

0044636018 763875978504 20 1213 06065 0045155684 656346126304 20 1209 06045 0044858364 7178679599


150 mm

04 20 1185 059251185666667 0592833333 0068498901

0064508686 582522521804 20 1178 0589 0063748809 693455239104 20 1194 0597 0065492285 4389290021


200 mm

04 20 1328 06641324666667 0662333333 0084187651

0081017277 376584191504 20 1332 0666 0081506068 318524499304 20 1314 0657 0079318085 5784180162


250 mm

04 20 1485 074251501 07505 0104358901

0101305821 362651766404 20 1509 07545 0104606813 029447598504 20 1509 07545 0104606813 0294475985


1634666667 0817333333 00487731420061356248 1494649851

04 20 1625 08125 0060608008 149464985104 20 1644 0822 0062033588 1494649851

25


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only beam045 20 1863 09315

1887666667 0943833333 00457214970043762855 4283854264

045 20 19 095 0045518416 04441707045 20 19 095 0045518416 04441707


045 20 13 0651286666667 0643333333 0048327651

0046102556 4604185043045 20 1282 0641 0044834709 7227626401045 20 1278 0639 0044555366 7805646014


150 mm

045 20 1268 06341264333333 0632166667 0068498901

006565495 4151819235045 20 1272 0636 0066069831 3546145034045 20 1253 06265 0064110788 6406107612


200 mm

045 20 1503 075151504 0752 0084187651

009224587 9571735346045 20 1506 0753 0092614485 10009584045 20 1503 07515 009224587 9571735346


250 mm

045 20 1575 078751580666667 0790333333 0104358901

0101295485 3638795325045 20 1585 07925 010258586 2106058254045 20 1582 0791 010219789 2566897967


1722333333 0861166667 00487731420061414329 1501548924

045 20 1716 0858 0060076597 1501548924045 20 1716 0858 0060076597 1501548924

26


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only beam048 20 1953 09765

1960333333 0980166667 00457214970045087453 1386753778

048 20 1956 0978 0045226077 1083561808048 20 1972 0986 0045968998 0541321699


048 20 1344 06721337666667 0668833333 0048327651

0046196398 4410006579048 20 1328 0664 0045103031 6672411469048 20 1341 06705 0045990395 4836271348


150 mm

048 20 1322 06611303333333 0651666667 0068498901

0066905706 2325869495048 20 1325 06625 0067209707 1882064854048 20 1263 06315 0061067051 1084959004


200 mm

048 20 1463 073151458666667 0729333333 0084187651

0081938668 2671393071048 20 1466 0733 0082275056 2271823432048 20 1447 07235 0080156235 4788607411


250 mm

048 20 1628 08141622 0811 0104358901

0101463315 3439442497048 20 1613 08065 0099602212 5650103475048 20 1625 08125 0101089716 3883211714


1751333333 0875666667 00487731420059651783 1292189689

048 20 1747 08735 0058375127 1292189689048 20 1741 08705 0057974842 1292189689

27


28

Trifilar Apparatus




Theoretical

I zz=25mr2

= 25

(4365 ) (0314 )2 = 048441024 m4

Experimental



L = C τ 2

29

C = L τ 2

τ = timecycles

= 29930

= 099667 (sec)

C= 04

(099667)2=iquest0388 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I p = (mp )R2 g

4 π2 C=

4365(0272)2(981)4 π 2(0417)

= 0457430617 (m4)



30


Theoretical





= 0513003156 m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

1653330 = 05511 (sec)

C= 04

(0569)2=iquest1317 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I+ I p = (m p+m )R2 g

4π 2C=0563225526m2

31





Theoretical

I=32mr2+ 1

2mro

2minus12mr i

2

=((32)43650272^2)+((12)138601^2)-((12)138600418^2)

= 0541601867m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

16330 = 054333 (sec)

32

C= 04

(05433)2=iquest13549 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )


4π 2C=0563225526m2




Experimental



L = C τ 2

C = L τ 2

τ = timecycles

= 2123

30= 0676 sec

C= 04

(0676)2=iquest0707667 m sec2

C = (m+m p )R2 g

4 π2(I + Ip )

33

I+ I p = (m p+m )R2 g

4π 2C=052239m4

Note



I=(( 32 )lowast4365lowast02722)+( 2

5 )lowast5656lowast0152=0535314 m4



34


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate04 30 298 149

2993333333 1496666667 0484410240445393497 805448343

04 30 298 149 0445393497 80544834304 30 302 151 0457430617 5569581401



04 30 164 0821653333333 0826666667 0513003156

0563225526 978987534704 30 165 0825 0570115071 111328582804 30 167 0835 0584019806 1384331624



04 30 157 0785163 0815 0541601867

05161714 469541704204 30 168 084 0591034996 912720797704 30 164 082 0563225526 399253776


the centroid

04 30 213 10652123333333 1061666667 053531424

052239302 24137636304 30 212 106 0517499435 332791537304 30 212 106 0517499435 3327915373

35


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate044 30 314 157

3156666667 1578333333 0484410240449549951 7196439229

044 30 316 158 045529494 6010463261044 30 317 1585 0458181114 5414651525



044 30 162 0811626666667 0813333333 0513003156

0499610983 2610544028044 30 162 081 0499610983 2610544028044 30 164 082 0512023205 0191022412



044 30 169 0845169 0845 0541601867

0543720061 0391098067044 30 167 0835 0530927096 1970962712044 30 171 0855 0556665324 2781278617


centroid

044 30 231 115523 115 053531424

0558559487 4342355492044 30 23 115 0553733942 344091388044 30 229 1145 0548929332 2543383077

36


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate05 30 343 1715

339 1695 0484410240472051706 2551253737

05 30 335 1675 0450288593 704395660505 30 339 1695 0461105951 4810857982



05 30 185 09251846666667 0923333333 051300315

6

0573360905 117655706105 30 187 0935 0585824908 141951859305 30 182 091 055491619 8170131796



05 30 186 0931820666667 0910333333 054160186

7

0579576154 701147639405 30 1792 0896 0537973187 06699902505 30 181 0905 0548834963 1335500582


centroid

05 30 238 1192396666667 1198333333 053531424

0521773549 252948448605 30 239 1195 0526167412 170868376705 30 242 121 0539459539 0774367463

37



-002-00100010020030040

005

01

015

02

025

03

035

04

045

log τ

log l



38

42 Discussion





39




I = md1 d2 gτ 2

4 π 2 L

uL=05480

=000104

ut=00051432

=000035

part Ipart L


(4 π2 L)2

part Ipart t


2π 2 L


uL)2

+( tIpart Ipart t

ut)2


40


41

Chapter 5

Conclusion







42

References


43


ϴ΄΄+(m+m p ) R2 g

L ( Ī+ I p )θ=0 (23)



(24)



L ( τ2π )

2

(25)


L = (m+m p )R2 g

4 π2 ( Ī + I p ) τ2 (26)


L = C τ 2 (27)

With C defined as

C = (m+mp )R2 g

4 π2 ( Ī + I p ) (28)

14

Chapter 3





31 Equipments




15

32 Procedure












16







17

Chapter 4




18


Bifilar Apparatus


19


(a) The beam only

Theoretical



= 045721497 (m4)

Experimental

20


τ = time

cycles = 1791

20 = 08955 (s)


I p = (m )d1 d2 gτ 2

4π 2 L=


= 045501 (m4)





21

Theoretical



= 0048327651 (m4)

Experimental


τ = timecycles =

121320

=06065(s)

I = (m )d1 d2 τ

2 g

4π 2 L

iquest (1444+1793 ) (052 )2(02525)(02525)(981)

4 π 2(04)

= 0045155684 (m4 iquest


theoritical=65634


Theoretical

22


=


= 0104358901 (m4 iquest

Experimental


τ = timecycles =

150920 = 07545 (s)

I = (m )d1 d2 gτ 2

4π 2 L

= (503 ) (05735 )2 (02525 ) (02525 ) ( 981 )

4 π2 (04 )

= 0104606813 (m4 iquest




23

24


SystemCord

Length L (m)

No of Oscillations

(cyclesrsquo)


(sec)

Period τ

Average time (s)

Average Period



Only beam04 20 1785 08925

1782666667 0891333333 00457214970045196926 1147318216

04 20 1772 0886 0044540993 258194598404 20 1791 08955 0045501282 0481645472


04 20 1206 06031209333333 0604666667 0048327651

0044636018 763875978504 20 1213 06065 0045155684 656346126304 20 1209 06045 0044858364 7178679599


150 mm

04 20 1185 059251185666667 0592833333 0068498901

0064508686 582522521804 20 1178 0589 0063748809 693455239104 20 1194 0597 0065492285 4389290021


200 mm

04 20 1328 06641324666667 0662333333 0084187651

0081017277 376584191504 20 1332 0666 0081506068 318524499304 20 1314 0657 0079318085 5784180162


250 mm

04 20 1485 074251501 07505 0104358901

0101305821 362651766404 20 1509 07545 0104606813 029447598504 20 1509 07545 0104606813 0294475985


1634666667 0817333333 00487731420061356248 1494649851

04 20 1625 08125 0060608008 149464985104 20 1644 0822 0062033588 1494649851

25


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only beam045 20 1863 09315

1887666667 0943833333 00457214970043762855 4283854264

045 20 19 095 0045518416 04441707045 20 19 095 0045518416 04441707


045 20 13 0651286666667 0643333333 0048327651

0046102556 4604185043045 20 1282 0641 0044834709 7227626401045 20 1278 0639 0044555366 7805646014


150 mm

045 20 1268 06341264333333 0632166667 0068498901

006565495 4151819235045 20 1272 0636 0066069831 3546145034045 20 1253 06265 0064110788 6406107612


200 mm

045 20 1503 075151504 0752 0084187651

009224587 9571735346045 20 1506 0753 0092614485 10009584045 20 1503 07515 009224587 9571735346


250 mm

045 20 1575 078751580666667 0790333333 0104358901

0101295485 3638795325045 20 1585 07925 010258586 2106058254045 20 1582 0791 010219789 2566897967


1722333333 0861166667 00487731420061414329 1501548924

045 20 1716 0858 0060076597 1501548924045 20 1716 0858 0060076597 1501548924

26


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only beam048 20 1953 09765

1960333333 0980166667 00457214970045087453 1386753778

048 20 1956 0978 0045226077 1083561808048 20 1972 0986 0045968998 0541321699


048 20 1344 06721337666667 0668833333 0048327651

0046196398 4410006579048 20 1328 0664 0045103031 6672411469048 20 1341 06705 0045990395 4836271348


150 mm

048 20 1322 06611303333333 0651666667 0068498901

0066905706 2325869495048 20 1325 06625 0067209707 1882064854048 20 1263 06315 0061067051 1084959004


200 mm

048 20 1463 073151458666667 0729333333 0084187651

0081938668 2671393071048 20 1466 0733 0082275056 2271823432048 20 1447 07235 0080156235 4788607411


250 mm

048 20 1628 08141622 0811 0104358901

0101463315 3439442497048 20 1613 08065 0099602212 5650103475048 20 1625 08125 0101089716 3883211714


1751333333 0875666667 00487731420059651783 1292189689

048 20 1747 08735 0058375127 1292189689048 20 1741 08705 0057974842 1292189689

27


28

Trifilar Apparatus




Theoretical

I zz=25mr2

= 25

(4365 ) (0314 )2 = 048441024 m4

Experimental



L = C τ 2

29

C = L τ 2

τ = timecycles

= 29930

= 099667 (sec)

C= 04

(099667)2=iquest0388 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I p = (mp )R2 g

4 π2 C=

4365(0272)2(981)4 π 2(0417)

= 0457430617 (m4)



30


Theoretical





= 0513003156 m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

1653330 = 05511 (sec)

C= 04

(0569)2=iquest1317 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I+ I p = (m p+m )R2 g

4π 2C=0563225526m2

31





Theoretical

I=32mr2+ 1

2mro

2minus12mr i

2

=((32)43650272^2)+((12)138601^2)-((12)138600418^2)

= 0541601867m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

16330 = 054333 (sec)

32

C= 04

(05433)2=iquest13549 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )


4π 2C=0563225526m2




Experimental



L = C τ 2

C = L τ 2

τ = timecycles

= 2123

30= 0676 sec

C= 04

(0676)2=iquest0707667 m sec2

C = (m+m p )R2 g

4 π2(I + Ip )

33

I+ I p = (m p+m )R2 g

4π 2C=052239m4

Note



I=(( 32 )lowast4365lowast02722)+( 2

5 )lowast5656lowast0152=0535314 m4



34


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate04 30 298 149

2993333333 1496666667 0484410240445393497 805448343

04 30 298 149 0445393497 80544834304 30 302 151 0457430617 5569581401



04 30 164 0821653333333 0826666667 0513003156

0563225526 978987534704 30 165 0825 0570115071 111328582804 30 167 0835 0584019806 1384331624



04 30 157 0785163 0815 0541601867

05161714 469541704204 30 168 084 0591034996 912720797704 30 164 082 0563225526 399253776


the centroid

04 30 213 10652123333333 1061666667 053531424

052239302 24137636304 30 212 106 0517499435 332791537304 30 212 106 0517499435 3327915373

35


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate044 30 314 157

3156666667 1578333333 0484410240449549951 7196439229

044 30 316 158 045529494 6010463261044 30 317 1585 0458181114 5414651525



044 30 162 0811626666667 0813333333 0513003156

0499610983 2610544028044 30 162 081 0499610983 2610544028044 30 164 082 0512023205 0191022412



044 30 169 0845169 0845 0541601867

0543720061 0391098067044 30 167 0835 0530927096 1970962712044 30 171 0855 0556665324 2781278617


centroid

044 30 231 115523 115 053531424

0558559487 4342355492044 30 23 115 0553733942 344091388044 30 229 1145 0548929332 2543383077

36


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate05 30 343 1715

339 1695 0484410240472051706 2551253737

05 30 335 1675 0450288593 704395660505 30 339 1695 0461105951 4810857982



05 30 185 09251846666667 0923333333 051300315

6

0573360905 117655706105 30 187 0935 0585824908 141951859305 30 182 091 055491619 8170131796



05 30 186 0931820666667 0910333333 054160186

7

0579576154 701147639405 30 1792 0896 0537973187 06699902505 30 181 0905 0548834963 1335500582


centroid

05 30 238 1192396666667 1198333333 053531424

0521773549 252948448605 30 239 1195 0526167412 170868376705 30 242 121 0539459539 0774367463

37



-002-00100010020030040

005

01

015

02

025

03

035

04

045

log τ

log l



38

42 Discussion





39




I = md1 d2 gτ 2

4 π 2 L

uL=05480

=000104

ut=00051432

=000035

part Ipart L


(4 π2 L)2

part Ipart t


2π 2 L


uL)2

+( tIpart Ipart t

ut)2


40


41

Chapter 5

Conclusion







42

References


43

Chapter 3





31 Equipments




15

32 Procedure












16







17

Chapter 4




18


Bifilar Apparatus


19


(a) The beam only

Theoretical



= 045721497 (m4)

Experimental

20


τ = time

cycles = 1791

20 = 08955 (s)


I p = (m )d1 d2 gτ 2

4π 2 L=


= 045501 (m4)





21

Theoretical



= 0048327651 (m4)

Experimental


τ = timecycles =

121320

=06065(s)

I = (m )d1 d2 τ

2 g

4π 2 L

iquest (1444+1793 ) (052 )2(02525)(02525)(981)

4 π 2(04)

= 0045155684 (m4 iquest


theoritical=65634


Theoretical

22


=


= 0104358901 (m4 iquest

Experimental


τ = timecycles =

150920 = 07545 (s)

I = (m )d1 d2 gτ 2

4π 2 L

= (503 ) (05735 )2 (02525 ) (02525 ) ( 981 )

4 π2 (04 )

= 0104606813 (m4 iquest




23

24


SystemCord

Length L (m)

No of Oscillations

(cyclesrsquo)


(sec)

Period τ

Average time (s)

Average Period



Only beam04 20 1785 08925

1782666667 0891333333 00457214970045196926 1147318216

04 20 1772 0886 0044540993 258194598404 20 1791 08955 0045501282 0481645472


04 20 1206 06031209333333 0604666667 0048327651

0044636018 763875978504 20 1213 06065 0045155684 656346126304 20 1209 06045 0044858364 7178679599


150 mm

04 20 1185 059251185666667 0592833333 0068498901

0064508686 582522521804 20 1178 0589 0063748809 693455239104 20 1194 0597 0065492285 4389290021


200 mm

04 20 1328 06641324666667 0662333333 0084187651

0081017277 376584191504 20 1332 0666 0081506068 318524499304 20 1314 0657 0079318085 5784180162


250 mm

04 20 1485 074251501 07505 0104358901

0101305821 362651766404 20 1509 07545 0104606813 029447598504 20 1509 07545 0104606813 0294475985


1634666667 0817333333 00487731420061356248 1494649851

04 20 1625 08125 0060608008 149464985104 20 1644 0822 0062033588 1494649851

25


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only beam045 20 1863 09315

1887666667 0943833333 00457214970043762855 4283854264

045 20 19 095 0045518416 04441707045 20 19 095 0045518416 04441707


045 20 13 0651286666667 0643333333 0048327651

0046102556 4604185043045 20 1282 0641 0044834709 7227626401045 20 1278 0639 0044555366 7805646014


