1

School of Technology

Division of Mechanical Engineering

Mechanical Laboratory Practical

ROLLING DISC ON AN INCLINED PLANE

Objectives:

1. To determine experimentally the moment of inertia of different disc

assemblies.

2. To compare the result with the theoretical values obtained from the mass

and the physical dimensions of disc assembly.

Theory:

Figure 1: Schematic Diagram on Rolling Disc on an Inclined Plane

A flywheel, with mass m and radius R, rolls from rest at top position and takes time

t(s), to reach bottom position.

Let the linear velocity of flywheel centre at the bottom position = v (m/s)

Then, the angular velocity of the fly wheel at this position = (rad/s) = v/R (rad/s)

Average linear velocity = ½ v (m/s) = L/t (m/s), where L is the linear distance

travelled

From conservation of energy,

L (m)

h (m) (rad/s)

v (m/s)

)/(2

smt

Lv

2

))(2

1(

2

1

2

1

2

1

)2

1

2

1(

22

22

22

JoulevghmI

mvmghI

Imvmgh

Potential energy (at highest position) = Kinetic energy (at lowest position)

Therefore, moment of inertia of flywheel,

))(12

(

)14

2(

)12

(

)2(

)2(

))(2

1(

2

2

2

22

2

22

2

2

2

2

2

2

2

22

2

kgmL

ghtmRI

L

ghtmRI

v

ghmRI

vghv

mRI

vghm

I

kgmvghm

I

For volumes calculation:

Volume of the disc, 1

2 lRVD

Volume of the spindle, 32

2 llrVS

l1 l2 l3 spindle with radius r

disc with radius R

3

Procedure:

1. Use the given weighing scale and vernier calliper, record the measurement in

Table 1.

2. Place the inclined plane apparatus on a level surface and ensure that the top

surfaces of the two rails are same level. Wipe off any grease and dirt, which

may be on the tops of the rails.

3. Set one end of the two flanking rails of apparatus at a level above that of the

other end. Set a distance of L (m) along the length of the plane (ex: 1m) and

at height h = 120 mm between the extremities of the distance traversed by the

centre of the disc.

4. Allow the spindle of the small disc assembly to rest on the two flanking rails

and release it so that it starts rolling unaided down the incline, ensuring that

the disc does not rub against the rails during its motion. Note the time t (s)

taken for the disc to traverse the distance L (m).

5. Carryout the procedures three times to get the average time taken. Tabulate

the results in Table 2.

6. Repeat procedure 4 and 5 for large disc assembly.

Result:

Table 1

Large disc (mm) Small disc (mm)

Diameter of disc,D

Thickness of disc, l1

Diameter of spindle, d

Length of spindle, l2 + l3

Mass, m

4

Time, t (s) Small Disc Large Disc

t1

t2

t3

Average, t = t/3

Table 2

Results and Calculation:

In order to determine the theoretical value of I from the mass and physical

dimensions of disc assembly; determine the volume of the disc VD and the volume of

the spindle VS, which may be considered as a single cylinder.

))(12

( 2

2

22 kgm

L

ghtmrI

Mass of the disc MD,

)(KgVV

VmM

SD

DD

Mass of the spindle, MS,

Theoretical moment of inertia of disc ID,

Theoretical moment of inertia of the spindle IS,

)(2

22

kgmr

MI SS

)(KgVV

VmM

SD

SS

)(2

22

KgmR

MI DD

5

Thus, theoretical total moment of inertia of the disc assembly,

SD III

Discussion and Conclusion:

1) Compare the experiment and theoretical results obtained from the both

equation respectively.

2) Comment on the accuracy of the experiment; discuss any possible

sources of error and then recommending methods for improving the

overall efficiency of the experiment.

