Oscilations and Waves Notes and MCQs

Notes 59 Pages

Joint Entrance Examination (Main)

Physics

Contributed by

Yash Kuruvilla

321
Unit - 10
Ocsillations And
Waves
Page 1
322
SUMMARY
1. Waves : The motion of the disturbance in the medium (or in free space) is called wave pulse or
generally a wave.
2. Amplitude of a wave : Amplitude of oscillation of particles of the medium is called the amplitude of
a wave.
3. Wavelength and frequency : The linear distance between any two points or particles having phase
difference of 2

rad is called the wavelength
(λ)
of the wave.
Frequency of wave is just the frequency of oscillation of particles of the medium. Relation between
wavelength and frequency :
v = f, =
k


where, v is the speed of wave in the medium.
4. Mechanical waves : The waves which require elastic medium for their transmission are called
mechanical waves, e.g. sound waves.
5. Transverse and longitudinal waves : Waves in which the oscillations are in a direction perpendicular
to the direction of wave propagation are called the transverse wave.
Waves in which the oscillations of the particles of medium are a!cng the direction of wave propagation
are called longitudinal waves.
6. Wave Equation : The equation which describe the displacement for any particle of medium at a required
time is called wave equation. Various forms of wave equations are as follows :
(i)
y A sin ( t - kx) 
(ii)
t x
y A sin –
T
 

 

 
(iii)
x
y A sin 2 t –
v
 
 
 
 
(iv)
 
2
y A sin vt x

 

The above equations are for the wave travelling in the direction of increasing value of x. If the wave
is travelling in the direction of decreasing value of x then put '+' instead of '—' in above equations.
7. The elasticity and inertia of the medium are necessary for the propagation of the mechanical waves.
8. The speed of the transverse waves in a medium like string kept under tension,
T
v 

where, T = Tension in the string and (I = mass per unit length of the string -- y
9. Speed of sound waves in elastic medium,
E
v 

where, E = Elastic constant of a medium,

= Density of the medium.
Speed of longitudinal waves in a fluid,
B P
v

 
 
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323
where, B — Bulk modulus of a medium
p
V
C
y
C

= 1.41 (for air)
Speed of longitudinal waves in a linear medium like a rod,
v



where,
= 
Young modulus,

= Density of a medium
At constant pressure and constant humidity, speed of sound waves in gas is directly proportional to
the square root of its absolute temperature.
RT
v v T
M

  
The speed of sound in a gas does not depend on the pressure variation.
10. Principle of Superposition : When a particle of medium comes under the influence of two or more
waves simultaneously, its net displacement is the vector sum of displacement that could occur under
the influence of the individual waves.
11. Stationary Waves : When two waves having same amplitude and frequency and travelling in mutually
opposite directions are superposed the resultant wave formed loses the property of propagation. Such
a wave is .called a stationary wave.
Equation of stationary wave : y = – 2 A sinkx cos t
Amplitude of stationary wave : 2 A sin kx
Position of nodes in stationary wave x
n
=
n
2

where, n = 1,2, 3.....At all these points the amplitude is zero.
Position of antinodes in stationary wave's,
x
n
= (2n – 1)
4

where , n — 1, 2, 3,....
The amplitude of all these points is 2A.
12. Frequencies corresponding to different normal modes of vibration in a stretched string of length L
fixed at both the ends are given by,

n
nv n T
f
2L 2L
 

where n — 1, 2, 3......
13. In a closed pipe the values of possible wavelengths required for stationary wave pattern are given by.
4L
n
(2n - 1)
 
and possible frequencies,
n 1
v
f (2n – 1) (2n – 1) f
4L
 
where, n = 1, 2, 3,..... and L = length of pipe.
In a closed pipe only odd harmonics f
1
, 3f
1
, 5f
1
.... are possible.
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14. In an open pipe the values of possible wavelength required for stationary waves are given by,
2L
n
n
 