150 mm

045 20 1268 06341264333333 0632166667 0068498901

006565495 4151819235045 20 1272 0636 0066069831 3546145034045 20 1253 06265 0064110788 6406107612


200 mm

045 20 1503 075151504 0752 0084187651

009224587 9571735346045 20 1506 0753 0092614485 10009584045 20 1503 07515 009224587 9571735346


250 mm

045 20 1575 078751580666667 0790333333 0104358901

0101295485 3638795325045 20 1585 07925 010258586 2106058254045 20 1582 0791 010219789 2566897967


1722333333 0861166667 00487731420061414329 1501548924

045 20 1716 0858 0060076597 1501548924045 20 1716 0858 0060076597 1501548924

26


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only beam048 20 1953 09765

1960333333 0980166667 00457214970045087453 1386753778

048 20 1956 0978 0045226077 1083561808048 20 1972 0986 0045968998 0541321699


048 20 1344 06721337666667 0668833333 0048327651

0046196398 4410006579048 20 1328 0664 0045103031 6672411469048 20 1341 06705 0045990395 4836271348


150 mm

048 20 1322 06611303333333 0651666667 0068498901

0066905706 2325869495048 20 1325 06625 0067209707 1882064854048 20 1263 06315 0061067051 1084959004


200 mm

048 20 1463 073151458666667 0729333333 0084187651

0081938668 2671393071048 20 1466 0733 0082275056 2271823432048 20 1447 07235 0080156235 4788607411


250 mm

048 20 1628 08141622 0811 0104358901

0101463315 3439442497048 20 1613 08065 0099602212 5650103475048 20 1625 08125 0101089716 3883211714


1751333333 0875666667 00487731420059651783 1292189689

048 20 1747 08735 0058375127 1292189689048 20 1741 08705 0057974842 1292189689

27


28

Trifilar Apparatus




Theoretical

I zz=25mr2

= 25

(4365 ) (0314 )2 = 048441024 m4

Experimental



L = C τ 2

29

C = L τ 2

τ = timecycles

= 29930

= 099667 (sec)

C= 04

(099667)2=iquest0388 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I p = (mp )R2 g

4 π2 C=

4365(0272)2(981)4 π 2(0417)

= 0457430617 (m4)



30


Theoretical





= 0513003156 m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

1653330 = 05511 (sec)

C= 04

(0569)2=iquest1317 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I+ I p = (m p+m )R2 g

4π 2C=0563225526m2

31





Theoretical

I=32mr2+ 1

2mro

2minus12mr i

2

=((32)43650272^2)+((12)138601^2)-((12)138600418^2)

= 0541601867m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

16330 = 054333 (sec)

32

C= 04

(05433)2=iquest13549 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )


4π 2C=0563225526m2




Experimental



L = C τ 2

C = L τ 2

τ = timecycles

= 2123

30= 0676 sec

C= 04

(0676)2=iquest0707667 m sec2

C = (m+m p )R2 g

4 π2(I + Ip )

33

I+ I p = (m p+m )R2 g

4π 2C=052239m4

Note



I=(( 32 )lowast4365lowast02722)+( 2

5 )lowast5656lowast0152=0535314 m4



34


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate04 30 298 149

2993333333 1496666667 0484410240445393497 805448343

04 30 298 149 0445393497 80544834304 30 302 151 0457430617 5569581401



04 30 164 0821653333333 0826666667 0513003156

0563225526 978987534704 30 165 0825 0570115071 111328582804 30 167 0835 0584019806 1384331624



04 30 157 0785163 0815 0541601867

05161714 469541704204 30 168 084 0591034996 912720797704 30 164 082 0563225526 399253776


the centroid

04 30 213 10652123333333 1061666667 053531424

052239302 24137636304 30 212 106 0517499435 332791537304 30 212 106 0517499435 3327915373

35


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate044 30 314 157

3156666667 1578333333 0484410240449549951 7196439229

044 30 316 158 045529494 6010463261044 30 317 1585 0458181114 5414651525



044 30 162 0811626666667 0813333333 0513003156

0499610983 2610544028044 30 162 081 0499610983 2610544028044 30 164 082 0512023205 0191022412



044 30 169 0845169 0845 0541601867

0543720061 0391098067044 30 167 0835 0530927096 1970962712044 30 171 0855 0556665324 2781278617


centroid

044 30 231 115523 115 053531424

0558559487 4342355492044 30 23 115 0553733942 344091388044 30 229 1145 0548929332 2543383077

36


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate05 30 343 1715

339 1695 0484410240472051706 2551253737

05 30 335 1675 0450288593 704395660505 30 339 1695 0461105951 4810857982



05 30 185 09251846666667 0923333333 051300315

6

0573360905 117655706105 30 187 0935 0585824908 141951859305 30 182 091 055491619 8170131796



05 30 186 0931820666667 0910333333 054160186

7

0579576154 701147639405 30 1792 0896 0537973187 06699902505 30 181 0905 0548834963 1335500582


centroid

05 30 238 1192396666667 1198333333 053531424

0521773549 252948448605 30 239 1195 0526167412 170868376705 30 242 121 0539459539 0774367463

37



-002-00100010020030040

005

01

015

02

025

03

035

04

045

log τ

log l



38

42 Discussion





39




I = md1 d2 gτ 2

4 π 2 L

uL=05480

=000104

ut=00051432

=000035

part Ipart L


(4 π2 L)2

part Ipart t


2π 2 L


uL)2

+( tIpart Ipart t

ut)2


40


41

Chapter 5

Conclusion







42

References


43

32 Procedure












16







17

Chapter 4




18


Bifilar Apparatus


19


(a) The beam only

Theoretical



= 045721497 (m4)

Experimental

20


τ = time

cycles = 1791

20 = 08955 (s)


I p = (m )d1 d2 gτ 2

4π 2 L=


= 045501 (m4)





21

Theoretical



= 0048327651 (m4)

Experimental


τ = timecycles =

121320

=06065(s)

I = (m )d1 d2 τ

2 g

4π 2 L

iquest (1444+1793 ) (052 )2(02525)(02525)(981)

4 π 2(04)

= 0045155684 (m4 iquest


theoritical=65634


Theoretical

22


=


= 0104358901 (m4 iquest

Experimental


τ = timecycles =

150920 = 07545 (s)

I = (m )d1 d2 gτ 2

4π 2 L

= (503 ) (05735 )2 (02525 ) (02525 ) ( 981 )

4 π2 (04 )

= 0104606813 (m4 iquest




23

24


SystemCord

Length L (m)

No of Oscillations

(cyclesrsquo)


(sec)

Period τ

Average time (s)

Average Period



Only beam04 20 1785 08925

1782666667 0891333333 00457214970045196926 1147318216

04 20 1772 0886 0044540993 258194598404 20 1791 08955 0045501282 0481645472


04 20 1206 06031209333333 0604666667 0048327651

0044636018 763875978504 20 1213 06065 0045155684 656346126304 20 1209 06045 0044858364 7178679599


150 mm

04 20 1185 059251185666667 0592833333 0068498901

0064508686 582522521804 20 1178 0589 0063748809 693455239104 20 1194 0597 0065492285 4389290021


200 mm

04 20 1328 06641324666667 0662333333 0084187651

0081017277 376584191504 20 1332 0666 0081506068 318524499304 20 1314 0657 0079318085 5784180162


250 mm

04 20 1485 074251501 07505 0104358901

0101305821 362651766404 20 1509 07545 0104606813 029447598504 20 1509 07545 0104606813 0294475985


1634666667 0817333333 00487731420061356248 1494649851

04 20 1625 08125 0060608008 149464985104 20 1644 0822 0062033588 1494649851

25


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only beam045 20 1863 09315

1887666667 0943833333 00457214970043762855 4283854264

045 20 19 095 0045518416 04441707045 20 19 095 0045518416 04441707


045 20 13 0651286666667 0643333333 0048327651

0046102556 4604185043045 20 1282 0641 0044834709 7227626401045 20 1278 0639 0044555366 7805646014


150 mm

045 20 1268 06341264333333 0632166667 0068498901

006565495 4151819235045 20 1272 0636 0066069831 3546145034045 20 1253 06265 0064110788 6406107612


200 mm

045 20 1503 075151504 0752 0084187651

009224587 9571735346045 20 1506 0753 0092614485 10009584045 20 1503 07515 009224587 9571735346


250 mm

045 20 1575 078751580666667 0790333333 0104358901

0101295485 3638795325045 20 1585 07925 010258586 2106058254045 20 1582 0791 010219789 2566897967


1722333333 0861166667 00487731420061414329 1501548924

045 20 1716 0858 0060076597 1501548924045 20 1716 0858 0060076597 1501548924

26


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only beam048 20 1953 09765

1960333333 0980166667 00457214970045087453 1386753778

048 20 1956 0978 0045226077 1083561808048 20 1972 0986 0045968998 0541321699


048 20 1344 06721337666667 0668833333 0048327651

0046196398 4410006579048 20 1328 0664 0045103031 6672411469048 20 1341 06705 0045990395 4836271348


150 mm

048 20 1322 06611303333333 0651666667 0068498901

0066905706 2325869495048 20 1325 06625 0067209707 1882064854048 20 1263 06315 0061067051 1084959004


200 mm

048 20 1463 073151458666667 0729333333 0084187651

0081938668 2671393071048 20 1466 0733 0082275056 2271823432048 20 1447 07235 0080156235 4788607411


250 mm

048 20 1628 08141622 0811 0104358901

0101463315 3439442497048 20 1613 08065 0099602212 5650103475048 20 1625 08125 0101089716 3883211714


1751333333 0875666667 00487731420059651783 1292189689

048 20 1747 08735 0058375127 1292189689048 20 1741 08705 0057974842 1292189689

27


28

Trifilar Apparatus




Theoretical

I zz=25mr2

= 25

(4365 ) (0314 )2 = 048441024 m4

Experimental



L = C τ 2

29

C = L τ 2

τ = timecycles

= 29930

= 099667 (sec)

C= 04

(099667)2=iquest0388 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I p = (mp )R2 g

4 π2 C=

4365(0272)2(981)4 π 2(0417)

= 0457430617 (m4)



30


Theoretical





= 0513003156 m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

1653330 = 05511 (sec)

C= 04

(0569)2=iquest1317 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I+ I p = (m p+m )R2 g

4π 2C=0563225526m2

31





Theoretical

I=32mr2+ 1

2mro

2minus12mr i

2

=((32)43650272^2)+((12)138601^2)-((12)138600418^2)

= 0541601867m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

16330 = 054333 (sec)

32

C= 04

(05433)2=iquest13549 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )


4π 2C=0563225526m2




Experimental



L = C τ 2

C = L τ 2

τ = timecycles

= 2123

30= 0676 sec

C= 04

(0676)2=iquest0707667 m sec2

C = (m+m p )R2 g

4 π2(I + Ip )

33

I+ I p = (m p+m )R2 g

4π 2C=052239m4

Note



I=(( 32 )lowast4365lowast02722)+( 2

5 )lowast5656lowast0152=0535314 m4



34


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate04 30 298 149

2993333333 1496666667 0484410240445393497 805448343

04 30 298 149 0445393497 80544834304 30 302 151 0457430617 5569581401



04 30 164 0821653333333 0826666667 0513003156

0563225526 978987534704 30 165 0825 0570115071 111328582804 30 167 0835 0584019806 1384331624



04 30 157 0785163 0815 0541601867

05161714 469541704204 30 168 084 0591034996 912720797704 30 164 082 0563225526 399253776


the centroid

04 30 213 10652123333333 1061666667 053531424

052239302 24137636304 30 212 106 0517499435 332791537304 30 212 106 0517499435 3327915373

35


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate044 30 314 157

3156666667 1578333333 0484410240449549951 7196439229

044 30 316 158 045529494 6010463261044 30 317 1585 0458181114 5414651525



044 30 162 0811626666667 0813333333 0513003156

0499610983 2610544028044 30 162 081 0499610983 2610544028044 30 164 082 0512023205 0191022412



044 30 169 0845169 0845 0541601867

0543720061 0391098067044 30 167 0835 0530927096 1970962712044 30 171 0855 0556665324 2781278617


centroid

044 30 231 115523 115 053531424

0558559487 4342355492044 30 23 115 0553733942 344091388044 30 229 1145 0548929332 2543383077

36


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate05 30 343 1715

339 1695 0484410240472051706 2551253737

05 30 335 1675 0450288593 704395660505 30 339 1695 0461105951 4810857982



05 30 185 09251846666667 0923333333 051300315

6

0573360905 117655706105 30 187 0935 0585824908 141951859305 30 182 091 055491619 8170131796



05 30 186 0931820666667 0910333333 054160186

7

0579576154 701147639405 30 1792 0896 0537973187 06699902505 30 181 0905 0548834963 1335500582


centroid

05 30 238 1192396666667 1198333333 053531424

0521773549 252948448605 30 239 1195 0526167412 170868376705 30 242 121 0539459539 0774367463

37



-002-00100010020030040

005

01

015

02

025

03

035

04

045

log τ

log l



38

42 Discussion





39




I = md1 d2 gτ 2

4 π 2 L

uL=05480

=000104

ut=00051432

=000035

part Ipart L


(4 π2 L)2

part Ipart t


2π 2 L


uL)2

+( tIpart Ipart t

ut)2


40


41

Chapter 5

Conclusion







42

References


43







17

Chapter 4




18


Bifilar Apparatus


19


(a) The beam only

Theoretical



= 045721497 (m4)

Experimental

20


τ = time

cycles = 1791

20 = 08955 (s)


I p = (m )d1 d2 gτ 2

4π 2 L=


= 045501 (m4)





21

Theoretical



= 0048327651 (m4)

Experimental


τ = timecycles =

121320

=06065(s)

I = (m )d1 d2 τ

2 g

4π 2 L

iquest (1444+1793 ) (052 )2(02525)(02525)(981)

4 π 2(04)

= 0045155684 (m4 iquest


theoritical=65634


Theoretical

22


=


= 0104358901 (m4 iquest

Experimental


τ = timecycles =

150920 = 07545 (s)

I = (m )d1 d2 gτ 2

4π 2 L

= (503 ) (05735 )2 (02525 ) (02525 ) ( 981 )

4 π2 (04 )

= 0104606813 (m4 iquest




23

24


SystemCord

Length L (m)

No of Oscillations

(cyclesrsquo)


(sec)

Period τ

Average time (s)

Average Period



Only beam04 20 1785 08925

1782666667 0891333333 00457214970045196926 1147318216

04 20 1772 0886 0044540993 258194598404 20 1791 08955 0045501282 0481645472


04 20 1206 06031209333333 0604666667 0048327651

0044636018 763875978504 20 1213 06065 0045155684 656346126304 20 1209 06045 0044858364 7178679599


150 mm

04 20 1185 059251185666667 0592833333 0068498901

0064508686 582522521804 20 1178 0589 0063748809 693455239104 20 1194 0597 0065492285 4389290021


200 mm

04 20 1328 06641324666667 0662333333 0084187651

0081017277 376584191504 20 1332 0666 0081506068 318524499304 20 1314 0657 0079318085 5784180162


250 mm

04 20 1485 074251501 07505 0104358901

0101305821 362651766404 20 1509 07545 0104606813 029447598504 20 1509 07545 0104606813 0294475985


1634666667 0817333333 00487731420061356248 1494649851

04 20 1625 08125 0060608008 149464985104 20 1644 0822 0062033588 1494649851

25


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only beam045 20 1863 09315

1887666667 0943833333 00457214970043762855 4283854264

045 20 19 095 0045518416 04441707045 20 19 095 0045518416 04441707


045 20 13 0651286666667 0643333333 0048327651

0046102556 4604185043045 20 1282 0641 0044834709 7227626401045 20 1278 0639 0044555366 7805646014


150 mm

045 20 1268 06341264333333 0632166667 0068498901

006565495 4151819235045 20 1272 0636 0066069831 3546145034045 20 1253 06265 0064110788 6406107612


200 mm

045 20 1503 075151504 0752 0084187651

009224587 9571735346045 20 1506 0753 0092614485 10009584045 20 1503 07515 009224587 9571735346


250 mm

045 20 1575 078751580666667 0790333333 0104358901

0101295485 3638795325045 20 1585 07925 010258586 2106058254045 20 1582 0791 010219789 2566897967


1722333333 0861166667 00487731420061414329 1501548924

045 20 1716 0858 0060076597 1501548924045 20 1716 0858 0060076597 1501548924

26


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only beam048 20 1953 09765

1960333333 0980166667 00457214970045087453 1386753778

048 20 1956 0978 0045226077 1083561808048 20 1972 0986 0045968998 0541321699


048 20 1344 06721337666667 0668833333 0048327651

0046196398 4410006579048 20 1328 0664 0045103031 6672411469048 20 1341 06705 0045990395 4836271348


150 mm

048 20 1322 06611303333333 0651666667 0068498901

0066905706 2325869495048 20 1325 06625 0067209707 1882064854048 20 1263 06315 0061067051 1084959004


200 mm

048 20 1463 073151458666667 0729333333 0084187651

0081938668 2671393071048 20 1466 0733 0082275056 2271823432048 20 1447 07235 0080156235 4788607411


