CLASS XI

NOTES
SCIENCE

Key Notes and Important Questions with Anwers

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 Mechanical
10-A Waves
Oscillation & Waves

10.1 Periodic Motion

A motion, which repeat itself over and over again after a regular interval of
time is called a periodic motion and the fixed interval of time after which
the motion is repeated is called period of the motion. Examples : Revolution
of earth around the sun (period one year).

10.2 Oscillatory or Vibratory Motion.

The motion in which a body moves to and fro or back and forth repeatedly
about a fixed point in a definite interval of time. Oscillatory motion is also
called as harmonic motion. Example : The motion of the pendulum of a wall
clock.

10.3 Harmonic and Non-harmonic Oscillation.

Harmonic oscillation is that oscillation which can be expressed in terms of
single harmonic function (i.e. sine or cosine function). Example : y = a sin
ωt or y = a cos ωt.
Non-harmonic oscillation is that oscillation which can not be expressed in
terms of single harmonic function. Example : y = a sin ωt + b sin 2 ωt.

10.4 Some Important Definitions.

(1) Time period : It is the least interval of time after which the periodic
motion of a body repeats itself. S.l. units of time period is second.
(2) Frequency : It is defined as the number of periodic motions executed
by body per second. S.l unit of frequency is hertz (Hz).
(3) Angular Frequency : 2πn
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(4) Displacement: Its deviation from the mean position.

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 (5) Phase : It is a physical quantity, which completely express the position
and direction of motion, of the particle at that instant with respect to its
mean position.
Y = a sin θ = a sin (ωt + φ0) here θ = ωt + φ0 = phase of vibrating
particle.
(i) Initial phase or epoch : It is the phase of a vibrating particle
at t= 0.
(ii) Same phase: Two vibrating particle are said to be in same phase,
if the phase difference between them is an even multiple of n or
path difference is an even multiple of (λ/2) or time interval is an
even multiple of (T/2).
(iii) Opposite phase : Opposite phase means the phase difference
between the particle is an odd multiple of or the path difference
is an odd multiple of λ or the time interval is an odd multiple of
(T/2).

(iv) Phase difference : If two particles performs S.H.M and their

equation are y1 = a sin (ωt + φ1) and y2 = a sin (ωt + φ2) then phase
difference ∆φ = (ωt + φ2) – (ωt + φ1) = φ2 – φ1

10.5 Simple Harmonic Motion.

Simple harmonic motion is a special type of periodic motion, in which
Restoring force ∝ Displacement of the particle from mean position.

F = – kx

Where k is known as force constant. Its S.l. unit is Newton/meter and

dimension is [MT–2].

10.6 Displacement in S.H.M.

Simple harmonic motion is defined as the projection of uniform circular
motion on any diameter of circle of reference

(i) y = a sin ωt when at t = 0 the vibrating particle is at mean position.

(ii) y = a cos ωt when at t = 0 the vibrating particle is at extreme position.

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(iii) y = a sin (ωt ± φ) when the vibrating particle is φ phase leading or

lagging from the mean position.

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 10.7 Comparative Study of Displacement, Velocity and
Acceleration.
Displacement y = a sin ωt

Velocity v = aω cos ωt

ωt =

Acceleration A = – aω2 sin ωt

ωt = aω2 sin (ωt + π)

(i) All the three quantities displacement, velocity and acceleration show
harmonic variation with time having same period.

(ii) The velocity amplitude is ω times the displacement amplitude

(iii) The acceleration amplitude is ω2 times the displacement amplitude

(iv) In S.H.M. the velocity is ahead of displacement by a phase angle π/2.

(v) In S.H.M. the acceleration is ahead of velocity by a phase angle π/2.

(vi) The acceleration is ahead of displacement by a phase angle of π.

