mechanical oscillation (shm)
TRANSCRIPT
![Page 1: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/1.jpg)
Mechanical Oscillation (SHM):
• Introduction and equation of SHM
• Energy in SHM
• Oscillation of mass-spring system
• Compound Pendulum
4
![Page 2: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/2.jpg)
5
Oscillation:• Motions that repeat themselves are called oscillations .• They occur almost everywhere around us.
Examples:• Oscillating guitar strings, drums• bells, diaphragms in telephones
and speaker systems, quartz crystals in wristwatches• Oscillations of the air molecules that transmit the
sensation of temperature• Oscillations of the electrons in the antennas of radio and
TV transmitters that convey information. ( Source: Fundamentals of Physics :David Halliday, Robert Resnick, Jearl Walker)
![Page 3: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/3.jpg)
Simple Harmonic Motion (SHM) :Definition:
The motion in which the restoring force(F) is directly proportional to the displacement(x) from the mean position and is opposite to it is called Simple Harmonic Motion.
i.e. F ∝ x
Or, F = - kx, where k is a constant called force constant and the negative sign is due to the opposite direction of F and x.
6
![Page 4: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/4.jpg)
Definition of k:
• Since F = - kx,
k = 𝐹
𝑋, in magnitude
So, the force constant is defined as the restoring force per unit displacement from the mean position.
Unit of k:
k = 𝐹
𝑋
𝑁
𝑚(In SI System) or
𝑑𝑦𝑛𝑒
𝑐𝑚(In CGS System)
7
![Page 5: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/5.jpg)
Differential Equation of SHM:
Since F = - k𝑥 and F = ma,
ma = -k𝑥
Now, a can be written as 𝑎 =𝑑2𝑥
𝑑𝑡2
So, m𝑑2𝑥
𝑑𝑡2= -k𝑥
Or, 𝑑2𝑥
𝑑𝑡2+ω2𝑥 = 0, which is the required differential equation of
SHM.
8
![Page 6: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/6.jpg)
Contd…
The solution of the differential equation
can be written as
The range of x lies between to
1
![Page 7: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/7.jpg)
1. Displacement: It is given by
2. Velocity: It is given by
2
![Page 8: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/8.jpg)
Again, squaring (2), we get
So,
where is the angular frequency3
![Page 9: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/9.jpg)
3. Acceleration:
It is given by
But (from eq. (1))
So,4
![Page 10: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/10.jpg)
5
![Page 11: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/11.jpg)
4. Time period (T) :
It is the time taken by the body to complete one oscillation and is given by
But (From eqn (5))
So, (In magnitude)
6
![Page 12: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/12.jpg)
5. Frequency ( ): It is given by
6. Phase: It is given by
7. Phase constant: It is given by
7
![Page 13: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/13.jpg)
A particle executing SHM has two types of energy, viz.(i) Kinetic Energy(ii) Potential Energy
Kinetic energy arises due to its motion about the mean position and is given by
8
where m is the mass of the particle and v is its velocity
![Page 14: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/14.jpg)
9/22/2020 1
S.H.M_3
Continued from SHM_2
created by Arun Devkota_NCIT
![Page 15: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/15.jpg)
Contd. from energy……
9/22/2020 2
Now the velocity is given by
So,
Now the Potential energy is given by
where
created by Arun Devkota_NCIT
![Page 16: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/16.jpg)
9/22/2020 3
So
Also since (angular velocity)
Or,
Then,
Now, the total energy is defined as the sum of the kinetic and potential energy.
created by Arun Devkota_NCIT
![Page 17: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/17.jpg)
9/22/2020 4
Then,Total Energy (T.E.) = K.E. + P.E.
1
k
Or,
created by Arun Devkota_NCIT
![Page 18: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/18.jpg)
9/22/2020 5
This shows that the total energy remains conserved or constant although the kinetic and the potential energies varywith time.
