Download Tejas Engineers Academy

XI / PHYSICS
OSCILLATIONS
11/PA
Periodic motion
The motion of a body is said to be periodic if it repeats its motion identically after a fixed interval of time.
For example motion of earth around sun is periodic with time period 365.25 days, motion of earth about its axis is
periodic with period 24 hrs, A simple pendulum swinging in vertical plane has periodic motion with period
1 l
T
2 g
Example-1: Suppose there is an insect that climbs up a wall and falls down, graph of its height with time will look
like as shown. It is periodic with time period T.
Example-2: Suppose there is a child who steps up and down repeatedly on a stair, its height from ground with
respect to time can be represented as shown. It is periodic with time period T.
Example-3: When we play bouncing a ball, between our palm and ground, in that case its height versus time graph
will look like as shown below. It is periodic with time period T.
Oscillatory motion
If a body moves to and fro repeatedly about a fixed position then its motion is said to be oscillatory. For example a
simple pendulum swinging in vertical plane about its position of rest, motion of a vertical spring when released
from any stretched position or the motion of a floating cylinder in a liquid at rest.
Note: An oscillatory motion is always periodic, but a periodic motion may or may not be oscillatory. For example
motion of earth about its axis is periodic but it is not the oscillatory one.
Harmonic Function
Any mathematical function which involves linear combination of sine or cosine functions is known as harmonic
function.
By: Amrish Maggo
1
XI / PHYSICS
OSCILLATIONS
11/PA
For example displacement of bob of a simple pendulum at any time can be represented by a harmonic function of
the form:
y  a sin t
Harmonic oscillations: The oscillations of a body are said to be harmonic if we can express its oscillations in terms
of a single harmonic function, such as sine or cosine function.
Following mathematical functions represents harmonic oscillations.




y  5 sin t   , y  3 cos t  
4
2


Simple pendulum always has harmonic oscillations as described above.
Simple Harmonic Motion
A body is said to be having simple harmonic motion if it experiences a restoring force directed toward the mean
position whose value at any instant of time is directly proportional to the displacement of particle from mean
position i.e.
F y
F = - ky
Where k is known as spring factor. Negative sign shows that F and y are always in opposite direction.
For example consider the case of a body tied with a spring. We observe that greater is the displacement of body (x)
from mean position greater, will the restoring force experienced by body, it is shown below.
Similarly in case of simple pendulum greater is the displacement of bob of pendulum from mean position greater
will be the restoring force experienced by body. It is shown below.
Geometrical Representation of SHM
Suppose there is a particle which is moving uniformly in a circular path of radius a as shown. Suppose we draw
Projections of particle P on X-axis at M and on Y axis at N. Now imagine if the particle moves uniformly along
circular path then how projection moves along X-axis and Y-axis respectively.
We observe that Projections M and N execute SHM with point O as the mean position.
By: Amrish Maggo
2
XI / PHYSICS
OSCILLATIONS
N
P’
11/PA
P
N’
M’
M
Thus geometrically simple harmonic motion can be represented as the projection of a particle executing SHM
about any one of its diameter.
Characteristics of SHM
1.DISPLACEMENT
Shortest distance of a particle executing SHM from mean position at any instant of time is known as displacement.
Consider a particle P moving uniformly along a circular path with uniform angular velocity ω, suppose its motion
is started from A, lying on the X-axis. It reaches to point P in time t.
Thus
ω=θ/t
θ = ωt ------------------------------------------------------(1)
From right triangle, OAP
Y
P
a
X’
x
O
y
A
X
Y’
Sinθ =
AP
OP
AP = (OP) sinθ
y = a sinθ
[  using 1.]
y = a sin(ωt) --------------------------------------------(2)
By: Amrish Maggo
3
XI / PHYSICS
OSCILLATIONS
11/PA
Similarly
cosθ =
AP
OP
x = a cosθ
x = a cos( t )
Case-1: If the motion of particle is started from Point A lying below at angle φ0, then from
Y
P
a
X’
x
O
y
A
X
Y’
Putting angle = θ - φ0
y = a sin (θ - φ0)
y = a sin (ωt - φ0) ------------------------------------------(3)
Case-2: If the motion of a particle is started from point A lying above the X- axis, then from equation 2.
Now angle is θ - φ0
y = a sin (θ + φ0)
y = a sin (ωt + φ0) -------------------------------------------(4)
Y
a
X’
x
O
P
A
X
Y’
2. AMPLITUDE
By: Amrish Maggo
4
XI / PHYSICS
OSCILLATIONS
11/PA
The maximum displacement of a particle executing SHM from the mean position is known as amplitude, it is
denoted by ‘a’
It is also defined as the separation between mean position and extreme position.
Amplitude of a particle is the coefficient of sine cosine term used to represent SHM. Following is the
representation of a body executing SHM with amplitude a, it has amplitude +a on positive X-axis, while
amplitude – a on negative X-axis.
3. TIME PERIOD
It is the time taken by a particle to complete one Oscillation. It is denoted by T.
4. FREQUENCY
It is the number of oscillations completed by a particle in one second.
1