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(vii) Various physical quantities in S.H.M. at different position :

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 Physical quantities Equilibrium position (y = 0) Extreme Position (y = ± a)

Displacement y = a sin ωt Minimum (Zero) Maximum (a)

Velocity Maximum (aω) Minimum (Zero)

Acceleration A = – ω2y Minimum (Zero) Maximum (ω2a)

10.8 Energy in S.H.M.

A particle executing S.H.M. possesses two types of energy : Potential energy
and Kinetic energy

(1) Potential energy :

(i) when y = ± a; ωt = π/2; t = T/4

(ii) when y = 0; ωt = 0; t = 0
(2) Kinetic energy :

or

(i) when y = 0; t = 0; ωt = 0

(ii) when y = a; t = T/4, ωt = π/2

(3) Total energy : Total mechanical energy
= Kinetic energy + Potential energy

Total energy is not a position function i.e. it always remains constant.
(4) Energy position graph :
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 (5) Kinetic energy and potential energy vary periodically double the
frequency of S.H.M.

10.9 Time Period and Frequency of S.H.M.

Time period (T) = as =

Frequency (n) = =
In general m is called inertia factor and k is called spring factor.

Thus T = 2π

10.10 Differential Equation of S.H.M.

For S.H.M. (linear) [As

For angular S.H.M.

10.11 Simple Pendulum

Mass of the bob = m

Effective length of simple pendulum = l ; T = 2π

(i) Time period of simple pendulum is independent of amplitude as long

as its motion is simple harmonic.
(ii) Time period of simple pendulum is also independent of mass of the
bob.
(iii) If the length of the pendulum is comparable to the radius of earth

then 2π

If l >> R (→∞) 1/l< 1/R so 2π 84.6 minutes

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 (iv) The time period of simple pendulum whose point of suspension moving
horizontally with acceleration,

a 2π and θ = tan–1 (a/g)

(v) Second’s Pendulum : It is that simple pendulum whose time period of

vibrations is two seconds.
(vi) Work done in giving an angular displacement θ to the pendulum from
its mean position.
W = U = mgl (1 – cos θ)
(vii) Kinetic energy of the bob at mean position = work done or potential
energy at extreme.

10.12 Spring Pendulum

A point mass suspended from a mass less spring or placed on a frictionless
horizontal plane attached with spring constitutes a linear harmonic spring
pendulum

Time period 2π

2π and Frequency

(i) Time of a spring pendulum is independent of acceleration due to gravity.

(ii) If the spring has a mass M and mass m is suspended from it, effective

mass is given by

So that

(iii) If two masses of mass m1 and m2 are connected

by a spring and made to oscillate on horizontal
surface, the reduced mass mr is given by
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 So that

(iv) If a spring pendulum, oscillating in a vertical plane is made to oscillate

on a horizontal surface, (or on inclined plane) time period will remain
unchanged.

(v) If the stretch in a vertically loaded spring is y0 then

Time period does not depends on ‘g’ because along with g, y0 will also
change in such a way that remains constant.

(vi) Series combination : If n springs of different force constant are

connected in series having force constant k1, k2, k3 ........ respectively

then

(vii) Parallel combination: If the springs are connected in parallel then

keff = k1 + k2 + k3 + ........

(viii) If the spring of force constamt k is divided in to n equal parts then

spring constant of each part will become nk.

(ix) The spring constant k is inversely proportional to the spring length.

As k α

(x) When a spring of length l is cut in two pieces of length l1 and l2 such
that l1 = nl2.

If the constant of a spring is k then spring constant of first part

Spring constant of second part k2 = (n + 1 ) k and ratio of spring constant

.
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 10.13 Various Formulae of S.H.M. .
S.H.M. of a liquid in U tube :
If a liquid of density ρ contained in
a vertical U tube performs S.H.M. in its
two limbs. Then time period

Where L = Total length of liquid

column, H = Height of undisturbed
liquid in each limb (L = 2h)
S.H.M. of a floating cylinder S.H.M. of ball in the neck of an air
chamber
If l is the length of cylinder dipping in
Image
liquid then time period

M = mass of the ball

V = volume of air
chamber
A = area of cross
section of neck
E = Bulk modulus
for Air
S.H.M. of a body in a tunnel dug along S.H.M. of body in the
any chord of earth tunnel dug along the
diameter of earth
minutes

T = 84.6 minutes
R = radius of the earth
= 6400 km
g = acceleration due
to gravity = 9.8 m/s2
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at earth’s surface

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 10.14 Free, Damped, Forced and Maintained Oscillation.
(1) Free oscillation
(i) The oscillation of a particle with fundamental frequency under the
influence of restoring force are defined as free oscillations
(ii) The amplitude, frequency and energy of oscillation remains constant
(iii) Frequency of free oscillation is called natural frequency.
(2) Damped oscillation
(i) The oscillation of a body whose amplitude goes on decreasing with
time are defined as damped oscillation.
(ii) Amplitude of oscillation decreases exponentially due to damping
forces like frictional force, viscous force, hystersis etc.