Variation of Energy with time: Variation of Energy with displacement:
?H.W.
created by Arun Devkota_NCIT
![Page 19: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/19.jpg)
9/22/2020 6
Homework/Assignment (SET 1):1. Draw the graph of variation of Energy versus
Displacement.2. Draw the graphs of displacement vs time, velocity vs
time and acceleration vs time.3. What is S.H.M? Write the differential equation of
S.H.M.4. What is S.H.M? Discuss the characteristics of S.H.M
with neat graphs. Also discuss the energy consideration in S.H.M with graphs. (9 marks/Long Q)
created by Arun Devkota_NCIT
![Page 20: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/20.jpg)
9/22/2020 7
Numericals:1. A body of mass 0.3 kg executes SHM with a period of 2.5 sec and
amplitude of 4 cm. Calculate the amplitude, velocity, accelerationand kinetic energy.
gET YOUR BRAIN EXERCISED !
created by Arun Devkota_NCIT
![Page 21: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/21.jpg)
9/22/2020 8
𝑆𝑜𝑙𝑛: Here, given:Mass of the body (𝑚) = 0.3 kgPeriod of oscillation (𝑇) = 2.5 secAmplitude (𝑥𝑚) = 4 cm = 0.04 mNow,(i) Max. velocity, 𝑣𝑚𝑎𝑥 = ω𝑥𝑚=
= 0.1 m/s
(ii) Max. Acceleration, 𝑎𝑚𝑎𝑥 =
Use this format to do numericals
= 0.252 m/𝑠2created by Arun Devkota_NCIT
![Page 22: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/22.jpg)
9/22/2020 9
(iii) Maximum kinetic energy,
created by Arun Devkota_NCIT
![Page 23: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/23.jpg)
9/22/2020 10
2. A small body of mass 0.1 kg is undergoing a SHM of amplitude 0.1 m and period 2 sec. (i) What is the maximum force on body? (ii) If the oscillations are produced in the spring, what should be the force constant?
Assignment:
created by Arun Devkota_NCIT
![Page 24: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/24.jpg)
9/22/2020 11
3. When the displacement is one-half the amplitude, what fraction of the total energy is the K.E. and what fraction is P.E. in S.H.M? At what displacement is the energy half K.E. and half P.E.?
Assignment:
created by Arun Devkota_NCIT
![Page 25: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/25.jpg)
Created by Arun Devkota_NCIT
Simple Pendulum:
A simple pendulum consists of a metallic bob suspended
with an extensible thread or rope at the point O. It is
displaced through a small angle ϴ .
The tension T balances the component mg cosϴ while
mg sinϴ provides the necessary restoring force F.
Then,
![Page 26: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/26.jpg)
Created by Arun Devkota_NCIT
Restoring force (F) = -mg sinϴ (The negative sign is
due to the opposite direction of the restoring force and
the displacement.) Or, ma = -mg sinϴ
Or, a = -g sinϴ
For small angle ϴ, sinϴ≈ϴ
Or, a = -gϴ
Or, a = -g (𝑥
𝑙) , where 𝑥 is the small linear displacement.
Or, a + (𝑔
𝑙)𝑥 = 0
which can be written as
Where the angular frequency is given by
The former equation is the differential equation of SHM.
Hence the motion of simple pendulum is .
![Page 27: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/27.jpg)
Created by Arun Devkota_NCIT
To find the time period (T):
Since the angular frequency of the simple pendulum is
given by
Also, the time period is given by
So, the time period of the simple pendulum is given by
Note:
(i) T is independent of the mass (m) of the
pendulum and the angular displacement (ϴ).
(ii) T solely depends upon the distance between
the point of suspension and the center of gravity
(C.G) of the pendulum.
(iii) If ϴ is relatively large, the approximation
sinϴ≈ϴ is no longer valid, hence the motion is
no longer simple harmonic. T depends upon ϴ
![Page 28: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/28.jpg)
Created by Arun Devkota_NCIT
and this type of motion is called anharmonic
motion.