250 mm

048 20 1628 08141622 0811 0104358901

0101463315 3439442497048 20 1613 08065 0099602212 5650103475048 20 1625 08125 0101089716 3883211714


1751333333 0875666667 00487731420059651783 1292189689

048 20 1747 08735 0058375127 1292189689048 20 1741 08705 0057974842 1292189689

27


28

Trifilar Apparatus




Theoretical

I zz=25mr2

= 25

(4365 ) (0314 )2 = 048441024 m4

Experimental



L = C τ 2

29

C = L τ 2

τ = timecycles

= 29930

= 099667 (sec)

C= 04

(099667)2=iquest0388 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I p = (mp )R2 g

4 π2 C=

4365(0272)2(981)4 π 2(0417)

= 0457430617 (m4)



30


Theoretical





= 0513003156 m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

1653330 = 05511 (sec)

C= 04

(0569)2=iquest1317 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I+ I p = (m p+m )R2 g

4π 2C=0563225526m2

31





Theoretical

I=32mr2+ 1

2mro

2minus12mr i

2

=((32)43650272^2)+((12)138601^2)-((12)138600418^2)

= 0541601867m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

16330 = 054333 (sec)

32

C= 04

(05433)2=iquest13549 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )


4π 2C=0563225526m2




Experimental



L = C τ 2

C = L τ 2

τ = timecycles

= 2123

30= 0676 sec

C= 04

(0676)2=iquest0707667 m sec2

C = (m+m p )R2 g

4 π2(I + Ip )

33

I+ I p = (m p+m )R2 g

4π 2C=052239m4

Note



I=(( 32 )lowast4365lowast02722)+( 2

5 )lowast5656lowast0152=0535314 m4



34


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate04 30 298 149

2993333333 1496666667 0484410240445393497 805448343

04 30 298 149 0445393497 80544834304 30 302 151 0457430617 5569581401



04 30 164 0821653333333 0826666667 0513003156

0563225526 978987534704 30 165 0825 0570115071 111328582804 30 167 0835 0584019806 1384331624



04 30 157 0785163 0815 0541601867

05161714 469541704204 30 168 084 0591034996 912720797704 30 164 082 0563225526 399253776


the centroid

04 30 213 10652123333333 1061666667 053531424

052239302 24137636304 30 212 106 0517499435 332791537304 30 212 106 0517499435 3327915373

35


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate044 30 314 157

3156666667 1578333333 0484410240449549951 7196439229

044 30 316 158 045529494 6010463261044 30 317 1585 0458181114 5414651525



044 30 162 0811626666667 0813333333 0513003156

0499610983 2610544028044 30 162 081 0499610983 2610544028044 30 164 082 0512023205 0191022412



044 30 169 0845169 0845 0541601867

0543720061 0391098067044 30 167 0835 0530927096 1970962712044 30 171 0855 0556665324 2781278617


centroid

044 30 231 115523 115 053531424

0558559487 4342355492044 30 23 115 0553733942 344091388044 30 229 1145 0548929332 2543383077

36


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate05 30 343 1715

339 1695 0484410240472051706 2551253737

05 30 335 1675 0450288593 704395660505 30 339 1695 0461105951 4810857982



05 30 185 09251846666667 0923333333 051300315

6

0573360905 117655706105 30 187 0935 0585824908 141951859305 30 182 091 055491619 8170131796



05 30 186 0931820666667 0910333333 054160186

7

0579576154 701147639405 30 1792 0896 0537973187 06699902505 30 181 0905 0548834963 1335500582


centroid

05 30 238 1192396666667 1198333333 053531424

0521773549 252948448605 30 239 1195 0526167412 170868376705 30 242 121 0539459539 0774367463

37



-002-00100010020030040

005

01

015

02

025

03

035

04

045

log τ

log l



38

42 Discussion





39




I = md1 d2 gτ 2

4 π 2 L

uL=05480

=000104

ut=00051432

=000035

part Ipart L


(4 π2 L)2

part Ipart t


2π 2 L


uL)2

+( tIpart Ipart t

ut)2


40


41

Chapter 5

Conclusion







42

References


43

Chapter 4




18


Bifilar Apparatus


19


(a) The beam only

Theoretical



= 045721497 (m4)

Experimental

20


τ = time

cycles = 1791

20 = 08955 (s)


I p = (m )d1 d2 gτ 2

4π 2 L=


= 045501 (m4)





21

Theoretical



= 0048327651 (m4)

Experimental


τ = timecycles =

121320

=06065(s)

I = (m )d1 d2 τ

2 g

4π 2 L

iquest (1444+1793 ) (052 )2(02525)(02525)(981)

4 π 2(04)

= 0045155684 (m4 iquest


theoritical=65634


Theoretical

22


=


= 0104358901 (m4 iquest

Experimental


τ = timecycles =

150920 = 07545 (s)

I = (m )d1 d2 gτ 2

4π 2 L

= (503 ) (05735 )2 (02525 ) (02525 ) ( 981 )

4 π2 (04 )

= 0104606813 (m4 iquest




23

24


SystemCord

Length L (m)

No of Oscillations

(cyclesrsquo)


(sec)

Period τ

Average time (s)

Average Period



Only beam04 20 1785 08925

1782666667 0891333333 00457214970045196926 1147318216

04 20 1772 0886 0044540993 258194598404 20 1791 08955 0045501282 0481645472


04 20 1206 06031209333333 0604666667 0048327651

0044636018 763875978504 20 1213 06065 0045155684 656346126304 20 1209 06045 0044858364 7178679599


150 mm

04 20 1185 059251185666667 0592833333 0068498901

0064508686 582522521804 20 1178 0589 0063748809 693455239104 20 1194 0597 0065492285 4389290021


200 mm

04 20 1328 06641324666667 0662333333 0084187651

0081017277 376584191504 20 1332 0666 0081506068 318524499304 20 1314 0657 0079318085 5784180162


250 mm

04 20 1485 074251501 07505 0104358901

0101305821 362651766404 20 1509 07545 0104606813 029447598504 20 1509 07545 0104606813 0294475985


1634666667 0817333333 00487731420061356248 1494649851

04 20 1625 08125 0060608008 149464985104 20 1644 0822 0062033588 1494649851

25


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only beam045 20 1863 09315

1887666667 0943833333 00457214970043762855 4283854264

045 20 19 095 0045518416 04441707045 20 19 095 0045518416 04441707


045 20 13 0651286666667 0643333333 0048327651

0046102556 4604185043045 20 1282 0641 0044834709 7227626401045 20 1278 0639 0044555366 7805646014


150 mm

045 20 1268 06341264333333 0632166667 0068498901

006565495 4151819235045 20 1272 0636 0066069831 3546145034045 20 1253 06265 0064110788 6406107612


200 mm

045 20 1503 075151504 0752 0084187651

009224587 9571735346045 20 1506 0753 0092614485 10009584045 20 1503 07515 009224587 9571735346


250 mm

045 20 1575 078751580666667 0790333333 0104358901

0101295485 3638795325045 20 1585 07925 010258586 2106058254045 20 1582 0791 010219789 2566897967


1722333333 0861166667 00487731420061414329 1501548924

045 20 1716 0858 0060076597 1501548924045 20 1716 0858 0060076597 1501548924

26


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only beam048 20 1953 09765

1960333333 0980166667 00457214970045087453 1386753778

048 20 1956 0978 0045226077 1083561808048 20 1972 0986 0045968998 0541321699


048 20 1344 06721337666667 0668833333 0048327651

0046196398 4410006579048 20 1328 0664 0045103031 6672411469048 20 1341 06705 0045990395 4836271348


150 mm

048 20 1322 06611303333333 0651666667 0068498901

0066905706 2325869495048 20 1325 06625 0067209707 1882064854048 20 1263 06315 0061067051 1084959004


200 mm

048 20 1463 073151458666667 0729333333 0084187651

0081938668 2671393071048 20 1466 0733 0082275056 2271823432048 20 1447 07235 0080156235 4788607411


250 mm

048 20 1628 08141622 0811 0104358901

0101463315 3439442497048 20 1613 08065 0099602212 5650103475048 20 1625 08125 0101089716 3883211714


1751333333 0875666667 00487731420059651783 1292189689

048 20 1747 08735 0058375127 1292189689048 20 1741 08705 0057974842 1292189689

27


28

Trifilar Apparatus




Theoretical

I zz=25mr2

= 25

(4365 ) (0314 )2 = 048441024 m4

Experimental



L = C τ 2

29

C = L τ 2

τ = timecycles

= 29930

= 099667 (sec)

C= 04

(099667)2=iquest0388 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I p = (mp )R2 g

4 π2 C=

4365(0272)2(981)4 π 2(0417)

= 0457430617 (m4)



30


Theoretical





= 0513003156 m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

1653330 = 05511 (sec)

C= 04

(0569)2=iquest1317 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I+ I p = (m p+m )R2 g

4π 2C=0563225526m2

31





Theoretical

I=32mr2+ 1

2mro

2minus12mr i

2

=((32)43650272^2)+((12)138601^2)-((12)138600418^2)

= 0541601867m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

16330 = 054333 (sec)

32

C= 04

(05433)2=iquest13549 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )


4π 2C=0563225526m2




Experimental



L = C τ 2

C = L τ 2

τ = timecycles

= 2123

30= 0676 sec

C= 04

(0676)2=iquest0707667 m sec2

C = (m+m p )R2 g

4 π2(I + Ip )

33

I+ I p = (m p+m )R2 g

4π 2C=052239m4

Note



I=(( 32 )lowast4365lowast02722)+( 2

5 )lowast5656lowast0152=0535314 m4



34


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate04 30 298 149

2993333333 1496666667 0484410240445393497 805448343

04 30 298 149 0445393497 80544834304 30 302 151 0457430617 5569581401



04 30 164 0821653333333 0826666667 0513003156

0563225526 978987534704 30 165 0825 0570115071 111328582804 30 167 0835 0584019806 1384331624



04 30 157 0785163 0815 0541601867

05161714 469541704204 30 168 084 0591034996 912720797704 30 164 082 0563225526 399253776


the centroid

04 30 213 10652123333333 1061666667 053531424

052239302 24137636304 30 212 106 0517499435 332791537304 30 212 106 0517499435 3327915373

35


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate044 30 314 157

3156666667 1578333333 0484410240449549951 7196439229

044 30 316 158 045529494 6010463261044 30 317 1585 0458181114 5414651525



044 30 162 0811626666667 0813333333 0513003156

0499610983 2610544028044 30 162 081 0499610983 2610544028044 30 164 082 0512023205 0191022412



044 30 169 0845169 0845 0541601867

0543720061 0391098067044 30 167 0835 0530927096 1970962712044 30 171 0855 0556665324 2781278617


centroid

044 30 231 115523 115 053531424

0558559487 4342355492044 30 23 115 0553733942 344091388044 30 229 1145 0548929332 2543383077

36


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate05 30 343 1715

339 1695 0484410240472051706 2551253737

05 30 335 1675 0450288593 704395660505 30 339 1695 0461105951 4810857982



05 30 185 09251846666667 0923333333 051300315

6

0573360905 117655706105 30 187 0935 0585824908 141951859305 30 182 091 055491619 8170131796



05 30 186 0931820666667 0910333333 054160186

7

0579576154 701147639405 30 1792 0896 0537973187 06699902505 30 181 0905 0548834963 1335500582


centroid

05 30 238 1192396666667 1198333333 053531424

0521773549 252948448605 30 239 1195 0526167412 170868376705 30 242 121 0539459539 0774367463

37



-002-00100010020030040

005

01

015

02

025

03

035

04

045

log τ

log l



38

42 Discussion





39




I = md1 d2 gτ 2

4 π 2 L

uL=05480

=000104

ut=00051432

=000035

part Ipart L


(4 π2 L)2

part Ipart t


2π 2 L


uL)2

+( tIpart Ipart t

ut)2


40


41

Chapter 5

Conclusion







42

References


43


Bifilar Apparatus


19


(a) The beam only

Theoretical



= 045721497 (m4)

Experimental

20


τ = time

cycles = 1791

20 = 08955 (s)


I p = (m )d1 d2 gτ 2

4π 2 L=


= 045501 (m4)





21

Theoretical



= 0048327651 (m4)

Experimental


τ = timecycles =

121320

=06065(s)

I = (m )d1 d2 τ

2 g

4π 2 L

iquest (1444+1793 ) (052 )2(02525)(02525)(981)

4 π 2(04)

= 0045155684 (m4 iquest


theoritical=65634


Theoretical

22


=


= 0104358901 (m4 iquest

Experimental


τ = timecycles =

150920 = 07545 (s)

I = (m )d1 d2 gτ 2

4π 2 L

= (503 ) (05735 )2 (02525 ) (02525 ) ( 981 )

4 π2 (04 )

= 0104606813 (m4 iquest




23

24


SystemCord

Length L (m)

No of Oscillations

(cyclesrsquo)


(sec)

Period τ

Average time (s)

Average Period



Only beam04 20 1785 08925

1782666667 0891333333 00457214970045196926 1147318216

04 20 1772 0886 0044540993 258194598404 20 1791 08955 0045501282 0481645472


04 20 1206 06031209333333 0604666667 0048327651

0044636018 763875978504 20 1213 06065 0045155684 656346126304 20 1209 06045 0044858364 7178679599


150 mm

04 20 1185 059251185666667 0592833333 0068498901

0064508686 582522521804 20 1178 0589 0063748809 693455239104 20 1194 0597 0065492285 4389290021


200 mm

04 20 1328 06641324666667 0662333333 0084187651

0081017277 376584191504 20 1332 0666 0081506068 318524499304 20 1314 0657 0079318085 5784180162


250 mm

04 20 1485 074251501 07505 0104358901

0101305821 362651766404 20 1509 07545 0104606813 029447598504 20 1509 07545 0104606813 0294475985


1634666667 0817333333 00487731420061356248 1494649851

04 20 1625 08125 0060608008 149464985104 20 1644 0822 0062033588 1494649851

25


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only beam045 20 1863 09315

1887666667 0943833333 00457214970043762855 4283854264

045 20 19 095 0045518416 04441707045 20 19 095 0045518416 04441707


045 20 13 0651286666667 0643333333 0048327651

0046102556 4604185043045 20 1282 0641 0044834709 7227626401045 20 1278 0639 0044555366 7805646014


150 mm

045 20 1268 06341264333333 0632166667 0068498901

006565495 4151819235045 20 1272 0636 0066069831 3546145034045 20 1253 06265 0064110788 6406107612


200 mm

045 20 1503 075151504 0752 0084187651

009224587 9571735346045 20 1506 0753 0092614485 10009584045 20 1503 07515 009224587 9571735346


250 mm

045 20 1575 078751580666667 0790333333 0104358901

0101295485 3638795325045 20 1585 07925 010258586 2106058254045 20 1582 0791 010219789 2566897967


1722333333 0861166667 00487731420061414329 1501548924

045 20 1716 0858 0060076597 1501548924045 20 1716 0858 0060076597 1501548924

26


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only beam048 20 1953 09765

1960333333 0980166667 00457214970045087453 1386753778

048 20 1956 0978 0045226077 1083561808048 20 1972 0986 0045968998 0541321699


048 20 1344 06721337666667 0668833333 0048327651

0046196398 4410006579048 20 1328 0664 0045103031 6672411469048 20 1341 06705 0045990395 4836271348


150 mm

048 20 1322 06611303333333 0651666667 0068498901

0066905706 2325869495048 20 1325 06625 0067209707 1882064854048 20 1263 06315 0061067051 1084959004


200 mm

048 20 1463 073151458666667 0729333333 0084187651

0081938668 2671393071048 20 1466 0733 0082275056 2271823432048 20 1447 07235 0080156235 4788607411


250 mm

048 20 1628 08141622 0811 0104358901

0101463315 3439442497048 20 1613 08065 0099602212 5650103475048 20 1625 08125 0101089716 3883211714


1751333333 0875666667 00487731420059651783 1292189689

048 20 1747 08735 0058375127 1292189689048 20 1741 08705 0057974842 1292189689

27


28

Trifilar Apparatus




Theoretical

I zz=25mr2

= 25

(4365 ) (0314 )2 = 048441024 m4

Experimental



L = C τ 2

29

C = L τ 2

τ = timecycles

= 29930

= 099667 (sec)

C= 04

(099667)2=iquest0388 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I p = (mp )R2 g

4 π2 C=

4365(0272)2(981)4 π 2(0417)

= 0457430617 (m4)



30


Theoretical





= 0513003156 m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

1653330 = 05511 (sec)

C= 04

(0569)2=iquest1317 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I+ I p = (m p+m )R2 g

4π 2C=0563225526m2

31





Theoretical

I=32mr2+ 1

2mro

2minus12mr i

2

=((32)43650272^2)+((12)138601^2)-((12)138600418^2)

= 0541601867m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

16330 = 054333 (sec)

32

C= 04

(05433)2=iquest13549 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )


4π 2C=0563225526m2




Experimental



L = C τ 2

C = L τ 2

τ = timecycles

= 2123

30= 0676 sec

C= 04

(0676)2=iquest0707667 m sec2

C = (m+m p )R2 g

4 π2(I + Ip )