and possible frequencies,
n 1
nv
f nf
2L
 
where, n — 1, 2, 3,......and
L - length of pipe.
In open pipe of the harmonics like f
1
, 2f
1
, 3f
1
..... are possible.
15. Beat: The phenomenon of the loudness of sound becoming maximum periodically due to superposition
of two sound waves of equal amplitude and slightly different frequencies is called the 'beats'.
Number of beats produced in unit time = f
1
– f
2
.
16. Doppler Effect : Whenever there is a relative motion between a source of sound and a listener with
respect to the medium in which the waves are propagating the frequency of sound experienced by the
listener is different from that which is emitted by the source. This phenomenon is called Doppler
effect.
Frequency listened by the listener,
L
L
S
S
v v
f f
v v



Where, v = velocity of sound, v
L
= velocity of a listener,
v
S
= velocity of a source, f
S
= frequency of sound emitted by the source.
17. If a body repeats its motion along a certain path, about a fixed point, at a definite interval of time, it is
said to have periodic motion.
18. If a body moves to and fro, back and forth, or up and down about a fixed point in a fixed interval of
time, such a motion is called an oscillatory motion.
19. When a body moves to and fro repeatedly about an equilibrium position under a restoring force, which
is always directed towards equilibrium position and whose magnitude at any instant is directly
proportional to the displacement of the body from the equilibrium position of that instant then such a
motion is known as simple harmonic motion.
20. The maximum displacement of the oscillator on cither side of mean position is called amplitude of the
oscillator.
21. The time taken by the oscillator to complete one oscillation is known as periodic time or time period or
period (T) of the oscillator.
22. The number of oscillation completed by the simple harmonic oscillator in one second is known as its
frcquency(f).
23. 2

times the frequency of oscillator is the angular frequency CO of that oscillator..
24.
1 2 1 2
T or f or =
f T T
 
   

25. For simple harmonic motion, the displacement y(t} of a particle from its equilibrium position is represented
by sine, cosine or its linear combination like
y( t ) = A sin ( t + )
y(t) = B cos ( t + )
y( t ) A' sin t + B' cos t
whereA' Acos and B' = Bsin
 


  
  
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325
26. The velocity of SHO is given by
2 2
v A – y  
27. The acceleration of SHO is given by a = -
2
y
28. A particle of mass m oscillating under the influence of Hook's Law exhibits simple harmonic motion
with
k
;
m
 
m
T 2
k
 
29. Differential equation for SHM is
2
2
d y
+ y = 0
dth

30. For scries combination of n spring of spring constants k
1
, k
2
, k
3
,..., k
n
, the equivalent spring constant is
1 2 n
1 1 1 1
= = ...
k k k k

the periodic time
m
T 2
k
 
31. For parallel combination of n springs of spring constants k ky k ...., kn, the equivalent spring
constant is
k = k
1
+ k
2
+ k
3
+ .... + k
n
and period
m
T 2
k
 
32. The kinetic energy of the SHO is K =
2
1
m
2

(A
2
- y
2
)
33. The potential energy of the SHO is U =
2
1
ky
2
34. The total mechanical energy of SHO is E = K + U =
2
1
m
2

A
2
=
2
1
kA
2
35. For SHO, at y — 0, the potential energy is minimum (U = 0) and the kinetic energy is maximum
2
1
( K kA E )
2
 
36. For SHO, at y =

A, the potential energy is maximum
2
1
(U kA E )
2
 
and the kinetic energygy
is minimum (K = 0)
37. Simple harmonic motion is the projection of uniform circular motion on a diameter of the reference
circle.
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326
38. For simple pendulum, for small angular displacement
1
T = 2 and
g

2 g
2 f =
T l

   
39. For simple pendulum, T is independent of the mass of the bob as well as the amplitude of the oscillaions.
40. The differential equaiton for damped harmonic oscillation is
with the displacement
2
2
d y dy
m b = + ky = 0
dt
dt

and angular frequency
2
2
k b
' –
m 4m
 
41.
2 –etlm
1
E( t ) kA e
2

gives the mechanical energy of damped oscillation at time t.
42. A system oscillates under the influence of external periodic force are forced oscillations.
43. The differential equation for forced oscillations is
0
2
2
d y b dy k F
+ y = sin t
m dt m m
dt
 