6

School of Technology

Division of Mechanical Engineering

Mechanical Laboratory Practical

ANGLED PROJECTION

Objective:

1. Determination of the distance as function of inclination angle.

2. Determination of the maximum height as function of the inclination angle.

Theory:

In the experiment a steel ball of mass m is projected at an angle α to the horizontal

with an initial velocity v0. The motion of the steel ball in the (constant) gravitational

field lies in a plane and can be described by the equation (Figure 1):

Figure 1 Movement of a mass in a constant gravitational field.

a) From equation of motion, s = ut + 2

1at2

For vertical motion, when the mass return to the ground, s = 0, with velocity in

vertical direction, uy = u sin , a = -g and time of flight from point O to B = t,

0 = (u sin )t - 2

1gt2

Time of flight, t = g

sin 2u

b) From the horizontal motion, the velocity of the horizontal component, ux = u cos . Since g is a vertical component, no horizontal component for g.

Range of OB, R = velocity × time.

y

g

path

u cos

O

u

u sin

x B

7

= u cos × (g

sin2u )

g

R2sinu2

(1)

where sin 2 = 2 sin cos

e) For vertical motion, the maximum point, time of flight

t1/2 = g

sin u

2

2

sin

2

1sinsin

2

1

g

ug

g

uu

atuth

22

sin2g

u (2)

Apparatus:

Figure 2

8

Item Description

c Angle scales, on the front and back, calibrated from 0° to 90°.

f Pointer (one on each side)

i Knurled nuts

j Spring-loaded plunger with 3 grooves and handle, provided at its

front end with a small magnet in order to hold the bal to be projected

and at its rear end with (k)

k Releasing lever for setting the required initial velocity by means of a

compression spring located within the guide barrel and for trigger the

throw.

Procedure:

1. Check to ensure the projection apparatus is mounted as depicted in Figure 2

with a sand tray on a table.

2. To measure the maximum height h of the trajectory, place the vertical scale in

the sand tray as shown in Figure 3.

Figure 3

3. To set the projectile angle, loosened and displaced the knurled nuts (i) (Figure

2) until the guide barrel can swung to the desired angle, start from = 10°.

Then tightened both the knurled nuts.

Figure 4

9

4. Moved the releasing lever (k) is to the right. Pull the plunger out of the guide

barrel and place the releasing lever in the first groove of the plunger by

turning the releasing lever (k) counter clockwise.

5. Place a small steel ball in the muzzle-piece of the guide barrel. Hit the

releasing lever as shown in Figure 4 to launch the steel ball.

(Note: Do NOT try until you are shown. Make sure no one is in front of the

launcher before the releasing lever is hit.)

6. Observe that the steel ball must just cross over the movable pointers of the

vertical scale at the maximum height. Otherwise, readjust the height of the

movable pointers and repeat step 4 to 6. (Note: this has to be done by trial

and error)

7. Record the height, h1 and the range s1 in Table 1 and Table 2.

8. Repeat steps 4 to 6 but this time place the releasing lever at second and third

groove respectively. Record h2, s2, h3 and s3 in Table 1 and Table 2

respectively.

9. Repeat step 3 to step 8 for = 20°, 30°,…80°.

Result:

s1/m s2/m s3/m

10°

20°

30°

40°

50°

60°

70°

80°

Table 1

1. Plot a graph of range, s, against inclination angle .

2. From graphs plotted, read the ranges, Rmax1, Rmax2, Rmax3 at α = 45° at three

different releasing lever.

3. Substitute each of this range into equation (1) to calculate the initial velocities

u1, u2 and u3.

4. Substitute u1, u2 and u3 into equation (2) to calculate respective theoretical

maximum height , hT1, hT2 and hT3.

10

Experimental Theoretical

h1/m h2/m h3/m hT1/m hT2/m hT3/m

10°

20°

30°

40°

50°

60°

70°

80°

Table 2

5. On three separate graph papers, plot graphs of h1 against and hT1 against

; h2 against and hT2 against ; h3 against and hT3 against ;

Discussion and Conclusion:

1) Based on the result, comment the relationship between the distance as function of

inclination angle.

2) Based on the result, comment the relationship of the maximum height as function

of the inclination angle.

3) Based on the graphs plotted, observe the deviation from the theoretical and

experimental result. Suggest any reason for the deviation.

11

School of Technology

Division of Mechanical Engineering

Mechanical Laboratory Practical

FRICTION ON INCLINED PLANE

Experiment 1:

Objective:

1. To measure the force F1, along the incline plane and the force F2 normal to the

incline plane of a body as a function of the angle of inclination, α.