• Remember that we come across the
same idea in Compound Pendulum. So
in dealing with the theory of Compound
Pendulum you are advised to through
the theory of Simple Pendulum.
![Page 29: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/29.jpg)
Created by Arun Devkota_NCIT
It consists of a block of mass m and a spring of spring constant k. It has two types:
(i) Horizontal M (ii) Vertical M
(i) Horizontal M
Consider a horizontal mass-spring system of a block of mass m and a spring of spring constant k, as shown in the figure below.
Let the system be displaced through a distance X. (Note: You are also free to use small letter ‘x’ for displacement for consistency through the chapter of SHM). Then a restoring force is developed such that the system has the tendency to come back to the mean position.
![Page 30: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/30.jpg)
Created by Arun Devkota_NCIT
It is found that the restoring force is directly proportional to the displacement from the mean position and is opposite to it .
i.e. F ∝ X
or, F = - kX, k being a constant of proportion, and is called the spring constant.
or, ma = - kX
Substituting the value of acceleration, we get
(the angular frequency)
Equation (1) is the required differential equation of SHM. Hence the motion of the horizontal mass-spring system is simple harmonic.
To find the time period:
Since the time period is given by
![Page 31: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/31.jpg)
Created by Arun Devkota_NCIT
And from above, we have
So,
![Page 32: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/32.jpg)
Mass-Spring
System:
Contd…from Simple Pendulum
created by Arun Devkota_NCIT 1
![Page 33: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/33.jpg)
Mass-Spring System
VerticalHorizontal
It consists of a block of mass m and a spring of spring constant k.
Mass-Spring System:
(i) Horizontal Mass-Spring System
(ii) Vertical Mass-Spring System
It has two types:
created by Arun Devkota_NCIT 2
![Page 34: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/34.jpg)
(i) Horizontal Mass-
Spring System:
created by Arun Devkota_NCIT 3
Consider a horizontal mass-spring system of a block of mass
m and a spring of spring constant k, as shown in the figure
below.
Let the system be displaced through a distance X. Then a restoring force is developed such that the system has the tendency to come back to the mean position.
![Page 35: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/35.jpg)
created by Arun Devkota_NCIT 4
ma = - k X
Diff. equation of SHM
1st task: To show motion is SHM
So motion of Mass-spring System is Simple Harmonic.
2nd task: To find T
![Page 36: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/36.jpg)
created by Arun Devkota_NCIT 5
Assignment:Vertical Mass-spring System
![Page 37: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/37.jpg)
created by Arun Devkota_NCIT6
Compound /Real/Physical Pendulum :
S
G
ϴ
G’A
![Page 38: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/38.jpg)
created by Arun Devkota_NCIT 7
Restoring torque is given by
Also,
![Page 39: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/39.jpg)
created by Arun Devkota_NCIT 8
Angular acceleration
Then,
So,
![Page 40: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/40.jpg)
created by Arun Devkota_NCIT 9
This is the differential equation of SHM. Hence the motion of Compound Pendulum is simple Harmonic
1st task is done. Mission accomplished!
2nd task: To find T
Previous slide
This is the angular frequency
![Page 41: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/41.jpg)
created by Arun Devkota_NCIT 10
So,
Previous slide
Combining these two equations,
Previous slide
So,
Dividing by
Very very important
![Page 42: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/42.jpg)
created by Arun Devkota_NCIT 11
Since for the compound pendulum, T is given by
And for the simple pendulum, T is given by
Let
called the length of the equivalent simple pendulum or the equivalent length of the simple pendulum or the reduced length of the compound pendulum.
![Page 43: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/43.jpg)
created by Arun Devkota_NCIT 12
Previous slide
Then, This is the Time Period of the Compound Pendulum in terms of the length of the equivalent simple pendulum or the equivalent length of the simple pendulum or the reduced length of the compound pendulum.