44. The amplitude for forced oscillation is
0
1
2 2 2 2 2
2
0
F
A
[m ( – ) + b ]

  
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327
MCQ
For the answer of the following questions choose the correct alternative from among the given ones.
SECTION – I:
1. If the equation for a particle performing S.H.M. is given by y = Sin2t +
3
Cos2t, its periodic time
will be ………..s.
(A) 21 ( B ) p (C) 2p (D) 4p.
2. The distance travelled by a particle performing S.H.M. during time interval equal to its periodic time
is ……..
(A) A (B) 2A (C) 4A (D) Zero.
3. A person standing in a stationary lift measures the periodic time of a simple pendulum inside the lift
to be equal to T. Now, if the lift moves along the vertically upward direction with an acceleration of
3
g
, then the periodic time of the lift will now be ………
(A)
T3
(B)
T
2
3
(C)
3
T
(D)
3
T
4. If the equation for displacement of two particles executing S.H.M. is given by y
1
= 2Sin(10t+è)
and y
2
= 3Cos10t respectively, then the phase difference between the velocity of two particles
will be ………..
(A) – è (B) è (C)
2



(D)
2



.
5. When a body having mass m is suspended from the free end of two
springs suspended from a rigid support, as shown in figure, its periodic
time of oscillation is T. If only one of the two springs are used, then
the periodic time would be ………
(A)
2
T
(B)
2
T
(C)
T2
(D) 2T
6. If the maximum velocity of two springs ( both has same mass ) executing S.H.M. and having force
constants k
1
and k
2
respectively are same, then the ratio of their amplitudes will be ……….
(A)
2
1
k
k
(B)
1
2
k
k
(C)
2
1
k
k
(D)
1
2
k
k
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328
7. As shown in figure, two masses of 3.0 kg and 1.0 kg are attached at the two ends of a spring having
force constant 300 N m
- 1
. The natural frequency of oscillation for the system will be …………hz.
( Ignore friction )
(A) ¼ (B) 1/3
(C) 4 (D) 3
8.
The bob of a simple pendulum having length ‘
l’ is displaced from its equilibrium position by an angle
of è and released. If the velocity of the bob, while passing through its equilibrium position is v, then
v = ………..
(A)
)1(2

Cosgl 
(B)
)1(2

Singl 
(C)
)1(2

Singl 
(D)
)1(2

Cosgl 
9. If
4
1
of a spring having length l is cutoff, then what will be the spring constant of remaining part?
(A) k (B) 4k (C)
3
4k
(D)
4
3k
10. The amplitude for a S.H.M. given by the equation x = 3Sin3pt + 4Cos3pt is ………m.
(A) 5 (B) 7 (C) 4 (D) 3.
11. When an elastic spring is given a displacement of 10mm, it gains an potential energy equal to U. If
this spring is given an additional displacement of 10 mm, then its potential energy will be ……..
(A) U (B) 2U (C) 4U (D) U/4.
12. The increase in periodic time of a simple pendulum executing S.H.M. is ………….when its length
is increased by 21%.
(A) 42 % (B) 10% (C) 11% (D) 21%.
13. A particle executing S.H.M. has an amplitude A and periodic time T. The minimum time required by
the particle to get displaced by
2
A
from its equilibrium position is …….. s.
(A) T (B) T/4` (C) T/8 (D) T/16.
14. If a body having mass M is suspended from the free ends of two springs A and B, their periodic time
are found to be T
1
and T
2
respectively. If both these springs are now connected in series and
if the same mass is suspended from the free end, then the periodic time is found to be T.
Therefore …………..
(A) T = T
1
+ T
2
(B)
21
111
TTT