Time Period
1

T
SI unit is Hz
5. ANGULAR FREQUENCY
It is the product of a factor 2π with the frequency of particle executing SHM i.e.
  2  (Frequency of SHM)
  2  
  2
We can put  
1
T
  2

2
T
1
T
SI unit is rad/s
6. VELOCITY
The rate of change of displacement of a particle is known as velocity.
dy
v=
---------------------------------------------(1)
dt
Suppose displacement is given by:
y = a sin( ωt) --------------------------------------------(2)
Putting in 1.
d
(a sin t )
v=
dt
By: Amrish Maggo
5
XI / PHYSICS
OSCILLATIONS
va
11/PA
d
(sin t )
dt
 d

 (sin t )   cos t 
 dt

v  a cos t
 cos t 
v  a 1  sin 2 t
1  sin 2 t

 Moving a inside square root 
v   a 2  (a sin t ) 2
Using 2.
v   a2  y2
Case-1: When body is at mean position we have y = 0
v   a 2  0 2   a 2  a
Therefore
This is the maximum velocity.
vmax  a
Case-2: When body is at extreme position we have y = a (maximum)
v   a2  a2  0
Therefore velocity is minimum at extreme position and is equal to zero. vmin  0
7. ACCELERATION:
The rate of change of velocity is known as acceleration.
dv
A=
dt
Putting
v  a cos(t )
d
A  a cos t 
dt
d
A  a cos t 
dt
A  a( sin t )
A   2 (a sin t )
A   2 y
using equation 1
Energies of particle executing SHM
There are two types of energies of a particle executing SHM, these are
1. Kinetic energy: It is the energy possessed by a particle because of its motion and is given by:
1
KE = mv2
2
Putting v   a 2  y 2
By: Amrish Maggo
6
XI / PHYSICS
OSCILLATIONS
KE =
Assuming k = mω2
11/PA
1
m 2 [a 2  y 2 ]
2
1
k[a 2  y 2 ] ----(4)
2
Case-1: At mean position y = 0, and particle has kinetic energy given by:
1
1
1
K.Emax = k[a 2  0 2 ] = ka 2 = m 2 a 2
2
2
2
Case-2: At extreme position y = a, and particle has kinetic energy given by:
1
1
K.Emin = k[a 2  a 2 ] = k[a 2  a 2 ] = 0
2
2
KE =
2. Potential energy
It is the work required to move a particle from mean position to any position.
Small work done in moving a particle over small displacement dy:
dW = F dy
Putting F = -ky
dW = (-ky)dy
Integrating
y
W  k  ydy
0
W
 
1
k y2
2
y
0


1
k y 2  02
2
This work done is the potential energy of particle executing SHM
1
PE  ky 2 -----------(5)
2
1
k = mω2
PE  m 2 y 2
2
Adding (4) and (5) we obtain the net expression for energy.
1
1
TE  KE  PE  m 2 (a 2  y 2 )  m 2 a 2
2
2
1
 m 2 [a 2  y 2  y 2 ]
2
1
 m 2 a 2
2
Thus total energy remains constant with time.
W
Plot of Energy with time:
By: Amrish Maggo
7
XI / PHYSICS
OSCILLATIONS
11/PA
Graph can be represented in short as:
Where U(x) is the potential energy and K(x) is the kinetic energy of particle executing SHM.
Time Period of Particle in SHM
Consider a particle of mass m executing SHM. Let y is the displacement of particle at point P. Suppose F is the
restoring force.
Restoring force acting on it
F = -ky
By Newton’s law this force is also
F = mA ----------------------(1)
Where A is the acceleration of particle. Putting A = - ω2y
F = m [ -ω2y]
---------------(2)
Equating (1) and (2) we get
-ky = -mω2y
Putting  
2
T
k = mω2
 2 
k  m