33

I+ I p = (m p+m )R2 g

4π 2C=052239m4

Note



I=(( 32 )lowast4365lowast02722)+( 2

5 )lowast5656lowast0152=0535314 m4



34


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate04 30 298 149

2993333333 1496666667 0484410240445393497 805448343

04 30 298 149 0445393497 80544834304 30 302 151 0457430617 5569581401



04 30 164 0821653333333 0826666667 0513003156

0563225526 978987534704 30 165 0825 0570115071 111328582804 30 167 0835 0584019806 1384331624



04 30 157 0785163 0815 0541601867

05161714 469541704204 30 168 084 0591034996 912720797704 30 164 082 0563225526 399253776


the centroid

04 30 213 10652123333333 1061666667 053531424

052239302 24137636304 30 212 106 0517499435 332791537304 30 212 106 0517499435 3327915373

35


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate044 30 314 157

3156666667 1578333333 0484410240449549951 7196439229

044 30 316 158 045529494 6010463261044 30 317 1585 0458181114 5414651525



044 30 162 0811626666667 0813333333 0513003156

0499610983 2610544028044 30 162 081 0499610983 2610544028044 30 164 082 0512023205 0191022412



044 30 169 0845169 0845 0541601867

0543720061 0391098067044 30 167 0835 0530927096 1970962712044 30 171 0855 0556665324 2781278617


centroid

044 30 231 115523 115 053531424

0558559487 4342355492044 30 23 115 0553733942 344091388044 30 229 1145 0548929332 2543383077

36


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate05 30 343 1715

339 1695 0484410240472051706 2551253737

05 30 335 1675 0450288593 704395660505 30 339 1695 0461105951 4810857982



05 30 185 09251846666667 0923333333 051300315

6

0573360905 117655706105 30 187 0935 0585824908 141951859305 30 182 091 055491619 8170131796



05 30 186 0931820666667 0910333333 054160186

7

0579576154 701147639405 30 1792 0896 0537973187 06699902505 30 181 0905 0548834963 1335500582


centroid

05 30 238 1192396666667 1198333333 053531424

0521773549 252948448605 30 239 1195 0526167412 170868376705 30 242 121 0539459539 0774367463

37



-002-00100010020030040

005

01

015

02

025

03

035

04

045

log τ

log l



38

42 Discussion





39




I = md1 d2 gτ 2

4 π 2 L

uL=05480

=000104

ut=00051432

=000035

part Ipart L


(4 π2 L)2

part Ipart t


2π 2 L


uL)2

+( tIpart Ipart t

ut)2


40


41

Chapter 5

Conclusion







42

References


43


(a) The beam only

Theoretical



= 045721497 (m4)

Experimental

20


τ = time

cycles = 1791

20 = 08955 (s)


I p = (m )d1 d2 gτ 2

4π 2 L=


= 045501 (m4)





21

Theoretical



= 0048327651 (m4)

Experimental


τ = timecycles =

121320

=06065(s)

I = (m )d1 d2 τ

2 g

4π 2 L

iquest (1444+1793 ) (052 )2(02525)(02525)(981)

4 π 2(04)

= 0045155684 (m4 iquest


theoritical=65634


Theoretical

22


=


= 0104358901 (m4 iquest

Experimental


τ = timecycles =

150920 = 07545 (s)

I = (m )d1 d2 gτ 2

4π 2 L

= (503 ) (05735 )2 (02525 ) (02525 ) ( 981 )

4 π2 (04 )

= 0104606813 (m4 iquest




23

24


SystemCord

Length L (m)

No of Oscillations

(cyclesrsquo)


(sec)

Period τ

Average time (s)

Average Period



Only beam04 20 1785 08925

1782666667 0891333333 00457214970045196926 1147318216

04 20 1772 0886 0044540993 258194598404 20 1791 08955 0045501282 0481645472


04 20 1206 06031209333333 0604666667 0048327651

0044636018 763875978504 20 1213 06065 0045155684 656346126304 20 1209 06045 0044858364 7178679599


150 mm

04 20 1185 059251185666667 0592833333 0068498901

0064508686 582522521804 20 1178 0589 0063748809 693455239104 20 1194 0597 0065492285 4389290021


200 mm

04 20 1328 06641324666667 0662333333 0084187651

0081017277 376584191504 20 1332 0666 0081506068 318524499304 20 1314 0657 0079318085 5784180162


250 mm

04 20 1485 074251501 07505 0104358901

0101305821 362651766404 20 1509 07545 0104606813 029447598504 20 1509 07545 0104606813 0294475985


1634666667 0817333333 00487731420061356248 1494649851

04 20 1625 08125 0060608008 149464985104 20 1644 0822 0062033588 1494649851

25


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only beam045 20 1863 09315

1887666667 0943833333 00457214970043762855 4283854264

045 20 19 095 0045518416 04441707045 20 19 095 0045518416 04441707


045 20 13 0651286666667 0643333333 0048327651

0046102556 4604185043045 20 1282 0641 0044834709 7227626401045 20 1278 0639 0044555366 7805646014


150 mm

045 20 1268 06341264333333 0632166667 0068498901

006565495 4151819235045 20 1272 0636 0066069831 3546145034045 20 1253 06265 0064110788 6406107612


200 mm

045 20 1503 075151504 0752 0084187651

009224587 9571735346045 20 1506 0753 0092614485 10009584045 20 1503 07515 009224587 9571735346


250 mm

045 20 1575 078751580666667 0790333333 0104358901

0101295485 3638795325045 20 1585 07925 010258586 2106058254045 20 1582 0791 010219789 2566897967


1722333333 0861166667 00487731420061414329 1501548924

045 20 1716 0858 0060076597 1501548924045 20 1716 0858 0060076597 1501548924

26


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only beam048 20 1953 09765

1960333333 0980166667 00457214970045087453 1386753778

048 20 1956 0978 0045226077 1083561808048 20 1972 0986 0045968998 0541321699


048 20 1344 06721337666667 0668833333 0048327651

0046196398 4410006579048 20 1328 0664 0045103031 6672411469048 20 1341 06705 0045990395 4836271348


150 mm

048 20 1322 06611303333333 0651666667 0068498901

0066905706 2325869495048 20 1325 06625 0067209707 1882064854048 20 1263 06315 0061067051 1084959004


200 mm

048 20 1463 073151458666667 0729333333 0084187651

0081938668 2671393071048 20 1466 0733 0082275056 2271823432048 20 1447 07235 0080156235 4788607411


250 mm

048 20 1628 08141622 0811 0104358901

0101463315 3439442497048 20 1613 08065 0099602212 5650103475048 20 1625 08125 0101089716 3883211714


1751333333 0875666667 00487731420059651783 1292189689

048 20 1747 08735 0058375127 1292189689048 20 1741 08705 0057974842 1292189689

27


28

Trifilar Apparatus




Theoretical

I zz=25mr2

= 25

(4365 ) (0314 )2 = 048441024 m4

Experimental



L = C τ 2

29

C = L τ 2

τ = timecycles

= 29930

= 099667 (sec)

C= 04

(099667)2=iquest0388 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I p = (mp )R2 g

4 π2 C=

4365(0272)2(981)4 π 2(0417)

= 0457430617 (m4)



30


Theoretical





= 0513003156 m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

1653330 = 05511 (sec)

C= 04

(0569)2=iquest1317 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I+ I p = (m p+m )R2 g

4π 2C=0563225526m2

31





Theoretical

I=32mr2+ 1

2mro

2minus12mr i

2

=((32)43650272^2)+((12)138601^2)-((12)138600418^2)

= 0541601867m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

16330 = 054333 (sec)

32

C= 04

(05433)2=iquest13549 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )


4π 2C=0563225526m2




Experimental



L = C τ 2

C = L τ 2

τ = timecycles

= 2123

30= 0676 sec

C= 04

(0676)2=iquest0707667 m sec2

C = (m+m p )R2 g

4 π2(I + Ip )

33

I+ I p = (m p+m )R2 g

4π 2C=052239m4

Note



I=(( 32 )lowast4365lowast02722)+( 2

5 )lowast5656lowast0152=0535314 m4



34


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate04 30 298 149

2993333333 1496666667 0484410240445393497 805448343

04 30 298 149 0445393497 80544834304 30 302 151 0457430617 5569581401



04 30 164 0821653333333 0826666667 0513003156

0563225526 978987534704 30 165 0825 0570115071 111328582804 30 167 0835 0584019806 1384331624



04 30 157 0785163 0815 0541601867

05161714 469541704204 30 168 084 0591034996 912720797704 30 164 082 0563225526 399253776


the centroid

04 30 213 10652123333333 1061666667 053531424

052239302 24137636304 30 212 106 0517499435 332791537304 30 212 106 0517499435 3327915373

35


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate044 30 314 157

3156666667 1578333333 0484410240449549951 7196439229

044 30 316 158 045529494 6010463261044 30 317 1585 0458181114 5414651525



044 30 162 0811626666667 0813333333 0513003156

0499610983 2610544028044 30 162 081 0499610983 2610544028044 30 164 082 0512023205 0191022412



044 30 169 0845169 0845 0541601867

0543720061 0391098067044 30 167 0835 0530927096 1970962712044 30 171 0855 0556665324 2781278617


centroid

044 30 231 115523 115 053531424

0558559487 4342355492044 30 23 115 0553733942 344091388044 30 229 1145 0548929332 2543383077

36


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate05 30 343 1715

339 1695 0484410240472051706 2551253737

05 30 335 1675 0450288593 704395660505 30 339 1695 0461105951 4810857982



05 30 185 09251846666667 0923333333 051300315

6

0573360905 117655706105 30 187 0935 0585824908 141951859305 30 182 091 055491619 8170131796



05 30 186 0931820666667 0910333333 054160186

7

0579576154 701147639405 30 1792 0896 0537973187 06699902505 30 181 0905 0548834963 1335500582


centroid

05 30 238 1192396666667 1198333333 053531424

0521773549 252948448605 30 239 1195 0526167412 170868376705 30 242 121 0539459539 0774367463

37



-002-00100010020030040

005

01

015

02

025

03

035

04

045

log τ

log l



38

42 Discussion





39




I = md1 d2 gτ 2

4 π 2 L

uL=05480

=000104

ut=00051432

=000035

part Ipart L


(4 π2 L)2

part Ipart t


2π 2 L


uL)2

+( tIpart Ipart t

ut)2


40


41

Chapter 5

Conclusion







42

References


43


τ = time

cycles = 1791

20 = 08955 (s)


I p = (m )d1 d2 gτ 2

4π 2 L=


= 045501 (m4)





21

Theoretical



= 0048327651 (m4)

Experimental


τ = timecycles =

121320

=06065(s)

I = (m )d1 d2 τ

2 g

4π 2 L

iquest (1444+1793 ) (052 )2(02525)(02525)(981)

4 π 2(04)

= 0045155684 (m4 iquest


theoritical=65634


Theoretical

22


=


= 0104358901 (m4 iquest

Experimental


τ = timecycles =

150920 = 07545 (s)

I = (m )d1 d2 gτ 2

4π 2 L

= (503 ) (05735 )2 (02525 ) (02525 ) ( 981 )

4 π2 (04 )

= 0104606813 (m4 iquest




23

24


SystemCord

Length L (m)

No of Oscillations

(cyclesrsquo)


(sec)

Period τ

Average time (s)

Average Period



Only beam04 20 1785 08925

1782666667 0891333333 00457214970045196926 1147318216

04 20 1772 0886 0044540993 258194598404 20 1791 08955 0045501282 0481645472


04 20 1206 06031209333333 0604666667 0048327651

0044636018 763875978504 20 1213 06065 0045155684 656346126304 20 1209 06045 0044858364 7178679599


150 mm

04 20 1185 059251185666667 0592833333 0068498901

0064508686 582522521804 20 1178 0589 0063748809 693455239104 20 1194 0597 0065492285 4389290021


200 mm

04 20 1328 06641324666667 0662333333 0084187651

0081017277 376584191504 20 1332 0666 0081506068 318524499304 20 1314 0657 0079318085 5784180162


250 mm

04 20 1485 074251501 07505 0104358901

0101305821 362651766404 20 1509 07545 0104606813 029447598504 20 1509 07545 0104606813 0294475985


1634666667 0817333333 00487731420061356248 1494649851

04 20 1625 08125 0060608008 149464985104 20 1644 0822 0062033588 1494649851

25


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only beam045 20 1863 09315

1887666667 0943833333 00457214970043762855 4283854264

045 20 19 095 0045518416 04441707045 20 19 095 0045518416 04441707


045 20 13 0651286666667 0643333333 0048327651

0046102556 4604185043045 20 1282 0641 0044834709 7227626401045 20 1278 0639 0044555366 7805646014


150 mm

045 20 1268 06341264333333 0632166667 0068498901

006565495 4151819235045 20 1272 0636 0066069831 3546145034045 20 1253 06265 0064110788 6406107612


200 mm

045 20 1503 075151504 0752 0084187651

009224587 9571735346045 20 1506 0753 0092614485 10009584045 20 1503 07515 009224587 9571735346


250 mm

045 20 1575 078751580666667 0790333333 0104358901

0101295485 3638795325045 20 1585 07925 010258586 2106058254045 20 1582 0791 010219789 2566897967


1722333333 0861166667 00487731420061414329 1501548924

045 20 1716 0858 0060076597 1501548924045 20 1716 0858 0060076597 1501548924

26


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only beam048 20 1953 09765

1960333333 0980166667 00457214970045087453 1386753778

048 20 1956 0978 0045226077 1083561808048 20 1972 0986 0045968998 0541321699


048 20 1344 06721337666667 0668833333 0048327651

0046196398 4410006579048 20 1328 0664 0045103031 6672411469048 20 1341 06705 0045990395 4836271348


150 mm

048 20 1322 06611303333333 0651666667 0068498901

0066905706 2325869495048 20 1325 06625 0067209707 1882064854048 20 1263 06315 0061067051 1084959004


200 mm

048 20 1463 073151458666667 0729333333 0084187651

0081938668 2671393071048 20 1466 0733 0082275056 2271823432048 20 1447 07235 0080156235 4788607411


250 mm

048 20 1628 08141622 0811 0104358901

0101463315 3439442497048 20 1613 08065 0099602212 5650103475048 20 1625 08125 0101089716 3883211714


1751333333 0875666667 00487731420059651783 1292189689

048 20 1747 08735 0058375127 1292189689048 20 1741 08705 0057974842 1292189689

27


28

Trifilar Apparatus




Theoretical

I zz=25mr2

= 25

(4365 ) (0314 )2 = 048441024 m4

Experimental



L = C τ 2

29

C = L τ 2

τ = timecycles

= 29930

= 099667 (sec)

C= 04

(099667)2=iquest0388 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I p = (mp )R2 g

4 π2 C=

4365(0272)2(981)4 π 2(0417)

= 0457430617 (m4)



30


Theoretical





= 0513003156 m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

1653330 = 05511 (sec)

C= 04

(0569)2=iquest1317 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I+ I p = (m p+m )R2 g

4π 2C=0563225526m2

31





Theoretical

I=32mr2+ 1

2mro

2minus12mr i

2

=((32)43650272^2)+((12)138601^2)-((12)138600418^2)

= 0541601867m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

16330 = 054333 (sec)

32

C= 04

(05433)2=iquest13549 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )


4π 2C=0563225526m2




Experimental



L = C τ 2

C = L τ 2

τ = timecycles

= 2123

30= 0676 sec

C= 04

(0676)2=iquest0707667 m sec2

C = (m+m p )R2 g

4 π2(I + Ip )

33

I+ I p = (m p+m )R2 g

4π 2C=052239m4

Note



I=(( 32 )lowast4365lowast02722)+( 2

5 )lowast5656lowast0152=0535314 m4



34


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate04 30 298 149

2993333333 1496666667 0484410240445393497 805448343

04 30 298 149 0445393497 80544834304 30 302 151 0457430617 5569581401



04 30 164 0821653333333 0826666667 0513003156

0563225526 978987534704 30 165 0825 0570115071 111328582804 30 167 0835 0584019806 1384331624



04 30 157 0785163 0815 0541601867

05161714 469541704204 30 168 084 0591034996 912720797704 30 164 082 0563225526 399253776


the centroid

04 30 213 10652123333333 1061666667 053531424

052239302 24137636304 30 212 106 0517499435 332791537304 30 212 106 0517499435 3327915373

35


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate044 30 314 157

3156666667 1578333333 0484410240449549951 7196439229

044 30 316 158 045529494 6010463261044 30 317 1585 0458181114 5414651525



044 30 162 0811626666667 0813333333 0513003156

0499610983 2610544028044 30 162 081 0499610983 2610544028044 30 164 082 0512023205 0191022412



044 30 169 0845169 0845 0541601867

0543720061 0391098067044 30 167 0835 0530927096 1970962712044 30 171 0855 0556665324 2781278617


centroid

044 30 231 115523 115 053531424

0558559487 4342355492044 30 23 115 0553733942 344091388044 30 229 1145 0548929332 2543383077

36


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate05 30 343 1715

339 1695 0484410240472051706 2551253737

05 30 335 1675 0450288593 704395660505 30 339 1695 0461105951 4810857982



05 30 185 09251846666667 0923333333 051300315

6

0573360905 117655706105 30 187 0935 0585824908 141951859305 30 182 091 055491619 8170131796