(C) T
2
= T
1
2
+ T
2
2
(D) 2
2
2
1
2
111
TT
T

.
3kg 1kg
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329
15. The displacement of a S.H.O. is given by the equation x = A Cos ( ùt +
8

). At what time will it attain
maximum velocity?
(A)


8
3
(B)


3
8
(C)


16
3
(D)


16
.
16. At what position will the potential energy of a S.H.O. become equal to one third its kinetic energy?
(A)
2
A

(B)
2
A

(C)
3
A

(D)
A3
.
17. Three identical springs are shown in figure. When a 4 kg
mass is suspended from spring A, its length increases by
1cm. Now if a 6 kg mass is suspended from the free end
of spring C, then increase in its length is ………cm.
(A) 1.5 (B) 3.0
(C) 4.5 (D) 6.0.
18. For particles A and B executing S.H.M., the equation for displacement is given by y
1
=
0.1Sin(100t+p/3) and y
2
= 0.1Cospt respectively. The phase difference between velocity of
particle A with respect to that of B is …………
(A)
3


(B)
6

(C)
6


(D)
3

19. The periodic time of a simple pendulum is T
1
. Now if the point of suspension of this pendulum
starts moving along the vertical direction according to the equation y = kt
2
, the periodic time
of the pendulum becomes T
2
. Therefore,
2
2
2
1
T
T
= …( k = 1 m/s
2
& g= 10 m/s
2
)
(A) 6/5 (B) 5/6 (C) 4/5 (D) 1
20. A hollow sphere is filled with water. There is a hole at the bottom of this sphere. This sphere is
suspended with a string from a rigid support and given an oscillation. During oscillation, the
hole is opened up and the periodic time of this oscillating system is measured. The periodic
time of the system………….
(A) will remain constant
(B) Will increase upto a certain time
(C) Increases initially and then decreases to attain its initial periodic time
(D) Initially decreases and then will attain the initial periodic time value.
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330
21. The periodic time of a S.H.O. oscillating about a fixed point is 2 s. After what time will the kinetic
energy of the oscillator become 25% of its total energy?
(A) 1/12 s (B) 1/6 s (C) ¼ s (D) 1/3 s.
22. A body having mass 5g is executing S.H.M. with an amplitude of 0.3 m. If the periodic time of the
system is
10

s, then the maximum force acting on body is ……….
(A) 0.6 N (B) 0.3 N (C) 6 N (D) 3 N
23. As shown in figure, a body having mass m is attached with two springs having spring constants k
1
and k
2
. The frequency of oscillation is f. Now, if the springs constants of both the springs are
increased 4 times, then the frequency of oscillation will be equal to ………….
(A) 2f (B) f/2
(C) f/4 (D) 4f
24. The figure shows a graph of displacement versus time for a particle executing S.H.M. The acceleration
of the S.H.O. at the end of time t =
3
4
second is ………..cm.s
– 2
(A)
2
32
3

(B)
32
2


(C)
32
2

(D)
2
32
3


25. As shown in figure, the object having mass M is executing S.H.M. with an amplitude A. The
amplitude of point P shown in figure will be …….
(A)
2
1
k
Ak
(B)
1
2
k
Ak
(C)
21
1
kk
Ak

(D)
21
2
kk
Ak

26. A particle is executing S.H.M. between x = - A and x = +A. If the time taken by the particle to travel
from x = 0 to A/2 is T
1
and that taken to travel from x = A/2 to x = A is T
2
, then ……….
(A) T
1
< T
2
( B ) T
1
> T
2
(C) T
1
= 2T
2
(D) T
1
= T
2
27. For a particle executing S.H.M., when the potential energy of the oscillator becomes 1/8 the maximum
potential energy, the displacement of the oscillator in terms of amplitude A will be ……….
(A)
2
A
(B)
22
A
(C)
2
A
(D)
23
A
.
Page 10

Oscilations and Waves Notes and MCQs

Oscilations and Waves Notes and MCQs

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