(3) Forced oscillation

(i) The oscillation in which a body oscillates under the influence of an

external periodic force are known as forced oscillation.

(ii) Resonance: When the frequency of external force is equal to the

natural frequency of the oscillator. Then this state is known as
the state of resonance. And this frequency is known as resonant
frequency.

(4) Maintained oscillation : The oscillation in which the loss of oscillator

is compensated by the supplying energy from an external source are
known as maintained oscillation.

10.15 Wave
A wave is a disturbance which propagates energy and momentum from one
place to the other without the transport of matter.

(1) Necessary properties of the medium for wave propagation :

(i) Elasticity : So that particles can return to their mean position, after
having been disturbed.
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(ii) Inertia : So that particles can store energy and overshoot their mean
position.

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 (iii) Minimum friction amongst the particles of the medium.

(iv) Uniform density of the medium.

(2) Mechanical waves : The waves which require medium for their
propagation are called mechanical waves.

Example : Waves on string and spring, waves on water surface, sound

waves, seismic waves.

(3) Non-mechanical waves : The waves which do not require medium for
their propagation are called non-mechanical or electromagnetic waves.

Examples : Light, heat (Infrared), radio waves, γ-rays. X-rays etc.

(4) Transverse waves : Particles of the medium execute simple harmonic

motion about their mean position in a direction perpendicular to the
direction of propagation of wave motion.
(i) It travels in the form of crests and troughs.
(ii) A crest is a portion of the medium which is raised temporarily.
(iii) A trough is a portion of the medium which is depressed temporarily.
(iv) Examples of transverse wave motion: Movement of string of a sitar,
waves on the surface of water.
(v) Transverse waves can not be transmitted into liquids and gases.

(5) Longitudinal waves: If the particles of a medium vibrate in the direction

of wave motion the wave is called longitudinal.
(i) It travels in the form of compression and rarefaction.
(ii) A compression (c) is a region of the medium in which particles are
compressed.
(iii) A rarefaction (R) is a region of the medium in which particles are
rarefied.
(iv) Examples sound waves travel through air in the form of longitudinal
waves.
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(v) These waves can be transmitted through solids, liquids and gases.

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 10.16 Important Terms
(1) Wavelength :

(i) It is the length of one wave.

(ii) Distance travelled by the wave in one time period is known as

wavelength.

λ = Distance between two consecutive crests or troughs.

(2) Frequency : Number of vibrations completed in one second.

(3) Time period : Time period of vibration of particle is defined as the time
taken by the particle to complete one vibration about its mean position.

(4) Relation between frequency and time period :

Time period = 1 /Frequency

⇒ T = 1/n

(5) Relation between velocity, frequency and wavelength : v = nλ.

10.17 Velocity of Sound (Wave motion)

(1) Speed of transverse wave motion :

(i) On a stretched string : , T = Tension in the string;

m = Linear density of string (mass per unit length).

(ii) In a solid body : (η = Modulus of rigidity; ρ = Density of

the material.)
(2) Speed of longitudinal wave motion :

(i) In a solid long bar (Y = Young’s modulus; ρ = Density)

(ii) In a liquid medium (k = Bulk modulus)

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(iii) In gases

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 10.18 Velocity of Sound in Elastic Medium
Velocity of sound in any medium is

(E = Elasticity of the medium; ρ = Density of the medium)

(1) vsteel > vwater > vair ⇒ 5000 m/s > 1500 m/s > 330 m/s
(2) Newton’s formula : He assumed that propagation of sound is isothermal

As K = Eθ = P; Eθ = Isothermal elasticity; P = Pressure.