05 30 186 0931820666667 0910333333 054160186

7

0579576154 701147639405 30 1792 0896 0537973187 06699902505 30 181 0905 0548834963 1335500582


centroid

05 30 238 1192396666667 1198333333 053531424

0521773549 252948448605 30 239 1195 0526167412 170868376705 30 242 121 0539459539 0774367463

37



-002-00100010020030040

005

01

015

02

025

03

035

04

045

log τ

log l



38

42 Discussion





39




I = md1 d2 gτ 2

4 π 2 L

uL=05480

=000104

ut=00051432

=000035

part Ipart L


(4 π2 L)2

part Ipart t


2π 2 L


uL)2

+( tIpart Ipart t

ut)2


40


41

Chapter 5

Conclusion







42

References


43

Theoretical



= 0048327651 (m4)

Experimental


τ = timecycles =

121320

=06065(s)

I = (m )d1 d2 τ

2 g

4π 2 L

iquest (1444+1793 ) (052 )2(02525)(02525)(981)

4 π 2(04)

= 0045155684 (m4 iquest


theoritical=65634


Theoretical

22


=


= 0104358901 (m4 iquest

Experimental


τ = timecycles =

150920 = 07545 (s)

I = (m )d1 d2 gτ 2

4π 2 L

= (503 ) (05735 )2 (02525 ) (02525 ) ( 981 )

4 π2 (04 )

= 0104606813 (m4 iquest




23

24


SystemCord

Length L (m)

No of Oscillations

(cyclesrsquo)


(sec)

Period τ

Average time (s)

Average Period



Only beam04 20 1785 08925

1782666667 0891333333 00457214970045196926 1147318216

04 20 1772 0886 0044540993 258194598404 20 1791 08955 0045501282 0481645472


04 20 1206 06031209333333 0604666667 0048327651

0044636018 763875978504 20 1213 06065 0045155684 656346126304 20 1209 06045 0044858364 7178679599


150 mm

04 20 1185 059251185666667 0592833333 0068498901

0064508686 582522521804 20 1178 0589 0063748809 693455239104 20 1194 0597 0065492285 4389290021


200 mm

04 20 1328 06641324666667 0662333333 0084187651

0081017277 376584191504 20 1332 0666 0081506068 318524499304 20 1314 0657 0079318085 5784180162


250 mm

04 20 1485 074251501 07505 0104358901

0101305821 362651766404 20 1509 07545 0104606813 029447598504 20 1509 07545 0104606813 0294475985


1634666667 0817333333 00487731420061356248 1494649851

04 20 1625 08125 0060608008 149464985104 20 1644 0822 0062033588 1494649851

25


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only beam045 20 1863 09315

1887666667 0943833333 00457214970043762855 4283854264

045 20 19 095 0045518416 04441707045 20 19 095 0045518416 04441707


045 20 13 0651286666667 0643333333 0048327651

0046102556 4604185043045 20 1282 0641 0044834709 7227626401045 20 1278 0639 0044555366 7805646014


150 mm

045 20 1268 06341264333333 0632166667 0068498901

006565495 4151819235045 20 1272 0636 0066069831 3546145034045 20 1253 06265 0064110788 6406107612


200 mm

045 20 1503 075151504 0752 0084187651

009224587 9571735346045 20 1506 0753 0092614485 10009584045 20 1503 07515 009224587 9571735346


250 mm

045 20 1575 078751580666667 0790333333 0104358901

0101295485 3638795325045 20 1585 07925 010258586 2106058254045 20 1582 0791 010219789 2566897967


1722333333 0861166667 00487731420061414329 1501548924

045 20 1716 0858 0060076597 1501548924045 20 1716 0858 0060076597 1501548924

26


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only beam048 20 1953 09765

1960333333 0980166667 00457214970045087453 1386753778

048 20 1956 0978 0045226077 1083561808048 20 1972 0986 0045968998 0541321699


048 20 1344 06721337666667 0668833333 0048327651

0046196398 4410006579048 20 1328 0664 0045103031 6672411469048 20 1341 06705 0045990395 4836271348


150 mm

048 20 1322 06611303333333 0651666667 0068498901

0066905706 2325869495048 20 1325 06625 0067209707 1882064854048 20 1263 06315 0061067051 1084959004


200 mm

048 20 1463 073151458666667 0729333333 0084187651

0081938668 2671393071048 20 1466 0733 0082275056 2271823432048 20 1447 07235 0080156235 4788607411


250 mm

048 20 1628 08141622 0811 0104358901

0101463315 3439442497048 20 1613 08065 0099602212 5650103475048 20 1625 08125 0101089716 3883211714


1751333333 0875666667 00487731420059651783 1292189689

048 20 1747 08735 0058375127 1292189689048 20 1741 08705 0057974842 1292189689

27


28

Trifilar Apparatus




Theoretical

I zz=25mr2

= 25

(4365 ) (0314 )2 = 048441024 m4

Experimental



L = C τ 2

29

C = L τ 2

τ = timecycles

= 29930

= 099667 (sec)

C= 04

(099667)2=iquest0388 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I p = (mp )R2 g

4 π2 C=

4365(0272)2(981)4 π 2(0417)

= 0457430617 (m4)



30


Theoretical





= 0513003156 m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

1653330 = 05511 (sec)

C= 04

(0569)2=iquest1317 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I+ I p = (m p+m )R2 g

4π 2C=0563225526m2

31





Theoretical

I=32mr2+ 1

2mro

2minus12mr i

2

=((32)43650272^2)+((12)138601^2)-((12)138600418^2)

= 0541601867m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

16330 = 054333 (sec)

32

C= 04

(05433)2=iquest13549 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )


4π 2C=0563225526m2




Experimental



L = C τ 2

C = L τ 2

τ = timecycles

= 2123

30= 0676 sec

C= 04

(0676)2=iquest0707667 m sec2

C = (m+m p )R2 g

4 π2(I + Ip )

33

I+ I p = (m p+m )R2 g

4π 2C=052239m4

Note



I=(( 32 )lowast4365lowast02722)+( 2

5 )lowast5656lowast0152=0535314 m4



34


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate04 30 298 149

2993333333 1496666667 0484410240445393497 805448343

04 30 298 149 0445393497 80544834304 30 302 151 0457430617 5569581401



04 30 164 0821653333333 0826666667 0513003156

0563225526 978987534704 30 165 0825 0570115071 111328582804 30 167 0835 0584019806 1384331624



04 30 157 0785163 0815 0541601867

05161714 469541704204 30 168 084 0591034996 912720797704 30 164 082 0563225526 399253776


the centroid

04 30 213 10652123333333 1061666667 053531424

052239302 24137636304 30 212 106 0517499435 332791537304 30 212 106 0517499435 3327915373

35


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate044 30 314 157

3156666667 1578333333 0484410240449549951 7196439229

044 30 316 158 045529494 6010463261044 30 317 1585 0458181114 5414651525



044 30 162 0811626666667 0813333333 0513003156

0499610983 2610544028044 30 162 081 0499610983 2610544028044 30 164 082 0512023205 0191022412



044 30 169 0845169 0845 0541601867

0543720061 0391098067044 30 167 0835 0530927096 1970962712044 30 171 0855 0556665324 2781278617


centroid

044 30 231 115523 115 053531424

0558559487 4342355492044 30 23 115 0553733942 344091388044 30 229 1145 0548929332 2543383077

36


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate05 30 343 1715

339 1695 0484410240472051706 2551253737

05 30 335 1675 0450288593 704395660505 30 339 1695 0461105951 4810857982



05 30 185 09251846666667 0923333333 051300315

6

0573360905 117655706105 30 187 0935 0585824908 141951859305 30 182 091 055491619 8170131796



05 30 186 0931820666667 0910333333 054160186

7

0579576154 701147639405 30 1792 0896 0537973187 06699902505 30 181 0905 0548834963 1335500582


centroid

05 30 238 1192396666667 1198333333 053531424

0521773549 252948448605 30 239 1195 0526167412 170868376705 30 242 121 0539459539 0774367463

37



-002-00100010020030040

005

01

015

02

025

03

035

04

045

log τ

log l



38

42 Discussion





39




I = md1 d2 gτ 2

4 π 2 L

uL=05480

=000104

ut=00051432

=000035

part Ipart L


(4 π2 L)2

part Ipart t


2π 2 L


uL)2

+( tIpart Ipart t

ut)2


40


41

Chapter 5

Conclusion







42

References


43


=


= 0104358901 (m4 iquest

Experimental


τ = timecycles =

150920 = 07545 (s)

I = (m )d1 d2 gτ 2

4π 2 L

= (503 ) (05735 )2 (02525 ) (02525 ) ( 981 )

4 π2 (04 )

= 0104606813 (m4 iquest




23

24


SystemCord

Length L (m)

No of Oscillations

(cyclesrsquo)


(sec)

Period τ

Average time (s)

Average Period



Only beam04 20 1785 08925

1782666667 0891333333 00457214970045196926 1147318216

04 20 1772 0886 0044540993 258194598404 20 1791 08955 0045501282 0481645472


04 20 1206 06031209333333 0604666667 0048327651

0044636018 763875978504 20 1213 06065 0045155684 656346126304 20 1209 06045 0044858364 7178679599


150 mm

04 20 1185 059251185666667 0592833333 0068498901

0064508686 582522521804 20 1178 0589 0063748809 693455239104 20 1194 0597 0065492285 4389290021


200 mm

04 20 1328 06641324666667 0662333333 0084187651

0081017277 376584191504 20 1332 0666 0081506068 318524499304 20 1314 0657 0079318085 5784180162


250 mm

04 20 1485 074251501 07505 0104358901

0101305821 362651766404 20 1509 07545 0104606813 029447598504 20 1509 07545 0104606813 0294475985


1634666667 0817333333 00487731420061356248 1494649851

04 20 1625 08125 0060608008 149464985104 20 1644 0822 0062033588 1494649851

25


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only beam045 20 1863 09315

1887666667 0943833333 00457214970043762855 4283854264

045 20 19 095 0045518416 04441707045 20 19 095 0045518416 04441707


045 20 13 0651286666667 0643333333 0048327651

0046102556 4604185043045 20 1282 0641 0044834709 7227626401045 20 1278 0639 0044555366 7805646014


150 mm

045 20 1268 06341264333333 0632166667 0068498901

006565495 4151819235045 20 1272 0636 0066069831 3546145034045 20 1253 06265 0064110788 6406107612


200 mm

045 20 1503 075151504 0752 0084187651

009224587 9571735346045 20 1506 0753 0092614485 10009584045 20 1503 07515 009224587 9571735346


250 mm

045 20 1575 078751580666667 0790333333 0104358901

0101295485 3638795325045 20 1585 07925 010258586 2106058254045 20 1582 0791 010219789 2566897967


1722333333 0861166667 00487731420061414329 1501548924

045 20 1716 0858 0060076597 1501548924045 20 1716 0858 0060076597 1501548924

26


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only beam048 20 1953 09765

1960333333 0980166667 00457214970045087453 1386753778

048 20 1956 0978 0045226077 1083561808048 20 1972 0986 0045968998 0541321699


048 20 1344 06721337666667 0668833333 0048327651

0046196398 4410006579048 20 1328 0664 0045103031 6672411469048 20 1341 06705 0045990395 4836271348


150 mm

048 20 1322 06611303333333 0651666667 0068498901

0066905706 2325869495048 20 1325 06625 0067209707 1882064854048 20 1263 06315 0061067051 1084959004


200 mm

048 20 1463 073151458666667 0729333333 0084187651

0081938668 2671393071048 20 1466 0733 0082275056 2271823432048 20 1447 07235 0080156235 4788607411


250 mm

048 20 1628 08141622 0811 0104358901

0101463315 3439442497048 20 1613 08065 0099602212 5650103475048 20 1625 08125 0101089716 3883211714


1751333333 0875666667 00487731420059651783 1292189689

048 20 1747 08735 0058375127 1292189689048 20 1741 08705 0057974842 1292189689

27


28

Trifilar Apparatus




Theoretical

I zz=25mr2

= 25

(4365 ) (0314 )2 = 048441024 m4

Experimental



L = C τ 2

29

C = L τ 2

τ = timecycles

= 29930

= 099667 (sec)

C= 04

(099667)2=iquest0388 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I p = (mp )R2 g

4 π2 C=

4365(0272)2(981)4 π 2(0417)

= 0457430617 (m4)



30


Theoretical





= 0513003156 m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

1653330 = 05511 (sec)

C= 04

(0569)2=iquest1317 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I+ I p = (m p+m )R2 g

4π 2C=0563225526m2

31





Theoretical

I=32mr2+ 1

2mro

2minus12mr i

2

=((32)43650272^2)+((12)138601^2)-((12)138600418^2)

= 0541601867m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

16330 = 054333 (sec)

32

C= 04

(05433)2=iquest13549 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )


4π 2C=0563225526m2




Experimental



L = C τ 2

C = L τ 2

τ = timecycles

= 2123

30= 0676 sec

C= 04

(0676)2=iquest0707667 m sec2

C = (m+m p )R2 g

4 π2(I + Ip )

33

I+ I p = (m p+m )R2 g

4π 2C=052239m4

Note



I=(( 32 )lowast4365lowast02722)+( 2

5 )lowast5656lowast0152=0535314 m4



34


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate04 30 298 149

2993333333 1496666667 0484410240445393497 805448343

04 30 298 149 0445393497 80544834304 30 302 151 0457430617 5569581401



04 30 164 0821653333333 0826666667 0513003156

0563225526 978987534704 30 165 0825 0570115071 111328582804 30 167 0835 0584019806 1384331624



04 30 157 0785163 0815 0541601867

05161714 469541704204 30 168 084 0591034996 912720797704 30 164 082 0563225526 399253776


the centroid

04 30 213 10652123333333 1061666667 053531424

052239302 24137636304 30 212 106 0517499435 332791537304 30 212 106 0517499435 3327915373

35


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate044 30 314 157

3156666667 1578333333 0484410240449549951 7196439229

044 30 316 158 045529494 6010463261044 30 317 1585 0458181114 5414651525



044 30 162 0811626666667 0813333333 0513003156

0499610983 2610544028044 30 162 081 0499610983 2610544028044 30 164 082 0512023205 0191022412



044 30 169 0845169 0845 0541601867

0543720061 0391098067044 30 167 0835 0530927096 1970962712044 30 171 0855 0556665324 2781278617


centroid

044 30 231 115523 115 053531424

0558559487 4342355492044 30 23 115 0553733942 344091388044 30 229 1145 0548929332 2543383077

36


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate05 30 343 1715

339 1695 0484410240472051706 2551253737

05 30 335 1675 0450288593 704395660505 30 339 1695 0461105951 4810857982



05 30 185 09251846666667 0923333333 051300315

6

0573360905 117655706105 30 187 0935 0585824908 141951859305 30 182 091 055491619 8170131796



05 30 186 0931820666667 0910333333 054160186

7

0579576154 701147639405 30 1792 0896 0537973187 06699902505 30 181 0905 0548834963 1335500582


centroid

05 30 238 1192396666667 1198333333 053531424

0521773549 252948448605 30 239 1195 0526167412 170868376705 30 242 121 0539459539 0774367463

37



-002-00100010020030040

005

01

015

02

025

03

035

04

045

log τ

log l



38

42 Discussion





39




I = md1 d2 gτ 2

4 π 2 L

uL=05480

=000104

ut=00051432

=000035

part Ipart L


(4 π2 L)2

part Ipart t


2π 2 L


uL)2

+( tIpart Ipart t

ut)2


40


41

Chapter 5

Conclusion







42

References


43

24


SystemCord

Length L (m)

No of Oscillations

(cyclesrsquo)


(sec)

Period τ

Average time (s)

Average Period



Only beam04 20 1785 08925

1782666667 0891333333 00457214970045196926 1147318216

04 20 1772 0886 0044540993 258194598404 20 1791 08955 0045501282 0481645472


04 20 1206 06031209333333 0604666667 0048327651

0044636018 763875978504 20 1213 06065 0045155684 656346126304 20 1209 06045 0044858364 7178679599


150 mm

04 20 1185 059251185666667 0592833333 0068498901

0064508686 582522521804 20 1178 0589 0063748809 693455239104 20 1194 0597 0065492285 4389290021


200 mm

04 20 1328 06641324666667 0662333333 0084187651

0081017277 376584191504 20 1332 0666 0081506068 318524499304 20 1314 0657 0079318085 5784180162


250 mm

04 20 1485 074251501 07505 0104358901

0101305821 362651766404 20 1509 07545 0104606813 029447598504 20 1509 07545 0104606813 0294475985


1634666667 0817333333 00487731420061356248 1494649851

04 20 1625 08125 0060608008 149464985104 20 1644 0822 0062033588 1494649851

25


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only beam045 20 1863 09315

1887666667 0943833333 00457214970043762855 4283854264

045 20 19 095 0045518416 04441707045 20 19 095 0045518416 04441707


045 20 13 0651286666667 0643333333 0048327651

0046102556 4604185043045 20 1282 0641 0044834709 7227626401045 20 1278 0639 0044555366 7805646014


150 mm

045 20 1268 06341264333333 0632166667 0068498901

006565495 4151819235045 20 1272 0636 0066069831 3546145034045 20 1253 06265 0064110788 6406107612