![Page 44: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/44.jpg)
created by Arun Devkota_NCIT 13
Assignment:
Q. Define moment of inertia and radius of gyration. (2 marks)
Q. Show that the motion of compound Pendulum is Simple Harmonic. Hence find its time period. (9 marks)
Q. Show that the motion of compound Pendulum is Simple Harmonic. Hence find its time period in terms of the length of the equivalent simple pendulum or the equivalent length of the simple pendulum or the reduced length of the compound pendulum.. (9 marks)
Equivalent questionOr
![Page 45: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/45.jpg)
created by Arun Devkota_NCIT 14
Previous Slide
Remember this for Simple Pendulum
The factor is the extra term in the Compound Pendulum
For Compound Pendulum
![Page 46: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/46.jpg)
created by Arun Devkota_NCIT 15
Previous Slide
This guy, THIS point which lies at a distance of
below CG is called the Point of Oscillation, & is denoted by the point ‘O’
![Page 47: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/47.jpg)
created by Arun Devkota_NCIT 16
Compound Pendulum _Contd…
To show that the point of suspension and the point of oscillation are interchangeable:
![Page 48: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/48.jpg)
created by Arun Devkota_NCIT 17
Contd…
For the Point of Suspension ‘S’, the time period is given by
Let
Then,
![Page 49: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/49.jpg)
created by Arun Devkota_NCIT 18
Previous Slide
Now, the time period for the Point of Oscillation ‘O’ is given by
But
So,
So,
Hence, the point of suspension and the point of oscillation are interchangeable:
![Page 50: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/50.jpg)
created by Arun Devkota_NCIT 19
Note:
# T will be maximum (i.e. ) for
(i)
(ii)
(iii)
point of suspension is the C.G itself
Minimum Time Period: V. Imp
Q. Show that the period of the Compound Pendulum is minimumatOR
The period is minimum when the point of suspension and the point of oscillation are equidistant from C.G.
![Page 51: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/51.jpg)
created by Arun Devkota_NCIT 20
Solution: Since
Squaring both sides,
Differentiating both sides, we get
Or,
![Page 52: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/52.jpg)
created by Arun Devkota_NCIT 21
For T to be minimum or maximum:
So,
(Taking the positive value only)
![Page 53: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/53.jpg)
created by Arun Devkota_NCIT 22
Hence for , T will be maximum or minimum.
If we show , T will be minimum at
Previous slide
Differentiate this one w.r.t. ‘l’
![Page 54: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/54.jpg)
created by Arun Devkota_NCIT 23
2 is cancelled out from LHS and RHS
Since
Or,
Since RHS>0 and T>0,
Hence the period of the Compound Pendulum is minimum at
![Page 55: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/55.jpg)
created by Arun Devkota_NCIT 24
So, The period is minimum when the point of suspension and the point of oscillation are equidistant from C.G.
Again, since at , period is minimum, the minimum period is given by
![Page 56: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/56.jpg)
Compound Pendulum contd…
+
Bar Pendulum
created by Arun Devkota_NCIT 1
![Page 57: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/57.jpg)
Question: Show that there are 4 colinear points in the compound
pendulum for which the period is same.Soln:
Let the pendulum be given as shown. S
and O are the point of suspension and
point of oscillation at distances 𝑙 and
𝑘2/𝑙 from the C.G., denoted by G.
Now, take radius equal to
GO = 𝑘2/𝑙 and cut the vertical axis above C.G at
O’ such that GO’ =𝑘2/𝑙 = 𝐺𝑂created by Arun Devkota_NCIT 2
![Page 58: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/58.jpg)
Then the time period remains same for the points O
and O’
Again , Now, take radius equal to
SG = 𝑙 and cut the vertical axis below C.G at O’ such
that S’G = 𝑙 = 𝑆𝐺.
Again there are two points for which the period is
same.
Thus in total there are 4 colinear points S, O, S’ and
O’ for which the period is same.created by Arun Devkota_NCIT 3
![Page 59: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/59.jpg)
Bar Pendulum:
G
P
𝜃
G’A
mg
Consider a bar Pendulum having
C.G. at the point G and
suspended at the point P.