By calculation vair = 279 m/sec.

However the experimental value of sound in air is 332 m/sec

(3) Laplace correction : He modified that propagation of sound in air is

adiabatic process.

(As k = Eφ = γρ = Adiabatic elasticity)

v = 331.3 m/s (γAir = 1.41)

(4) Effect of density :

(5) Effect of pressure : Velocity of sound is independent of the pressure

(when T = constant)

(6) Effect of temperature :

When the temperature change is small then vt = v0 (1 + αt)

Value of (Approx.)

(7) Effect of humidity : With rise in humidity velocity of sound increases.

(8) Sound of any frequency or wavelength travels through a given medium

ith the same velocity.
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(9) Sound of any frequency or wavelength travels through a given medium

with the same velocity.

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 10.19 Reflection of Mechanical
Medium Longitudinal Transverse Change in Phase Time Path
wave wave direction change change change

Reflection Compression as Crest as Reversed π

from rigid rarefaction and crest and
end/denser vice-versa Trough as
medium trough

Reflection Compression Crest as No change Zero Zero Zero

from free as compression trough and
end/rarer and rarefaction trough as
medium as rarefaction crest

10.20 Progressive Wave

(1) These waves propagate in the forward direction of medium with a finite
velocity.
(2) Energy and momentum are transmitted in the direction of propagation
of waves.
(3) In progressive waves, equal changes in pressure and density occurs at
all points of medium.
(4) Various forms of progressive wave function.
(i) y = A sin (ωt – kx) Where y = displacement
A = amplitude
ω = angular frequency
n = frequency
k = propagation constant
T = time period
λ = wave length
v = wave velocity
t = instantaneous time
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(ii)

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 (iii)

(iv)

(v)

(a) If the sign between t and x terms is negative the wave is propagating
along positive X-axis and if the sign is positive then the wave moves
in negative X-axis direction.
(b) The Argument of sin or cos function i.e. (ωt – kx) = Phase.
(c) The coefficient of t gives angular frequency

(d) The coefficient of x gives propagation constant or wave number

(e) The ratio of coefficient of t to that of x gives wave or phase velocity,
i.e.
(f) When a given wave passes from one medium to another its frequency
does not change.

(g) From n = constant

(5) Some terms related to progressive waves

(i) Wave number : The number of waves present in unit length.

(ii) Propagation constant (k) :

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and

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 (iii) Wave velocity (v) :

(iv) Phase and phase difference

(v) Phase difference Time difference.

(vi) Phase difference Path difference

⇒ Time difference Path difference.

10.21 Principle of Superposition

If are the displacements at a particular time at a particular

position, due to individual waves, then the resultant displacement,

Important applications of superposition principle: (a) Stationary waves,

(b) Beats.

10.22 Standing Waves or Stationary Waves

When two sets of progressive wave trains of same type (both longitudinal
or both transverse) having the same amplitude and same time period/frequency/
wavelength travelling with same speed along the same straight line in opposite
directions superimpose, a new set of waves are formed. These are called stationary
waves or standing waves.
Characteristics of standing waves :
(1) The disturbance confined to a particular region
(2) There is no forward motion of the disturbance beyond this particular
region.
(3) The total energy is twice the energy of each wave.
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(4) Points of zero amplitude are known as nodes.

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 The distance between two consecutive nodes is .

(5) Points of maximum amplitude is known as antinodes. The distance

between two consecutive antinodes is also λ/2. The distance between a
node and adjoining antinode is λ/4.
(6) The medium splits up into a number of segments.
(7) All the particles in one segment vibrate in the same phase. Particles in
two consecutive segments differ in phase by 180°.
(8) Twice during each vibration, all the particles of the medium pass
simultaneously through their mean position.