200 mm

045 20 1503 075151504 0752 0084187651

009224587 9571735346045 20 1506 0753 0092614485 10009584045 20 1503 07515 009224587 9571735346


250 mm

045 20 1575 078751580666667 0790333333 0104358901

0101295485 3638795325045 20 1585 07925 010258586 2106058254045 20 1582 0791 010219789 2566897967


1722333333 0861166667 00487731420061414329 1501548924

045 20 1716 0858 0060076597 1501548924045 20 1716 0858 0060076597 1501548924

26


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only beam048 20 1953 09765

1960333333 0980166667 00457214970045087453 1386753778

048 20 1956 0978 0045226077 1083561808048 20 1972 0986 0045968998 0541321699


048 20 1344 06721337666667 0668833333 0048327651

0046196398 4410006579048 20 1328 0664 0045103031 6672411469048 20 1341 06705 0045990395 4836271348


150 mm

048 20 1322 06611303333333 0651666667 0068498901

0066905706 2325869495048 20 1325 06625 0067209707 1882064854048 20 1263 06315 0061067051 1084959004


200 mm

048 20 1463 073151458666667 0729333333 0084187651

0081938668 2671393071048 20 1466 0733 0082275056 2271823432048 20 1447 07235 0080156235 4788607411


250 mm

048 20 1628 08141622 0811 0104358901

0101463315 3439442497048 20 1613 08065 0099602212 5650103475048 20 1625 08125 0101089716 3883211714


1751333333 0875666667 00487731420059651783 1292189689

048 20 1747 08735 0058375127 1292189689048 20 1741 08705 0057974842 1292189689

27


28

Trifilar Apparatus




Theoretical

I zz=25mr2

= 25

(4365 ) (0314 )2 = 048441024 m4

Experimental



L = C τ 2

29

C = L τ 2

τ = timecycles

= 29930

= 099667 (sec)

C= 04

(099667)2=iquest0388 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I p = (mp )R2 g

4 π2 C=

4365(0272)2(981)4 π 2(0417)

= 0457430617 (m4)



30


Theoretical





= 0513003156 m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

1653330 = 05511 (sec)

C= 04

(0569)2=iquest1317 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I+ I p = (m p+m )R2 g

4π 2C=0563225526m2

31





Theoretical

I=32mr2+ 1

2mro

2minus12mr i

2

=((32)43650272^2)+((12)138601^2)-((12)138600418^2)

= 0541601867m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

16330 = 054333 (sec)

32

C= 04

(05433)2=iquest13549 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )


4π 2C=0563225526m2




Experimental



L = C τ 2

C = L τ 2

τ = timecycles

= 2123

30= 0676 sec

C= 04

(0676)2=iquest0707667 m sec2

C = (m+m p )R2 g

4 π2(I + Ip )

33

I+ I p = (m p+m )R2 g

4π 2C=052239m4

Note



I=(( 32 )lowast4365lowast02722)+( 2

5 )lowast5656lowast0152=0535314 m4



34


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate04 30 298 149

2993333333 1496666667 0484410240445393497 805448343

04 30 298 149 0445393497 80544834304 30 302 151 0457430617 5569581401



04 30 164 0821653333333 0826666667 0513003156

0563225526 978987534704 30 165 0825 0570115071 111328582804 30 167 0835 0584019806 1384331624



04 30 157 0785163 0815 0541601867

05161714 469541704204 30 168 084 0591034996 912720797704 30 164 082 0563225526 399253776


the centroid

04 30 213 10652123333333 1061666667 053531424

052239302 24137636304 30 212 106 0517499435 332791537304 30 212 106 0517499435 3327915373

35


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate044 30 314 157

3156666667 1578333333 0484410240449549951 7196439229

044 30 316 158 045529494 6010463261044 30 317 1585 0458181114 5414651525



044 30 162 0811626666667 0813333333 0513003156

0499610983 2610544028044 30 162 081 0499610983 2610544028044 30 164 082 0512023205 0191022412



044 30 169 0845169 0845 0541601867

0543720061 0391098067044 30 167 0835 0530927096 1970962712044 30 171 0855 0556665324 2781278617


centroid

044 30 231 115523 115 053531424

0558559487 4342355492044 30 23 115 0553733942 344091388044 30 229 1145 0548929332 2543383077

36


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate05 30 343 1715

339 1695 0484410240472051706 2551253737

05 30 335 1675 0450288593 704395660505 30 339 1695 0461105951 4810857982



05 30 185 09251846666667 0923333333 051300315

6

0573360905 117655706105 30 187 0935 0585824908 141951859305 30 182 091 055491619 8170131796



05 30 186 0931820666667 0910333333 054160186

7

0579576154 701147639405 30 1792 0896 0537973187 06699902505 30 181 0905 0548834963 1335500582


centroid

05 30 238 1192396666667 1198333333 053531424

0521773549 252948448605 30 239 1195 0526167412 170868376705 30 242 121 0539459539 0774367463

37



-002-00100010020030040

005

01

015

02

025

03

035

04

045

log τ

log l



38

42 Discussion





39




I = md1 d2 gτ 2

4 π 2 L

uL=05480

=000104

ut=00051432

=000035

part Ipart L


(4 π2 L)2

part Ipart t


2π 2 L


uL)2

+( tIpart Ipart t

ut)2


40


41

Chapter 5

Conclusion







42

References


43


SystemCord

Length L (m)

No of Oscillations

(cyclesrsquo)


(sec)

Period τ

Average time (s)

Average Period



Only beam04 20 1785 08925

1782666667 0891333333 00457214970045196926 1147318216

04 20 1772 0886 0044540993 258194598404 20 1791 08955 0045501282 0481645472


04 20 1206 06031209333333 0604666667 0048327651

0044636018 763875978504 20 1213 06065 0045155684 656346126304 20 1209 06045 0044858364 7178679599


150 mm

04 20 1185 059251185666667 0592833333 0068498901

0064508686 582522521804 20 1178 0589 0063748809 693455239104 20 1194 0597 0065492285 4389290021


200 mm

04 20 1328 06641324666667 0662333333 0084187651

0081017277 376584191504 20 1332 0666 0081506068 318524499304 20 1314 0657 0079318085 5784180162


250 mm

04 20 1485 074251501 07505 0104358901

0101305821 362651766404 20 1509 07545 0104606813 029447598504 20 1509 07545 0104606813 0294475985


1634666667 0817333333 00487731420061356248 1494649851

04 20 1625 08125 0060608008 149464985104 20 1644 0822 0062033588 1494649851

25


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only beam045 20 1863 09315

1887666667 0943833333 00457214970043762855 4283854264

045 20 19 095 0045518416 04441707045 20 19 095 0045518416 04441707


045 20 13 0651286666667 0643333333 0048327651

0046102556 4604185043045 20 1282 0641 0044834709 7227626401045 20 1278 0639 0044555366 7805646014


150 mm

045 20 1268 06341264333333 0632166667 0068498901

006565495 4151819235045 20 1272 0636 0066069831 3546145034045 20 1253 06265 0064110788 6406107612


200 mm

045 20 1503 075151504 0752 0084187651

009224587 9571735346045 20 1506 0753 0092614485 10009584045 20 1503 07515 009224587 9571735346


250 mm

045 20 1575 078751580666667 0790333333 0104358901

0101295485 3638795325045 20 1585 07925 010258586 2106058254045 20 1582 0791 010219789 2566897967


1722333333 0861166667 00487731420061414329 1501548924

045 20 1716 0858 0060076597 1501548924045 20 1716 0858 0060076597 1501548924

26


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only beam048 20 1953 09765

1960333333 0980166667 00457214970045087453 1386753778

048 20 1956 0978 0045226077 1083561808048 20 1972 0986 0045968998 0541321699


048 20 1344 06721337666667 0668833333 0048327651

0046196398 4410006579048 20 1328 0664 0045103031 6672411469048 20 1341 06705 0045990395 4836271348


150 mm

048 20 1322 06611303333333 0651666667 0068498901

0066905706 2325869495048 20 1325 06625 0067209707 1882064854048 20 1263 06315 0061067051 1084959004


200 mm

048 20 1463 073151458666667 0729333333 0084187651

0081938668 2671393071048 20 1466 0733 0082275056 2271823432048 20 1447 07235 0080156235 4788607411


250 mm

048 20 1628 08141622 0811 0104358901

0101463315 3439442497048 20 1613 08065 0099602212 5650103475048 20 1625 08125 0101089716 3883211714


1751333333 0875666667 00487731420059651783 1292189689

048 20 1747 08735 0058375127 1292189689048 20 1741 08705 0057974842 1292189689

27


28

Trifilar Apparatus




Theoretical

I zz=25mr2

= 25

(4365 ) (0314 )2 = 048441024 m4

Experimental



L = C τ 2

29

C = L τ 2

τ = timecycles

= 29930

= 099667 (sec)

C= 04

(099667)2=iquest0388 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I p = (mp )R2 g

4 π2 C=

4365(0272)2(981)4 π 2(0417)

= 0457430617 (m4)



30


Theoretical





= 0513003156 m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

1653330 = 05511 (sec)

C= 04

(0569)2=iquest1317 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I+ I p = (m p+m )R2 g

4π 2C=0563225526m2

31





Theoretical

I=32mr2+ 1

2mro

2minus12mr i

2

=((32)43650272^2)+((12)138601^2)-((12)138600418^2)

= 0541601867m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

16330 = 054333 (sec)

32

C= 04

(05433)2=iquest13549 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )


4π 2C=0563225526m2




Experimental



L = C τ 2

C = L τ 2

τ = timecycles

= 2123

30= 0676 sec

C= 04

(0676)2=iquest0707667 m sec2

C = (m+m p )R2 g

4 π2(I + Ip )

33

I+ I p = (m p+m )R2 g

4π 2C=052239m4

Note



I=(( 32 )lowast4365lowast02722)+( 2

5 )lowast5656lowast0152=0535314 m4



34


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate04 30 298 149

2993333333 1496666667 0484410240445393497 805448343

04 30 298 149 0445393497 80544834304 30 302 151 0457430617 5569581401



04 30 164 0821653333333 0826666667 0513003156

0563225526 978987534704 30 165 0825 0570115071 111328582804 30 167 0835 0584019806 1384331624



04 30 157 0785163 0815 0541601867

05161714 469541704204 30 168 084 0591034996 912720797704 30 164 082 0563225526 399253776


the centroid

04 30 213 10652123333333 1061666667 053531424

052239302 24137636304 30 212 106 0517499435 332791537304 30 212 106 0517499435 3327915373

35


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate044 30 314 157

3156666667 1578333333 0484410240449549951 7196439229

044 30 316 158 045529494 6010463261044 30 317 1585 0458181114 5414651525



044 30 162 0811626666667 0813333333 0513003156

0499610983 2610544028044 30 162 081 0499610983 2610544028044 30 164 082 0512023205 0191022412



044 30 169 0845169 0845 0541601867

0543720061 0391098067044 30 167 0835 0530927096 1970962712044 30 171 0855 0556665324 2781278617


centroid

044 30 231 115523 115 053531424

0558559487 4342355492044 30 23 115 0553733942 344091388044 30 229 1145 0548929332 2543383077

36


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate05 30 343 1715

339 1695 0484410240472051706 2551253737

05 30 335 1675 0450288593 704395660505 30 339 1695 0461105951 4810857982



05 30 185 09251846666667 0923333333 051300315

6

0573360905 117655706105 30 187 0935 0585824908 141951859305 30 182 091 055491619 8170131796



05 30 186 0931820666667 0910333333 054160186

7

0579576154 701147639405 30 1792 0896 0537973187 06699902505 30 181 0905 0548834963 1335500582


centroid

05 30 238 1192396666667 1198333333 053531424

0521773549 252948448605 30 239 1195 0526167412 170868376705 30 242 121 0539459539 0774367463

37



-002-00100010020030040

005

01

015

02

025

03

035

04

045

log τ

log l



38

42 Discussion





39




I = md1 d2 gτ 2

4 π 2 L

uL=05480

=000104

ut=00051432

=000035

part Ipart L


(4 π2 L)2

part Ipart t


2π 2 L


uL)2

+( tIpart Ipart t

ut)2


40


41

Chapter 5

Conclusion







42

References


43


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only beam045 20 1863 09315

1887666667 0943833333 00457214970043762855 4283854264

045 20 19 095 0045518416 04441707045 20 19 095 0045518416 04441707


045 20 13 0651286666667 0643333333 0048327651

0046102556 4604185043045 20 1282 0641 0044834709 7227626401045 20 1278 0639 0044555366 7805646014


150 mm

045 20 1268 06341264333333 0632166667 0068498901

006565495 4151819235045 20 1272 0636 0066069831 3546145034045 20 1253 06265 0064110788 6406107612


200 mm

045 20 1503 075151504 0752 0084187651

009224587 9571735346045 20 1506 0753 0092614485 10009584045 20 1503 07515 009224587 9571735346


250 mm

045 20 1575 078751580666667 0790333333 0104358901

0101295485 3638795325045 20 1585 07925 010258586 2106058254045 20 1582 0791 010219789 2566897967


1722333333 0861166667 00487731420061414329 1501548924

045 20 1716 0858 0060076597 1501548924045 20 1716 0858 0060076597 1501548924

26


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only beam048 20 1953 09765

1960333333 0980166667 00457214970045087453 1386753778

048 20 1956 0978 0045226077 1083561808048 20 1972 0986 0045968998 0541321699


048 20 1344 06721337666667 0668833333 0048327651

0046196398 4410006579048 20 1328 0664 0045103031 6672411469048 20 1341 06705 0045990395 4836271348


150 mm

048 20 1322 06611303333333 0651666667 0068498901

0066905706 2325869495048 20 1325 06625 0067209707 1882064854048 20 1263 06315 0061067051 1084959004


200 mm

048 20 1463 073151458666667 0729333333 0084187651

0081938668 2671393071048 20 1466 0733 0082275056 2271823432048 20 1447 07235 0080156235 4788607411


250 mm

048 20 1628 08141622 0811 0104358901

0101463315 3439442497048 20 1613 08065 0099602212 5650103475048 20 1625 08125 0101089716 3883211714


1751333333 0875666667 00487731420059651783 1292189689

048 20 1747 08735 0058375127 1292189689048 20 1741 08705 0057974842 1292189689

27


28

Trifilar Apparatus




Theoretical

I zz=25mr2

= 25

(4365 ) (0314 )2 = 048441024 m4

Experimental



L = C τ 2

29

C = L τ 2

τ = timecycles

= 29930

= 099667 (sec)

C= 04

(099667)2=iquest0388 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I p = (mp )R2 g

4 π2 C=

4365(0272)2(981)4 π 2(0417)

= 0457430617 (m4)



30


Theoretical





= 0513003156 m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

1653330 = 05511 (sec)

C= 04

(0569)2=iquest1317 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I+ I p = (m p+m )R2 g

4π 2C=0563225526m2

31





Theoretical

I=32mr2+ 1

2mro

2minus12mr i

2

=((32)43650272^2)+((12)138601^2)-((12)138600418^2)

= 0541601867m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

16330 = 054333 (sec)

32

C= 04

(05433)2=iquest13549 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )


4π 2C=0563225526m2




Experimental



L = C τ 2

C = L τ 2

τ = timecycles

= 2123

30= 0676 sec

C= 04

(0676)2=iquest0707667 m sec2

C = (m+m p )R2 g

4 π2(I + Ip )

33

I+ I p = (m p+m )R2 g

4π 2C=052239m4

Note



I=(( 32 )lowast4365lowast02722)+( 2

5 )lowast5656lowast0152=0535314 m4



34


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate04 30 298 149

2993333333 1496666667 0484410240445393497 805448343

04 30 298 149 0445393497 80544834304 30 302 151 0457430617 5569581401



04 30 164 0821653333333 0826666667 0513003156

0563225526 978987534704 30 165 0825 0570115071 111328582804 30 167 0835 0584019806 1384331624



04 30 157 0785163 0815 0541601867

05161714 469541704204 30 168 084 0591034996 912720797704 30 164 082 0563225526 399253776


the centroid

04 30 213 10652123333333 1061666667 053531424

052239302 24137636304 30 212 106 0517499435 332791537304 30 212 106 0517499435 3327915373

35


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate044 30 314 157

3156666667 1578333333 0484410240449549951 7196439229

044 30 316 158 045529494 6010463261044 30 317 1585 0458181114 5414651525



044 30 162 0811626666667 0813333333 0513003156

0499610983 2610544028044 30 162 081 0499610983 2610544028044 30 164 082 0512023205 0191022412



044 30 169 0845169 0845 0541601867

0543720061 0391098067044 30 167 0835 0530927096 1970962712044 30 171 0855 0556665324 2781278617


centroid

044 30 231 115523 115 053531424

0558559487 4342355492044 30 23 115 0553733942 344091388044 30 229 1145 0548929332 2543383077

36


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate05 30 343 1715

339 1695 0484410240472051706 2551253737

05 30 335 1675 0450288593 704395660505 30 339 1695 0461105951 4810857982



05 30 185 09251846666667 0923333333 051300315

6

0573360905 117655706105 30 187 0935 0585824908 141951859305 30 182 091 055491619 8170131796