It is displaced through a small
angle 𝜃.Due to the restoring torque, it
comes back to the mean position.
Now the restoring torque is given
by
𝜏 = Force . Perpendicular distancecreated by Arun Devkota_NCIT 4
![Page 60: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/60.jpg)
i.e. 𝜏 = 𝑚𝑔. 𝑙 sin 𝜃
But, 𝜏 = 𝐼𝛼 = −𝑚𝑔𝑙 sin 𝜃
Now, 𝐼𝛼 = −𝑚𝑔𝑙 sin 𝜃
or, 𝐼ⅆ2𝜃
ⅆ𝑡2= −𝑚𝑔𝑙 sin 𝜃
ⅆ2𝜃
ⅆ𝑡2+ 𝜔2𝜃 = 0
This is the differential
equation of SHM. Hence
the motion of the Bar
Pendulum is Simple
Harmoniccreated by Arun Devkota_NCIT 5
![Page 61: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/61.jpg)
Here, 𝜔2 =𝑚𝑔𝑙
ഥ⊥
called the angular frequency
𝑇 =2𝜋
𝜔
Now,
is the time period
So,
𝜔 =𝑚𝑔𝑙
𝐼
created by Arun Devkota_NCIT 6
![Page 62: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/62.jpg)
Now,
Squaring both sides,
Or,
which is quadratic in
Or,
created by Arun Devkota_NCIT 7
![Page 63: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/63.jpg)
If and are the roots of the
above equation,
Then,
And
Or,
Or,
Therefore,
created by Arun Devkota_NCIT 8
Thus knowing the value
of
We can determine the
value of and .
and ,
![Page 64: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/64.jpg)
created by Arun Devkota_NCIT 9
1st method to find g & k:
We plot a graph of T vs l and get curves as shown below .
Side B
T
Side A
![Page 65: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/65.jpg)
created by Arun Devkota_NCIT 10
Side B
T
Side A
Now, we draw straight lines like ABOCD.
A B O C D
![Page 66: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/66.jpg)
created by Arun Devkota_NCIT 11
In the graph, &
&
Thus, or
&
E
In this way, the value of
g & k are determined
using the formulas
&
Notes:(i) If A is the point of suspension, C is the
point of oscillation. Similarly, if D is the point
of suspension, B is the corresponding point
of oscillation. # Here, A & C or B & D lie on
the opposite sides of C.G.
(ii) L = AC or BD is the length of the
equivalent simple pendulum or the reduced
length of the compound pendulum.
(iii) There are 4 colinear points: A,B,C & D
for which the period(T = OE) is same.
S T
(iv) Here, the points S & T are the points
for which the period is minimum
correspond to l=k & are equidistant from
C.G.
![Page 67: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/67.jpg)
Alternative method to find g and k:
created by Arun Devkota_NCIT 12
O
D
E
Plot the graph of vs. for
both sides A & B separately.
![Page 68: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/68.jpg)
created by Arun Devkota_NCIT 13
Then, from previous equation,
Comparing above equation
with , we have
From graph,
![Page 69: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/69.jpg)
created by Arun Devkota_NCIT 14
Thus, the values of OD & OE are determined from graph, hence the slope
(m) is determined which helps to find the value of g. Further the y-intercept
(c) and the value of g help to find the value of k.
![Page 70: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/70.jpg)
Derive the non-differential form of SHM.
Question:
Hint:
In the beginning of the chapter we defined
SHM.
And wrote directly
Then,
And arrived at the differential equation of
SHM
the solution of the differential equation
above can be written as
Now, we are
arriving above
equation step by
step
created by Arun Devkota_NCIT 15
![Page 71: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/71.jpg)
Derive the non-differential form of
SHM.Solution:
The motion in which the restoring
force is directly proportional to the
displacement from the mean
position and is opposite to it is
called SHM.
i.e.