10.23 Comparative Study of Stretched Strings, Open Organ

Pipe and Closed Organ Pipe
S. Parameter Stretched string Open organ Closed organ Pipe
No. Pipe

(1) Fundamental
frequency or
1st harmonic
(1st mode of
vibration)
(2) Frequency n2 = 2n1 n2 = 2n1 Missing
of 1st
overtone
or 2nd
harmonic
(2nd mode
of vibration)
(3) Frequency n3 = 3n1 n3 = 3n1 n3 = 3n1
of 2nd
overtone
or 3rd
harmonic
(3rd mode of
vibration)
(4) Frequency 2:3:4: . . . 2:3:4: . . . 3:5:7: . . .
ratio of
overtones
(5) Frequency 1:2:3:4: . . . 1:2:3:4: . . . 1:3:5:7: . . .
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ratio of
harmonics

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 (6) Nature of Transverse Longitudinal Longitudinal
waves stationary stationary stationary

(7) General
formula for
wavelength

(8) Position of x=0,

nodes
(9) Position of x= x=

antinodes

(i) Harmonics are the notes/sounds of frequency equal to or an integral

multiple of fundamental frequency (n).
(ii) Overtones are the notes/sounds of frequency twice/thrice/ four times
the fundamental frequency (n).
(iii) In organ pipe an antinode is not formed exactly at the open end rather
it is formed a little distance away from the open end outside it. The
distance of antinode from the open end of the pipe is = 0.6r (where r is
radius of organ pipe). This is known as end correction.

10.24 Vibration of a String

General formula of frequency

L = Length of string, T = Tension in the string

m = Mass per unit length (linear density), p = mode of vibration

(1) The string will be in resonance with the given body if any of its natural
frequencies concides with the body.

(2) If M is the mass of the string of length L,

So (r = Radius, ρ = Density)
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 10.25 Beats
When two sound waves of slightly different frequencies, travelling in a
medium along the same direction, superimpose on each other, the intensity of the
resultant sound at a particular position rises and falls regularly with time. This
phenomenon is called beats.
(1) Beat period : The time interval between two successive beats (i.e. two
successive maxima of sound) is called beat period.
(2) Beat frequency : The number of beats produced per second is called
beat frequency.
(3) Persistence of hearing : The impression of sound heard by our ears
persist in our mind for 1/10th of a second.
So for the formation of distinct beats, frequencies of two sources of
sound should be nearly equal (difference of frequencies less than 10)
(4) Equation of beats : If two waves of equal amplitudes ‘a’ and slightly
different frequencies n1 and n2 travelling in a medium in the same
direction then equation of beats is given by
y = A sin π (n1 – n2)t where A = 2a cos π (n1 – n2)t = Amplitude of
resultant wave.
Amplitude of resultant wave.
(5) Beat frequency : n = n1 – n2.

(6) Beat period:

n1 – n2

10.26 Doppler Effect

Whenever there is a relative motion between a source of sound and the
listener, the apparent frequency of sound heard by the listener is different
from the actual frequency of sound emitted by the source.

Apparent frequency

Here n = Actual frequency; vL = Velocity of listener; vs = Velocity of source

vm = Velocity of medium and v = Velocity of sound wave

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Sign convention: All velocities along the direction S to L are taken as positive
and all velocities along the direction L to S are taken as negative. If the

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 medium is stationary vm = 0 then

(1) No Doppler effect takes place (n′ = n) when relative motion between
source and listener is zero.
(2) Source and listener moves at right angle to the direction of wave
propagation. (n′ = n)
(i) If the velocity of source and listener is equal to or greater than the
sound velocity then Doppler effect is not observed.
(ii) Doppler effect does not say about intensity of sound.
(iii) Doppler effect in sound is asymmetric but in light it is symmetric.

QUESTIONS
ONE MARK QUESTIONS

1. How is the time period effected, if the amplitude of a simple pendulum is

increased?
2. Define force constant of a spring.
3. At what distance from the mean position, is the kinetic energy in simple
harmonic oscillator equal to potential energy ?
4. How is the frequency of oscillation related with the frequency of change in
the K.E. and P.E. of the body in S.H.M.?
5. What is the frequency of total energy of a particle in S.H.M. ?
6. How is the length of seconds pendulum related with acceleration due to
gravity of any planet ?
7. If the bob of a simple pendulum is made to oscillate in some fluid of density
greater than the density of air (density of the bob > density of the fluid), then
time period of the pendulum increased or decrease.
8. How is the time period of the pendulum effected when pendulum is taken
to hills or in mines ?
9. A transverse wave travels along x-axis. The particles of the medium must
move in which direction ?
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10. Define angular frequency. Give its S.I. unit.