05 30 186 0931820666667 0910333333 054160186

7

0579576154 701147639405 30 1792 0896 0537973187 06699902505 30 181 0905 0548834963 1335500582


centroid

05 30 238 1192396666667 1198333333 053531424

0521773549 252948448605 30 239 1195 0526167412 170868376705 30 242 121 0539459539 0774367463

37



-002-00100010020030040

005

01

015

02

025

03

035

04

045

log τ

log l



38

42 Discussion





39




I = md1 d2 gτ 2

4 π 2 L

uL=05480

=000104

ut=00051432

=000035

part Ipart L


(4 π2 L)2

part Ipart t


2π 2 L


uL)2

+( tIpart Ipart t

ut)2


40


41

Chapter 5

Conclusion







42

References


43


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only beam048 20 1953 09765

1960333333 0980166667 00457214970045087453 1386753778

048 20 1956 0978 0045226077 1083561808048 20 1972 0986 0045968998 0541321699


048 20 1344 06721337666667 0668833333 0048327651

0046196398 4410006579048 20 1328 0664 0045103031 6672411469048 20 1341 06705 0045990395 4836271348


150 mm

048 20 1322 06611303333333 0651666667 0068498901

0066905706 2325869495048 20 1325 06625 0067209707 1882064854048 20 1263 06315 0061067051 1084959004


200 mm

048 20 1463 073151458666667 0729333333 0084187651

0081938668 2671393071048 20 1466 0733 0082275056 2271823432048 20 1447 07235 0080156235 4788607411


250 mm

048 20 1628 08141622 0811 0104358901

0101463315 3439442497048 20 1613 08065 0099602212 5650103475048 20 1625 08125 0101089716 3883211714


1751333333 0875666667 00487731420059651783 1292189689

048 20 1747 08735 0058375127 1292189689048 20 1741 08705 0057974842 1292189689

27


28

Trifilar Apparatus




Theoretical

I zz=25mr2

= 25

(4365 ) (0314 )2 = 048441024 m4

Experimental



L = C τ 2

29

C = L τ 2

τ = timecycles

= 29930

= 099667 (sec)

C= 04

(099667)2=iquest0388 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I p = (mp )R2 g

4 π2 C=

4365(0272)2(981)4 π 2(0417)

= 0457430617 (m4)



30


Theoretical





= 0513003156 m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

1653330 = 05511 (sec)

C= 04

(0569)2=iquest1317 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I+ I p = (m p+m )R2 g

4π 2C=0563225526m2

31





Theoretical

I=32mr2+ 1

2mro

2minus12mr i

2

=((32)43650272^2)+((12)138601^2)-((12)138600418^2)

= 0541601867m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

16330 = 054333 (sec)

32

C= 04

(05433)2=iquest13549 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )


4π 2C=0563225526m2




Experimental



L = C τ 2

C = L τ 2

τ = timecycles

= 2123

30= 0676 sec

C= 04

(0676)2=iquest0707667 m sec2

C = (m+m p )R2 g

4 π2(I + Ip )

33

I+ I p = (m p+m )R2 g

4π 2C=052239m4

Note



I=(( 32 )lowast4365lowast02722)+( 2

5 )lowast5656lowast0152=0535314 m4



34


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate04 30 298 149

2993333333 1496666667 0484410240445393497 805448343

04 30 298 149 0445393497 80544834304 30 302 151 0457430617 5569581401



04 30 164 0821653333333 0826666667 0513003156

0563225526 978987534704 30 165 0825 0570115071 111328582804 30 167 0835 0584019806 1384331624



04 30 157 0785163 0815 0541601867

05161714 469541704204 30 168 084 0591034996 912720797704 30 164 082 0563225526 399253776


the centroid

04 30 213 10652123333333 1061666667 053531424

052239302 24137636304 30 212 106 0517499435 332791537304 30 212 106 0517499435 3327915373

35


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate044 30 314 157

3156666667 1578333333 0484410240449549951 7196439229

044 30 316 158 045529494 6010463261044 30 317 1585 0458181114 5414651525



044 30 162 0811626666667 0813333333 0513003156

0499610983 2610544028044 30 162 081 0499610983 2610544028044 30 164 082 0512023205 0191022412



044 30 169 0845169 0845 0541601867

0543720061 0391098067044 30 167 0835 0530927096 1970962712044 30 171 0855 0556665324 2781278617


centroid

044 30 231 115523 115 053531424

0558559487 4342355492044 30 23 115 0553733942 344091388044 30 229 1145 0548929332 2543383077

36


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate05 30 343 1715

339 1695 0484410240472051706 2551253737

05 30 335 1675 0450288593 704395660505 30 339 1695 0461105951 4810857982



05 30 185 09251846666667 0923333333 051300315

6

0573360905 117655706105 30 187 0935 0585824908 141951859305 30 182 091 055491619 8170131796



05 30 186 0931820666667 0910333333 054160186

7

0579576154 701147639405 30 1792 0896 0537973187 06699902505 30 181 0905 0548834963 1335500582


centroid

05 30 238 1192396666667 1198333333 053531424

0521773549 252948448605 30 239 1195 0526167412 170868376705 30 242 121 0539459539 0774367463

37



-002-00100010020030040

005

01

015

02

025

03

035

04

045

log τ

log l



38

42 Discussion





39




I = md1 d2 gτ 2

4 π 2 L

uL=05480

=000104

ut=00051432

=000035

part Ipart L


(4 π2 L)2

part Ipart t


2π 2 L


uL)2

+( tIpart Ipart t

ut)2


40


41

Chapter 5

Conclusion







42

References


43


28

Trifilar Apparatus




Theoretical

I zz=25mr2

= 25

(4365 ) (0314 )2 = 048441024 m4

Experimental



L = C τ 2

29

C = L τ 2

τ = timecycles

= 29930

= 099667 (sec)

C= 04

(099667)2=iquest0388 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I p = (mp )R2 g

4 π2 C=

4365(0272)2(981)4 π 2(0417)

= 0457430617 (m4)



30


Theoretical





= 0513003156 m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

1653330 = 05511 (sec)

C= 04

(0569)2=iquest1317 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I+ I p = (m p+m )R2 g

4π 2C=0563225526m2

31





Theoretical

I=32mr2+ 1

2mro

2minus12mr i

2

=((32)43650272^2)+((12)138601^2)-((12)138600418^2)

= 0541601867m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

16330 = 054333 (sec)

32

C= 04

(05433)2=iquest13549 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )


4π 2C=0563225526m2




Experimental



L = C τ 2

C = L τ 2

τ = timecycles

= 2123

30= 0676 sec

C= 04

(0676)2=iquest0707667 m sec2

C = (m+m p )R2 g

4 π2(I + Ip )

33

I+ I p = (m p+m )R2 g

4π 2C=052239m4

Note



I=(( 32 )lowast4365lowast02722)+( 2

5 )lowast5656lowast0152=0535314 m4



34


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate04 30 298 149

2993333333 1496666667 0484410240445393497 805448343

04 30 298 149 0445393497 80544834304 30 302 151 0457430617 5569581401



04 30 164 0821653333333 0826666667 0513003156

0563225526 978987534704 30 165 0825 0570115071 111328582804 30 167 0835 0584019806 1384331624



04 30 157 0785163 0815 0541601867

05161714 469541704204 30 168 084 0591034996 912720797704 30 164 082 0563225526 399253776


the centroid

04 30 213 10652123333333 1061666667 053531424

052239302 24137636304 30 212 106 0517499435 332791537304 30 212 106 0517499435 3327915373

35


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate044 30 314 157

3156666667 1578333333 0484410240449549951 7196439229

044 30 316 158 045529494 6010463261044 30 317 1585 0458181114 5414651525



044 30 162 0811626666667 0813333333 0513003156

0499610983 2610544028044 30 162 081 0499610983 2610544028044 30 164 082 0512023205 0191022412



044 30 169 0845169 0845 0541601867

0543720061 0391098067044 30 167 0835 0530927096 1970962712044 30 171 0855 0556665324 2781278617


centroid

044 30 231 115523 115 053531424

0558559487 4342355492044 30 23 115 0553733942 344091388044 30 229 1145 0548929332 2543383077

36


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate05 30 343 1715

339 1695 0484410240472051706 2551253737

05 30 335 1675 0450288593 704395660505 30 339 1695 0461105951 4810857982



05 30 185 09251846666667 0923333333 051300315

6

0573360905 117655706105 30 187 0935 0585824908 141951859305 30 182 091 055491619 8170131796



05 30 186 0931820666667 0910333333 054160186

7

0579576154 701147639405 30 1792 0896 0537973187 06699902505 30 181 0905 0548834963 1335500582


centroid

05 30 238 1192396666667 1198333333 053531424

0521773549 252948448605 30 239 1195 0526167412 170868376705 30 242 121 0539459539 0774367463

37



-002-00100010020030040

005

01

015

02

025

03

035

04

045

log τ

log l



38

42 Discussion





39




I = md1 d2 gτ 2

4 π 2 L

uL=05480

=000104

ut=00051432

=000035

part Ipart L


(4 π2 L)2

part Ipart t


2π 2 L


uL)2

+( tIpart Ipart t

ut)2


40


41

Chapter 5

Conclusion







42

References


43

Trifilar Apparatus




Theoretical

I zz=25mr2

= 25

(4365 ) (0314 )2 = 048441024 m4

Experimental



L = C τ 2

29

C = L τ 2

τ = timecycles

= 29930

= 099667 (sec)

C= 04

(099667)2=iquest0388 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I p = (mp )R2 g

4 π2 C=

4365(0272)2(981)4 π 2(0417)

= 0457430617 (m4)



30


Theoretical





= 0513003156 m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

1653330 = 05511 (sec)

C= 04

(0569)2=iquest1317 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I+ I p = (m p+m )R2 g

4π 2C=0563225526m2

31





Theoretical

I=32mr2+ 1

2mro

2minus12mr i

2

=((32)43650272^2)+((12)138601^2)-((12)138600418^2)

= 0541601867m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

16330 = 054333 (sec)

32

C= 04

(05433)2=iquest13549 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )


4π 2C=0563225526m2




Experimental



L = C τ 2

C = L τ 2

τ = timecycles

= 2123

30= 0676 sec

C= 04

(0676)2=iquest0707667 m sec2

C = (m+m p )R2 g

4 π2(I + Ip )

33

I+ I p = (m p+m )R2 g

4π 2C=052239m4

Note



I=(( 32 )lowast4365lowast02722)+( 2

5 )lowast5656lowast0152=0535314 m4



34


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate04 30 298 149

2993333333 1496666667 0484410240445393497 805448343

04 30 298 149 0445393497 80544834304 30 302 151 0457430617 5569581401



04 30 164 0821653333333 0826666667 0513003156

0563225526 978987534704 30 165 0825 0570115071 111328582804 30 167 0835 0584019806 1384331624



04 30 157 0785163 0815 0541601867

05161714 469541704204 30 168 084 0591034996 912720797704 30 164 082 0563225526 399253776


the centroid

04 30 213 10652123333333 1061666667 053531424

052239302 24137636304 30 212 106 0517499435 332791537304 30 212 106 0517499435 3327915373

35


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate044 30 314 157

3156666667 1578333333 0484410240449549951 7196439229

044 30 316 158 045529494 6010463261044 30 317 1585 0458181114 5414651525



044 30 162 0811626666667 0813333333 0513003156

0499610983 2610544028044 30 162 081 0499610983 2610544028044 30 164 082 0512023205 0191022412



044 30 169 0845169 0845 0541601867

0543720061 0391098067044 30 167 0835 0530927096 1970962712044 30 171 0855 0556665324 2781278617


centroid

044 30 231 115523 115 053531424

0558559487 4342355492044 30 23 115 0553733942 344091388044 30 229 1145 0548929332 2543383077

36


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate05 30 343 1715

339 1695 0484410240472051706 2551253737

05 30 335 1675 0450288593 704395660505 30 339 1695 0461105951 4810857982



05 30 185 09251846666667 0923333333 051300315

6

0573360905 117655706105 30 187 0935 0585824908 141951859305 30 182 091 055491619 8170131796



05 30 186 0931820666667 0910333333 054160186

7

0579576154 701147639405 30 1792 0896 0537973187 06699902505 30 181 0905 0548834963 1335500582


centroid

05 30 238 1192396666667 1198333333 053531424

0521773549 252948448605 30 239 1195 0526167412 170868376705 30 242 121 0539459539 0774367463

37



-002-00100010020030040

005

01

015

02

025

03

035

04

045

log τ

log l



38

42 Discussion





39




I = md1 d2 gτ 2

4 π 2 L

uL=05480

=000104

ut=00051432

=000035

part Ipart L


(4 π2 L)2

part Ipart t


2π 2 L


uL)2

+( tIpart Ipart t

ut)2


40


41

Chapter 5

Conclusion







42

References


43

C = L τ 2

τ = timecycles

= 29930

= 099667 (sec)

C= 04

(099667)2=iquest0388 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I p = (mp )R2 g

4 π2 C=

4365(0272)2(981)4 π 2(0417)

= 0457430617 (m4)



30


Theoretical





= 0513003156 m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

1653330 = 05511 (sec)

C= 04

(0569)2=iquest1317 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I+ I p = (m p+m )R2 g

4π 2C=0563225526m2

31





Theoretical

I=32mr2+ 1

2mro

2minus12mr i

2

=((32)43650272^2)+((12)138601^2)-((12)138600418^2)

= 0541601867m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

16330 = 054333 (sec)

32

C= 04

(05433)2=iquest13549 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )


4π 2C=0563225526m2




Experimental



L = C τ 2

C = L τ 2

τ = timecycles

= 2123

30= 0676 sec

C= 04

(0676)2=iquest0707667 m sec2

C = (m+m p )R2 g

4 π2(I + Ip )

33

I+ I p = (m p+m )R2 g

4π 2C=052239m4

Note



I=(( 32 )lowast4365lowast02722)+( 2

5 )lowast5656lowast0152=0535314 m4



34


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate04 30 298 149

2993333333 1496666667 0484410240445393497 805448343

04 30 298 149 0445393497 80544834304 30 302 151 0457430617 5569581401



04 30 164 0821653333333 0826666667 0513003156

0563225526 978987534704 30 165 0825 0570115071 111328582804 30 167 0835 0584019806 1384331624



04 30 157 0785163 0815 0541601867

05161714 469541704204 30 168 084 0591034996 912720797704 30 164 082 0563225526 399253776


the centroid

04 30 213 10652123333333 1061666667 053531424

052239302 24137636304 30 212 106 0517499435 332791537304 30 212 106 0517499435 3327915373

35


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate044 30 314 157

3156666667 1578333333 0484410240449549951 7196439229

044 30 316 158 045529494 6010463261044 30 317 1585 0458181114 5414651525



044 30 162 0811626666667 0813333333 0513003156

0499610983 2610544028044 30 162 081 0499610983 2610544028044 30 164 082 0512023205 0191022412



044 30 169 0845169 0845 0541601867

0543720061 0391098067044 30 167 0835 0530927096 1970962712044 30 171 0855 0556665324 2781278617


centroid

044 30 231 115523 115 053531424

0558559487 4342355492044 30 23 115 0553733942 344091388044 30 229 1145 0548929332 2543383077

36


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate05 30 343 1715

339 1695 0484410240472051706 2551253737

05 30 335 1675 0450288593 704395660505 30 339 1695 0461105951 4810857982



05 30 185 09251846666667 0923333333 051300315

6

0573360905 117655706105 30 187 0935 0585824908 141951859305 30 182 091 055491619 8170131796



05 30 186 0931820666667 0910333333 054160186

7

0579576154 701147639405 30 1792 0896 0537973187 06699902505 30 181 0905 0548834963 1335500582


centroid

05 30 238 1192396666667 1198333333 053531424

0521773549 252948448605 30 239 1195 0526167412 170868376705 30 242 121 0539459539 0774367463

37



-002-00100010020030040

005

01

015

02

025

03

035

04

045

log τ

log l



38

42 Discussion





39




I = md1 d2 gτ 2

4 π 2 L

uL=05480

=000104

ut=00051432

=000035

part Ipart L


(4 π2 L)2

part Ipart t


2π 2 L


uL)2

+( tIpart Ipart t

ut)2


40


41

Chapter 5

Conclusion







42

References


43


Theoretical





= 0513003156 m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

1653330 = 05511 (sec)

C= 04

(0569)2=iquest1317 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )

I+ I p = (m p+m )R2 g

4π 2C=0563225526m2

31





Theoretical

I=32mr2+ 1

2mro

2minus12mr i

2

=((32)43650272^2)+((12)138601^2)-((12)138600418^2)

= 0541601867m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

16330 = 054333 (sec)

32

C= 04

(05433)2=iquest13549 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )


4π 2C=0563225526m2




Experimental



L = C τ 2

C = L τ 2

τ = timecycles

= 2123

30= 0676 sec

C= 04

(0676)2=iquest0707667 m sec2

C = (m+m p )R2 g

4 π2(I + Ip )