Since
Let
Or,
(the angular frequency)
created by Arun Devkota_NCIT 16
![Page 72: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/72.jpg)
So,
Multiplying both sides
by
Or,
omit dt from both sides
Integrating both sides, we
get
(1)
where C is the constant of
integration.
created by Arun Devkota_NCIT 17
![Page 73: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/73.jpg)
Previous Slide
But is the velocity
which is maximum at the mean position.
Then,
(2)
From (1) & (2),
(3)
created by Arun Devkota_NCIT 18
![Page 74: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/74.jpg)
Previous Slide
So from (1) & (3),
created by Arun Devkota_NCIT 19
![Page 75: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/75.jpg)
Previous Slide
Taking square root, we have
Taking positive sign,
or,
Integrating both sides, we have
Again, taking negative sign,
Upon integrating, we
get,
(4)
(5)
Equation (4) or (5) is
the required non-
differential form of
SHM.
created by Arun Devkota_NCIT 20
![Page 76: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/76.jpg)
created by Arun Devkota_NCIT 21
Numerical:
Q.1Show that if a uniform stick of length ‘l’
horizontal axis perpendicular to the stick
and at a distance ‘d’ from the center of massperiod has a minimum value when d = 0.289l.
is mounted so as to rotate about a
or mark,
![Page 77: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/77.jpg)
created by Arun Devkota_NCIT 22
Solution:
C.G/C.M/Mark.
Point of
suspensiond
The stick ,here, is a compound
pendulum having total length
whose time period is given by
where
is the distance between the point
of suspension and the C.G.
& k is the radius of gyration
![Page 78: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/78.jpg)
created by Arun Devkota_NCIT 23
For the stick, the radius of gyration is given by
![Page 79: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/79.jpg)
created by Arun Devkota_NCIT 24
For T to be minimum,
Distance between the point of suspension
and C.G. = radius of gyration
So, period has a minimum value when d = 0.289l.
![Page 80: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/80.jpg)
created by Arun Devkota_NCIT 25
Numerical:
A small body of mass 0.1 kg is undergoing a SHM of
amplitude 0.1 m and period 2 sec. (i) what is the
maximum force on body? (ii) If the oscillations are
produced in the spring, what should be the force
constant?
Q.2
Assignment:
![Page 81: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/81.jpg)
created by Arun Devkota_NCIT 26
Numerical:
Q.3
A meter stick suspended from one end swings as a physical
pendulum (i) What is the period of oscillation? (ii) What would
be the length of the simple pendulum that would have the
same period?
![Page 82: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/82.jpg)
created by Arun Devkota_NCIT 27
Q.4
Numerical:
A physical pendulum consists of a meter stick that is
pivoted at a small hole drilled through the stick at a
distance x from mark. The period of oscillation is
observed to be 2.5 s. Find the distance x. (Ans: 5.57 cm)
![Page 83: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/83.jpg)
created by Arun Devkota_NCIT 28
Numerical:
Q.5What is the mechanical energy of the linear
oscillator so that the initial position of the
block is 11 cm at rest? The spring constant is
65 N/m. Calculate the K.E. and P.E. of the
oscillator for its displacement being half of its
amplitude.
![Page 84: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/84.jpg)
created by Arun Devkota_NCIT 29
Numerical:
Q.6 A uniform circular disc of radius R oscillates in a vertical
plane about a horizontal axis. Find the distance of the axis
of rotation from the center for which the period is minimum.
What is the value of this period?
Ans: ,
![Page 85: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/85.jpg)
created by Arun Devkota_NCIT 30
Q.6Show that the displacement equation represented by
or
represents S.H.M. or is a solution of S.H.M.
![Page 86: Mechanical Oscillation (SHM)](https://reader030.vdocuments.net/reader030/viewer/2022012804/61bd301261276e740b1033d8/html5/thumbnails/86.jpg)
created by Arun Devkota_NCIT 31