T 
k 2

m
T
2
By: Amrish Maggo
8
XI / PHYSICS
OSCILLATIONS
T  2
11/PA
m
k
k is called spring factor and m is called inertia factor.
T  2
Inertial Factor
Spring Factor
Simple Pendulum
A simple pendulum consists of light inextensible thread whose lower end is connected to a point mass object, and
upper end is fixed with a rigid support.
Suppose ‘m’ is the mass of bob. It experiences two forces, ‘mg’ is the downward gravitational force and T is
tension in string.
Resolving mg into two components we get:
1. mgsinθ [Towards mean position]
2. mgcosθ [Opposite to T]
Here mgcosθ is balanced by T while mgsinθ provides necessary restoring force [F]
F  mg sin 
y
F  mg
l
 mg 
F  
 y -------------------------------(1)
 l 
Can be written as
F  ky -------------------------------------(2)
Where k is called spring factor pendulum
mg
k
l
We know that
m
T  2
k
By: Amrish Maggo
9
XI / PHYSICS
OSCILLATIONS
T  2
T  2
11/PA
m
(mg / l )
l
g
This shows that oscillations of pendulum are SHM.
Factors affecting time period of pendulum
Time period of pendulum depends upon following factors
1. Length of pendulum: Time period is directly proportional to square root of length of pendulum.
T l
2. Time period is inversely proportional to square root of acceleration due to gravity
1
T
g
Oscillations of a mass connected to spring
Suppose there is a mass m connected to a massless spring of constant k, as shown.
If displaced by value y, mass experiences a restoring force of
F = -kx
F x
It shows that oscillations are SHM
Time period
m
T  2
k
Oscillations of a liquid in a U-tube
Consider a liquid of density ρ in a U-tube filled till length L. we observe that liquid oscillates freely when it is
compressed by displacement y and then left. We observe that restoring force acting on liquid will be
F = mg
Where m is the mass of liquid contained in height 2y
F = Vρg
[As m = Vρ]
Putting V = A(2y)
F = A(2y) ρg
Where A is the corossectional area of tube.
F= (2Aρg) y
By: Amrish Maggo
10
XI / PHYSICS
OSCILLATIONS
11/PA
Or
F = -2Aρg y ----------------------------(1)
Where negative sign shows that F and y are opposite in direction.
We observe that
Fy
This shows that oscillations of liquid in U-Tube are simple harmonic i.e. SHM
F  ky ------------------------------------------------(2)
Comparing (1) and (2) we observe that k  2 Ag
We know expression for time period
T  2
m
k
T  2
V
2 Ag
T  2
AL
2 Ag
L
2g
Springs in series
If a no. of springs are connected in series then combination behaves as a single spring. Suppose there are n springs
connected in series with spring constants k1, k2, ……. kn etc. Equivalent spring constant is given by
1 1
1
1
 