33

I+ I p = (m p+m )R2 g

4π 2C=052239m4

Note



I=(( 32 )lowast4365lowast02722)+( 2

5 )lowast5656lowast0152=0535314 m4



34


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate04 30 298 149

2993333333 1496666667 0484410240445393497 805448343

04 30 298 149 0445393497 80544834304 30 302 151 0457430617 5569581401



04 30 164 0821653333333 0826666667 0513003156

0563225526 978987534704 30 165 0825 0570115071 111328582804 30 167 0835 0584019806 1384331624



04 30 157 0785163 0815 0541601867

05161714 469541704204 30 168 084 0591034996 912720797704 30 164 082 0563225526 399253776


the centroid

04 30 213 10652123333333 1061666667 053531424

052239302 24137636304 30 212 106 0517499435 332791537304 30 212 106 0517499435 3327915373

35


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate044 30 314 157

3156666667 1578333333 0484410240449549951 7196439229

044 30 316 158 045529494 6010463261044 30 317 1585 0458181114 5414651525



044 30 162 0811626666667 0813333333 0513003156

0499610983 2610544028044 30 162 081 0499610983 2610544028044 30 164 082 0512023205 0191022412



044 30 169 0845169 0845 0541601867

0543720061 0391098067044 30 167 0835 0530927096 1970962712044 30 171 0855 0556665324 2781278617


centroid

044 30 231 115523 115 053531424

0558559487 4342355492044 30 23 115 0553733942 344091388044 30 229 1145 0548929332 2543383077

36


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate05 30 343 1715

339 1695 0484410240472051706 2551253737

05 30 335 1675 0450288593 704395660505 30 339 1695 0461105951 4810857982



05 30 185 09251846666667 0923333333 051300315

6

0573360905 117655706105 30 187 0935 0585824908 141951859305 30 182 091 055491619 8170131796



05 30 186 0931820666667 0910333333 054160186

7

0579576154 701147639405 30 1792 0896 0537973187 06699902505 30 181 0905 0548834963 1335500582


centroid

05 30 238 1192396666667 1198333333 053531424

0521773549 252948448605 30 239 1195 0526167412 170868376705 30 242 121 0539459539 0774367463

37



-002-00100010020030040

005

01

015

02

025

03

035

04

045

log τ

log l



38

42 Discussion





39




I = md1 d2 gτ 2

4 π 2 L

uL=05480

=000104

ut=00051432

=000035

part Ipart L


(4 π2 L)2

part Ipart t


2π 2 L


uL)2

+( tIpart Ipart t

ut)2


40


41

Chapter 5

Conclusion







42

References


43





Theoretical

I=32mr2+ 1

2mro

2minus12mr i

2

=((32)43650272^2)+((12)138601^2)-((12)138600418^2)

= 0541601867m4

Experimental



L = C τ 2

C = L τ 2

τ = timecycles =

16330 = 054333 (sec)

32

C= 04

(05433)2=iquest13549 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )


4π 2C=0563225526m2




Experimental



L = C τ 2

C = L τ 2

τ = timecycles

= 2123

30= 0676 sec

C= 04

(0676)2=iquest0707667 m sec2

C = (m+m p )R2 g

4 π2(I + Ip )

33

I+ I p = (m p+m )R2 g

4π 2C=052239m4

Note



I=(( 32 )lowast4365lowast02722)+( 2

5 )lowast5656lowast0152=0535314 m4



34


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate04 30 298 149

2993333333 1496666667 0484410240445393497 805448343

04 30 298 149 0445393497 80544834304 30 302 151 0457430617 5569581401



04 30 164 0821653333333 0826666667 0513003156

0563225526 978987534704 30 165 0825 0570115071 111328582804 30 167 0835 0584019806 1384331624



04 30 157 0785163 0815 0541601867

05161714 469541704204 30 168 084 0591034996 912720797704 30 164 082 0563225526 399253776


the centroid

04 30 213 10652123333333 1061666667 053531424

052239302 24137636304 30 212 106 0517499435 332791537304 30 212 106 0517499435 3327915373

35


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate044 30 314 157

3156666667 1578333333 0484410240449549951 7196439229

044 30 316 158 045529494 6010463261044 30 317 1585 0458181114 5414651525



044 30 162 0811626666667 0813333333 0513003156

0499610983 2610544028044 30 162 081 0499610983 2610544028044 30 164 082 0512023205 0191022412



044 30 169 0845169 0845 0541601867

0543720061 0391098067044 30 167 0835 0530927096 1970962712044 30 171 0855 0556665324 2781278617


centroid

044 30 231 115523 115 053531424

0558559487 4342355492044 30 23 115 0553733942 344091388044 30 229 1145 0548929332 2543383077

36


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate05 30 343 1715

339 1695 0484410240472051706 2551253737

05 30 335 1675 0450288593 704395660505 30 339 1695 0461105951 4810857982



05 30 185 09251846666667 0923333333 051300315

6

0573360905 117655706105 30 187 0935 0585824908 141951859305 30 182 091 055491619 8170131796



05 30 186 0931820666667 0910333333 054160186

7

0579576154 701147639405 30 1792 0896 0537973187 06699902505 30 181 0905 0548834963 1335500582


centroid

05 30 238 1192396666667 1198333333 053531424

0521773549 252948448605 30 239 1195 0526167412 170868376705 30 242 121 0539459539 0774367463

37



-002-00100010020030040

005

01

015

02

025

03

035

04

045

log τ

log l



38

42 Discussion





39




I = md1 d2 gτ 2

4 π 2 L

uL=05480

=000104

ut=00051432

=000035

part Ipart L


(4 π2 L)2

part Ipart t


2π 2 L


uL)2

+( tIpart Ipart t

ut)2


40


41

Chapter 5

Conclusion







42

References


43

C= 04

(05433)2=iquest13549 (m sec2)

C = (m+m p )R2 g

4 π2(I + Ip )


4π 2C=0563225526m2




Experimental



L = C τ 2

C = L τ 2

τ = timecycles

= 2123

30= 0676 sec

C= 04

(0676)2=iquest0707667 m sec2

C = (m+m p )R2 g

4 π2(I + Ip )

33

I+ I p = (m p+m )R2 g

4π 2C=052239m4

Note



I=(( 32 )lowast4365lowast02722)+( 2

5 )lowast5656lowast0152=0535314 m4



34


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate04 30 298 149

2993333333 1496666667 0484410240445393497 805448343

04 30 298 149 0445393497 80544834304 30 302 151 0457430617 5569581401



04 30 164 0821653333333 0826666667 0513003156

0563225526 978987534704 30 165 0825 0570115071 111328582804 30 167 0835 0584019806 1384331624



04 30 157 0785163 0815 0541601867

05161714 469541704204 30 168 084 0591034996 912720797704 30 164 082 0563225526 399253776


the centroid

04 30 213 10652123333333 1061666667 053531424

052239302 24137636304 30 212 106 0517499435 332791537304 30 212 106 0517499435 3327915373

35


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate044 30 314 157

3156666667 1578333333 0484410240449549951 7196439229

044 30 316 158 045529494 6010463261044 30 317 1585 0458181114 5414651525



044 30 162 0811626666667 0813333333 0513003156

0499610983 2610544028044 30 162 081 0499610983 2610544028044 30 164 082 0512023205 0191022412



044 30 169 0845169 0845 0541601867

0543720061 0391098067044 30 167 0835 0530927096 1970962712044 30 171 0855 0556665324 2781278617


centroid

044 30 231 115523 115 053531424

0558559487 4342355492044 30 23 115 0553733942 344091388044 30 229 1145 0548929332 2543383077

36


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate05 30 343 1715

339 1695 0484410240472051706 2551253737

05 30 335 1675 0450288593 704395660505 30 339 1695 0461105951 4810857982



05 30 185 09251846666667 0923333333 051300315

6

0573360905 117655706105 30 187 0935 0585824908 141951859305 30 182 091 055491619 8170131796



05 30 186 0931820666667 0910333333 054160186

7

0579576154 701147639405 30 1792 0896 0537973187 06699902505 30 181 0905 0548834963 1335500582


centroid

05 30 238 1192396666667 1198333333 053531424

0521773549 252948448605 30 239 1195 0526167412 170868376705 30 242 121 0539459539 0774367463

37



-002-00100010020030040

005

01

015

02

025

03

035

04

045

log τ

log l



38

42 Discussion





39




I = md1 d2 gτ 2

4 π 2 L

uL=05480

=000104

ut=00051432

=000035

part Ipart L


(4 π2 L)2

part Ipart t


2π 2 L


uL)2

+( tIpart Ipart t

ut)2


40


41

Chapter 5

Conclusion







42

References


43

I+ I p = (m p+m )R2 g

4π 2C=052239m4

Note



I=(( 32 )lowast4365lowast02722)+( 2

5 )lowast5656lowast0152=0535314 m4



34


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate04 30 298 149

2993333333 1496666667 0484410240445393497 805448343

04 30 298 149 0445393497 80544834304 30 302 151 0457430617 5569581401



04 30 164 0821653333333 0826666667 0513003156

0563225526 978987534704 30 165 0825 0570115071 111328582804 30 167 0835 0584019806 1384331624



04 30 157 0785163 0815 0541601867

05161714 469541704204 30 168 084 0591034996 912720797704 30 164 082 0563225526 399253776


the centroid

04 30 213 10652123333333 1061666667 053531424

052239302 24137636304 30 212 106 0517499435 332791537304 30 212 106 0517499435 3327915373

35


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate044 30 314 157

3156666667 1578333333 0484410240449549951 7196439229

044 30 316 158 045529494 6010463261044 30 317 1585 0458181114 5414651525



044 30 162 0811626666667 0813333333 0513003156

0499610983 2610544028044 30 162 081 0499610983 2610544028044 30 164 082 0512023205 0191022412



044 30 169 0845169 0845 0541601867

0543720061 0391098067044 30 167 0835 0530927096 1970962712044 30 171 0855 0556665324 2781278617


centroid

044 30 231 115523 115 053531424

0558559487 4342355492044 30 23 115 0553733942 344091388044 30 229 1145 0548929332 2543383077

36


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate05 30 343 1715

339 1695 0484410240472051706 2551253737

05 30 335 1675 0450288593 704395660505 30 339 1695 0461105951 4810857982



05 30 185 09251846666667 0923333333 051300315

6

0573360905 117655706105 30 187 0935 0585824908 141951859305 30 182 091 055491619 8170131796



05 30 186 0931820666667 0910333333 054160186

7

0579576154 701147639405 30 1792 0896 0537973187 06699902505 30 181 0905 0548834963 1335500582


centroid

05 30 238 1192396666667 1198333333 053531424

0521773549 252948448605 30 239 1195 0526167412 170868376705 30 242 121 0539459539 0774367463

37



-002-00100010020030040

005

01

015

02

025

03

035

04

045

log τ

log l



38

42 Discussion





39




I = md1 d2 gτ 2

4 π 2 L

uL=05480

=000104

ut=00051432

=000035

part Ipart L


(4 π2 L)2

part Ipart t


2π 2 L


uL)2

+( tIpart Ipart t

ut)2


40


41

Chapter 5

Conclusion







42

References


43


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate04 30 298 149

2993333333 1496666667 0484410240445393497 805448343

04 30 298 149 0445393497 80544834304 30 302 151 0457430617 5569581401



04 30 164 0821653333333 0826666667 0513003156

0563225526 978987534704 30 165 0825 0570115071 111328582804 30 167 0835 0584019806 1384331624



04 30 157 0785163 0815 0541601867

05161714 469541704204 30 168 084 0591034996 912720797704 30 164 082 0563225526 399253776


the centroid

04 30 213 10652123333333 1061666667 053531424

052239302 24137636304 30 212 106 0517499435 332791537304 30 212 106 0517499435 3327915373

35


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate044 30 314 157

3156666667 1578333333 0484410240449549951 7196439229

044 30 316 158 045529494 6010463261044 30 317 1585 0458181114 5414651525



044 30 162 0811626666667 0813333333 0513003156

0499610983 2610544028044 30 162 081 0499610983 2610544028044 30 164 082 0512023205 0191022412



044 30 169 0845169 0845 0541601867

0543720061 0391098067044 30 167 0835 0530927096 1970962712044 30 171 0855 0556665324 2781278617


centroid

044 30 231 115523 115 053531424

0558559487 4342355492044 30 23 115 0553733942 344091388044 30 229 1145 0548929332 2543383077

36


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate05 30 343 1715

339 1695 0484410240472051706 2551253737

05 30 335 1675 0450288593 704395660505 30 339 1695 0461105951 4810857982



05 30 185 09251846666667 0923333333 051300315

6

0573360905 117655706105 30 187 0935 0585824908 141951859305 30 182 091 055491619 8170131796



05 30 186 0931820666667 0910333333 054160186

7

0579576154 701147639405 30 1792 0896 0537973187 06699902505 30 181 0905 0548834963 1335500582


centroid

05 30 238 1192396666667 1198333333 053531424

0521773549 252948448605 30 239 1195 0526167412 170868376705 30 242 121 0539459539 0774367463

37



-002-00100010020030040

005

01

015

02

025

03

035

04

045

log τ

log l



38

42 Discussion





39




I = md1 d2 gτ 2

4 π 2 L

uL=05480

=000104

ut=00051432

=000035

part Ipart L


(4 π2 L)2

part Ipart t


2π 2 L


uL)2

+( tIpart Ipart t

ut)2


40


41

Chapter 5

Conclusion







42

References


43


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate044 30 314 157

3156666667 1578333333 0484410240449549951 7196439229

044 30 316 158 045529494 6010463261044 30 317 1585 0458181114 5414651525



044 30 162 0811626666667 0813333333 0513003156

0499610983 2610544028044 30 162 081 0499610983 2610544028044 30 164 082 0512023205 0191022412



044 30 169 0845169 0845 0541601867

0543720061 0391098067044 30 167 0835 0530927096 1970962712044 30 171 0855 0556665324 2781278617


centroid

044 30 231 115523 115 053531424

0558559487 4342355492044 30 23 115 0553733942 344091388044 30 229 1145 0548929332 2543383077

36


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate05 30 343 1715

339 1695 0484410240472051706 2551253737

05 30 335 1675 0450288593 704395660505 30 339 1695 0461105951 4810857982



05 30 185 09251846666667 0923333333 051300315

6

0573360905 117655706105 30 187 0935 0585824908 141951859305 30 182 091 055491619 8170131796



05 30 186 0931820666667 0910333333 054160186

7

0579576154 701147639405 30 1792 0896 0537973187 06699902505 30 181 0905 0548834963 1335500582


centroid

05 30 238 1192396666667 1198333333 053531424

0521773549 252948448605 30 239 1195 0526167412 170868376705 30 242 121 0539459539 0774367463

37



-002-00100010020030040

005

01

015

02

025

03

035

04

045

log τ

log l



38

42 Discussion





39




I = md1 d2 gτ 2

4 π 2 L

uL=05480

=000104

ut=00051432

=000035

part Ipart L


(4 π2 L)2

part Ipart t


2π 2 L


uL)2

+( tIpart Ipart t

ut)2


40


41

Chapter 5

Conclusion







42

References


43


SystemCord

Length L (m)

No of Oscillations

(cycles)


(sec)

Period τ

Average time (s)

Average Period



Only Plate05 30 343 1715

339 1695 0484410240472051706 2551253737

05 30 335 1675 0450288593 704395660505 30 339 1695 0461105951 4810857982



05 30 185 09251846666667 0923333333 051300315

6

0573360905 117655706105 30 187 0935 0585824908 141951859305 30 182 091 055491619 8170131796



05 30 186 0931820666667 0910333333 054160186

7

0579576154 701147639405 30 1792 0896 0537973187 06699902505 30 181 0905 0548834963 1335500582


centroid

05 30 238 1192396666667 1198333333 053531424

0521773549 252948448605 30 239 1195 0526167412 170868376705 30 242 121 0539459539 0774367463

37



-002-00100010020030040

005

01

015

02

025

03

035

04

045

log τ

log l



38

42 Discussion





39




I = md1 d2 gτ 2

4 π 2 L

uL=05480

=000104

ut=00051432

=000035

part Ipart L


(4 π2 L)2

part Ipart t


2π 2 L


uL)2

+( tIpart Ipart t

ut)2


40


41

Chapter 5

Conclusion







42

References


43



-002-00100010020030040

005

01

015

02

025

03

035

04

045

log τ

log l



38

42 Discussion





39




I = md1 d2 gτ 2

4 π 2 L

uL=05480

=000104

ut=00051432

=000035

part Ipart L


(4 π2 L)2

part Ipart t


2π 2 L


uL)2

+( tIpart Ipart t

ut)2


40


41

Chapter 5

Conclusion







42

References


43

42 Discussion





39




I = md1 d2 gτ 2

4 π 2 L

uL=05480

=000104

ut=00051432

=000035

part Ipart L


(4 π2 L)2

part Ipart t


2π 2 L


uL)2

+( tIpart Ipart t

ut)2


40


41

Chapter 5

Conclusion







42

References


43




I = md1 d2 gτ 2

4 π 2 L

uL=05480

=000104

ut=00051432

=000035

part Ipart L


(4 π2 L)2

part Ipart t


2π 2 L


uL)2

+( tIpart Ipart t

ut)2


40


41

Chapter 5

Conclusion







42

References


43


41

Chapter 5

Conclusion







42

References


43

Chapter 5

Conclusion







42

References


43

References


43

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