2. To compare the measured force F1 and F2 with the theoretical values obtained

from the vectorial resolution of the force of gravity G.

Theory:

The motion of a body on an inclined plane can be described most easily when the

force exerted by the weight G (force of gravity) on the body is vectorially resolved

into a force F1 along the plane and a force F2 normal to the plane. The force along

the plane acts parallel to a plane inclined at an angle α , and the force normal to the

plane acts perpendicular to the plane.

F1 = G sin α F2 = G cos α

Two forces F1 and F2 are measured for various angles of inclination using precision

dynamometer. The height of the support, h = 5cm.

Since

From the equation (i) and (iii),

F1 = G

From the equation (ii) and (iv), F1 = G

12

Figure 1 Vectorial resolution of the force of gravity G into the force F1 along the plane and

the force F2 normal to the plane on an inclined plane

Apparatus:

Inclined plane with trolley and screw model, precision dynamometer 1.0N

Procedures:

1. Lay out dynamometer F1 horizontally and correct the zero point. Hold

dynamometer F1 vertically downward and correct the zero point.

2. Determine the weight G of the trolley.

3. Set up the inclined plane and position the support at s = 50cm.

4. Place the trolley on the incline plane and hook it to dynamometer F1, support the

dynamometer with block.

5. Carefully arrange dynamometer F2 as nearly perpendicular as possible to the

inclined plane and lift the trolley until it is just barely touching the plane surface.

6. Read off and write down forces F1 and F2.

7. Move the ramp support to the positions S = 40cm, 30cm, 20cm, 15cm and 10cm

one after another. Each time arrange the dynamometer perpendicular to the

inclined plane and read off and write down forces F1 and F2.

13

Figure 2 Experiment setup for determining the force along the plane and force to the plane

Results and calculation:

S/cm F1 /N F2 /N

50

40

30

20

15

10

Discussion:

Q1) Comment on the accuracy of the experiment; discuss any possible sources of

error and then recommending methods for improving the overall efficiency of the

experiment.

14

Experiment 2:

Objectives:

1. To determine the coefficient of static friction µ from the equilibrium between the

force along the plane and the static friction on an inclined plane.

Theory:

A body on an incline plane with the weight G is subject to a force along the plane

(parallel to the plane) of

F1 = G sin α --------------------------- (i)

and to a force normal (perpendicular) to the plane of

F2 = G cos α ---------------------------- (ii)

The angle of inclination α of the plane is increased by moving the support until the

body just begins to slide. The force along the plane and the static friction force F are

in equilibrium. The height of the support h = 5cm, and its distance s from the pivot of

the inclined plane are measured.

--------------------------- (iii)

The static friction forces F is generally taken to be proportional to the force F2 along

the plane:

------------------------------- (iv)

From the equilibrium of forces F1 = F , we can deduce:

----------------------------- (v)

Where µ = coefficient of friction

From equation (i), (ii) & (iii),

15

Figure 3 Equilibrium between the force F1 along the plane and the static friction force F on

an inclined plane

Apparatus:

Inclined plane with trolley and screw model, pair of wooden blocks for friction

experiments

Procedures:

1. Set up the inclined plane and move the support to the farthest possible point from

the pivot.

2. Place block 1 (6 cm thick) on the inclined plane with the plastic-coated side down

and slowly move the support inward until the block start to slide.

3. Measure the distance between the pivot and the support using the tape measure

and calculate the coefficient of static friction using equation (iv).

4. Place block 1 on the plane with the wooden side down and repeat the

experienment.

5. Place block 2 (3 cm thick) on the inclined plane with the plastic-coated side down

and repeat the experiment.

6. Turn the wooden surface with the area A = 12 x 6cm2 down and repeat the

experiment.

7. Turn the wooden surface with the area A = 12 x 3cm2 down and repeat the

experiment.

16

Figure 4 Experiment setup for determining the coefficient of friction on an inclined plane

Results and calculation:

Block Material A/cm2 S/cm µ

1 Plastic

1 Wood

2 Plastic

2 Wood

2 Wood

Discussion:

Q1) What is the factors affecting the coefficient of static friction?