k k1 k 2
kn
T  2
Springs in parallel
A no. of springs as shown below are connected in parallel. Net spring constant is
k  k1  k 2      k n
Damped Oscillations
The oscillations are said to be damped if a system oscillates with continuously decreasing amplitude under the
action of a damping forces, such viscous forces or frictional forces.
Following is the graph of damped oscillations with time.
By: Amrish Maggo
11
XI / PHYSICS
OSCILLATIONS
11/PA
Undamped Oscillations
If a harmonic system oscillates with constant amplitude under the action of no damping or frictional forces, then its
oscillations are said to be undamped oscillations.
If a constant periodic force is exerted on the system for maintaining its constant amplitude, then its oscillations are
said to be forced oscillations.
Resonance
If the natural frequency of oscillation of a body is equal to the frequency of oscillation of a body then system starts
oscillating with continuously increasing amplitudes and such condition is called resonance.
In this case amplitude of the system increases indefinitely with time.
Soldiers are not advised to move in steps on hanging bridges. This is because if soldiers were allowed to move in
steps then bridge will start oscillating with same frequency as that of frequency of steps of soldiers. This will
constitute resonance oscillations of bridge. This will lead to increase in amplitude of oscillations of bridge and it
will be unsafe for soldiers.
ASSIGNMENT
1.
(a)
(b)
(c)
(d)
Which of the following examples represent periodic motion?
[NCERT]
A swimmer completing one trip from one bank of a river to the other and back.
A freely suspended bar magnet displaced from its N-S direction and released.
A hydrogen molecule rotating about its centre of mass.
An arrow released from a bow.
2. Which of the following examples represent (nearly) simple harmonic motion and which represent periodic, but not
simple harmonic motion?
[NCERT]
(a) The rotation of Earth about its axis.
(b) Motion of an oscillating mercury column in a U tube.
(c) Motion of a ball bearing inside a smooth curved bowl, when released from a point slightly above the lower most
point.
(d) General vibrations of a polyatomic molecule about its equilibrium position.
3. Given below are four x-t plots for linear motion of a particle. Which of the plots represent periodic motion? What is
the period of motion (in case of periodic motion)?
[NCERT]
By: Amrish Maggo
12
XI / PHYSICS
OSCILLATIONS
11/PA
4. A body executing SHM makes 100 vibrations per minute. If its speed at mean position is 15 m/s, calculate its path
length. What is its velocity when It is half way between its mean position and extreme position.
[Ans:2.864
m, 13 m/s]
5. Which of the following functions of time represent (a) simple harmonic (b) periodic but not simple harmonic and (c)
non periodic motion? Give the period for each case of periodic motion (ω is any positive constant).
[NCERT]
(a) sinωt – cosωt
(b) sin3ωt
(c) 3cos[π/4 – 2ωt]
(d) cosωt + cos3ωt + cos5ωt
(e) exp(-ω2t2)
(f) 1 + ωt + ω2t2.
6. A particle is moving in a st line with SHM. Its velocity has the values 3 m/s and 2m/s when its distance from the
mean positions is 1 m and 2 m respectively. Find the period of its motion and length of its path. [Ans: 4.86s, 5.06
m]
7. If the displacement of a moving particle at any time be given by an equation of the form
y = acosωt
+bsinωt, show that the motion is a SHM. If a=3, b=4, ω=2 s-1, determine the period, amplitude, maximum veleocity,
and maximum acceleration of the motion.
[Ans: 3.14s, 5 units, 10 unit, -20 unit]
8. A bob of simple pendulum executes SHM of period 20 s. Its velocity is 5 m/s, two seconds after it has passed through
its mean position. Determine the amplitude of SHM.
[Ans:19.67 m]
9. A body is vibrating SHM of amplitude 15 cm and frequency 4 Hz. Compute the maximum values of the acceleration
and velocity.
[Ans: 94.48ms-2, 3.77 ms-1]
10. An oscillating simple pendulum has the amplitude of oscillations as 7.07 m and the time period 5 s. Calculate the
maximum speed during the oscillation.
[Ans: 0.088 m/s]
11. Identify periodic, non periodic and SHM out of the following
(a) Fluttering tree leaves due to wind.
(b) Fluttering of honeybee’s wings for a microsecond.
(c) Rinsing and falling of water drops at the bottom of a water fall
(d) Sound produced by a motor cycle in a small interval of time.
(e) Motion of the wheel of a canon during rapid fire.
(f) Jerks in the machine gun during rapid fire.
(g) Jerks in a machine gun during rapid fire.
(h) Vibrations of a drum membrane in a music band.
12. A man is standing on a platform executing SHM in vertical direction and attached to a weighing machine. Will there
be any change in his weight.
13. Can you give some example where a harmonic oscillation does not posses potential energy?
radiation enclosed in a black body]
By: Amrish Maggo
[Ans: Thermal
13
XI / PHYSICS
OSCILLATIONS
11/PA
14. A simple pendulum is made with a table tennis ball filled with sand. A small opening is made at its bottom and it is
made to oscillate. How will ht time period of oscillation be affected as the sand falls down gradually out of the bob?
15. If the amplitude of vibration of an oscillator is made half how will its time period and energy be affected?
16. When a fast railway train passes over a river bridge we hear loud sound why?
17. Where in its motion, the tension is (a) maximum and (b) minimum for a simple pendulum?
18. If amplitude of vibration is not small in the simple pendulum experiment will it not be executing periodic harmonic
oscillations? How will the time period be affected?
19. A particle executing SHM along a straight line has a velocity of 4 m/s, when at a distance of 3 m from the mean
position and 3 m/a when at a distance of 4 m from it. Find the time it takes to travel 2 m from the positive extreme of
its oscillation.
[Ans: 0.93 s]
20. Which of the following relationships between the acceleration a and the displacement x of a particle involve simple
harmonic motion. [NCERT]
(a) a=0.7x
(b) a = -200x2
(c) a = -10x
(d) a = 100x3.
21. A particle moving with SHM in a straight line has a speed of 6 m/s when 4 m from the centre of oscillation ad a speed
of 8 m/s when 3 m from the centre of oscillation. Find the amplitude of oscillation and the shortest time taken by the
particle in moving from the extreme potion to a point mid way between the extreme potion and the centre. [Ans: 5
m, π/6 s]
22. The motion of a particle executing SHM is described by the displacement function: x(t) = A cos(ωt + φ)
If the initial (t=0) position of the particle is 1 cm
and its initial velocity is ω cm/s, what are its amplitude and initial phase angle? The angular frequency of the particle
is π s-1. If instead of the cosine function, we choose the sine function to describe the SHM x = B sin (ωt +α), what are
the amplitude and initial phase of the particle with the above initial conditions?
[NCERT]
23. A spring balance has a scale that reads from 0 to 50 kg. The length of the scale is 20 cm. A body suspended from this
balance when displaced and released, oscillates with a period of 0.6 s. What is the weight of the body? [NCERT]
24. A spring having with a spring constant 1200 N/m is mounted on a horizontal table as shown. A mass of 3 kg is
attached to the free end of the spring. ? The mass is then pulled sideways to a distance of 2 cm and released.
Determine
(i) The frequency of oscillations.
(j) The maximum acceleration of the mass and
(k) The maximum speed of the mass?
[NCERT]
25. Figure Below corresponding to two circular motions. The radius of the circle, the period of revolution, the initial
position and the sense of revolution (i.e. clockwise or anticlockwise) are indicated on each figure. Obtain the
corresponding simple harmonic motion of the x projection of the radius vector of the revolving particle P in each
case.
By: Amrish Maggo
14
XI / PHYSICS
OSCILLATIONS
11/PA
26. The piston in the cylinder head of a locomotive has a stroke (twice the amplitude) of 1 m. If the piston moves with
simple harmonic motion with an angular frequency of 200 rpm, what is its maximum speed? [NCERT]
27. The acceleration due to gravity on the surface of moon is 1.7 ms-2. What is the time period of a simple pendulum on
the surface moon if its time period on the surface of Earth is 3.5 s? ( g on the surface of earth is 9.8 ms-2)
[NCERT]
28. Answer the following question:
(a) Time period of a particle in SHM depends on the force constant and mass m of the particle. A simple
pendulum executes SHM approximately. Why then are the time periods of a pendulum independent of mass
of the pendulum?
(b) The motion of a simple pendulum is approximately simple harmonic for small angle oscillations. For larger
angles of oscillation, a more involved analysis shows that T is greater that 2π√l/g. Think of a qualitative
argument to appreciate this result.
(c) A man with a wristwatch on his hand falls from the top of a tower. Does the watch give the correct time
during free fall?
(d) What is the frequency of oscillation of a simple pendulum mounted in a cabin that is freely falling under
gravity?
29. A simple pendulum of length l and having a bob of mass M is suspended in a car. The car is moving on a circular
track of radius R with a uniform speed v. If the pendulum makes small oscillation in a radial direction about its
equilibrium position, what will be its time period?
30. A cylindrical piece of cork of base area A and height h floats in a liquid of density ρL. The cork is depressed slightly
and then released. Show that the cork oscillates up down simple harmonically with a period T = 2π√hρ/ρg, where ρ
is the density of the cork (Ignore damping due to viscosity of the liquid.)
31. A U tube of uniform cross section contains water upto 100 cm on each side. Find the frequency of oscillation if water
on one side is displaced and released. [0.49 Hz]
32. The velocity of a particle in SHM in a straight line is 3 m/s at a distance of 2 m from the mean potion and 2 m/s at 3
m from it. Find the time period of the particle.
[2πs]
33. Write the equation of SHM for particle whose amplitude along X direction is 3 m, frequency 20 Hz and initial phase
π/4.
34. Two linear simple harmonic motions of equal amplitudes and frequencies w and 3w are impressed on a particle along
the ages of X and Y respectively. If the initial phase difference between them is π/2 find the resultant path followed
by the particle.
By: Amrish Maggo
15
XI / PHYSICS
OSCILLATIONS
By: Amrish Maggo
11/PA
16