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 11. Sound waves from a point source are propagating in all directions. What
will be the ratio of amplitudes at distances of x meter and y meter from the
source ?
12. Does the direction of acceleration at various points during the oscillation of
a simple pendulum remain towards mean position ?
13. What is the time period for the function f(t) = sin ωt + cos ωt may represent
the simple harmonic motion ?
14. When is the swinging of simple pendulum considered approximately
SHM ?
15. Can the motion of an artificial satellite around the earth be taken as SHM?
16. What is the phase relationship between displacement, velocity and
acceleration in SHM ?
17. What forces keep the simple pendulum in motion ?
18. How will the time period of a simple pendulum change when its length is
doubled ?
19. What is a harmonic wave function ?
20. If the motion of revolving particle is periodic in nature, give the nature of
motion or projection of the revolving particle along the diameter.
21. In a forced oscillation of a particle, the amplitude is maximum for a frequency
w1 of the force, while the energy is maximum for a frequency w2 of the force.
What is the relation between w1 and w2 ?
22. Which property of the medium are responsible for propagation of waves
through it ?
23. What is the nature of the thermal change in air, when a sound wave propagates
through it ?
24. Why does sound travel faster in iron than in water or air ?
25. When will the motion of a simple pendulum be simple harmonic ?
26. A simple harmonic motion of acceleration ‘a’ and displacement ‘x’ is
represented by a + 4π2x = 0. What is the time period of S.H.M ?
27. What is the main difference between forced oscillations and resonance ?
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28. Define amplitude of S.H.M.

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 29. What is the condition to be satisfied by a mathematical relation between time
and displacement to describe a periodic motion ?
30. Why the pitch of an organ pipe on a hot summer day is higher ?
31. Under what conditions does a sudden phase reversal of waves on reflection
takes place ?
32. The speed of sound does not depend upon its frequency. Give an example
in support of this statement.
33. If an explosion takes place at the bottom of lake or sea, will the shock waves
in water be longitudinal or transverse ?
34. Frequency is the most fundamental property of wave, why ?
35. How do wave velocity and particle velocity differ from each other ?
36. If any liquid of density higher than the density of water is used in a resonance
tube, how will the frequency change ?
37. Under what condition, the Doppler effect will not be observed, if the source
of sound moves towards the listener ?
38. What physical change occurs when a source of sound moves and the listener
is stationary ?
39. What physical change occurs when a source of sound is stationary and the
listener moves ?
40. If two sound waves of frequencies 480 Hz and 536 Hz superpose, will they
produce beats? Would you hear the beats ?
41. Define non dissipative medium.

2 MARKS QUESTIONS
42. Which of the following condition is not sufficient for simple harmonic motion
and why ?
(i) acceleration and displacement
(ii) restoring force and displacement
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43. The formula for time period T for a loaded spring, T =

Does the time period depend on length of the spring ?

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 44. Water in a U-tube executes S.H.M. Will the time period for mercury filled
up to the same height in the tube be lesser of greater than that in case of
water ?
45. There are two springs, one delicate and another hard or stout one. For which
spring, the frequency of the oscillator will be more ?
46. Time period of a particle in S.H.M. depends on the force constant K and

mass m of the particle A simple pendulum for small angular

displacement executes S.H.M. approximately. Why then is the time period

of a pendulum independent of the mass of the pendulum ?
47. What is the frequency of oscillation of a simple pendulum mounted in a
cabin that is falling freely ?
48. The maximum acceleration of simple harmonic oscillator is A0. While the
maximum velocity is v0, calculate amplitude of motion.
49. The velocity of sound in a tube containing air at 27°C and pressure of 76
cm of Hg is 330 ms–1. What will be its velocity, when pressure is increased
to 152 cm of mercury and temperature is kept constant ?
50. Even after the breakup of one prong of tunning fork it produces a round
of same frequency, then what is the use of having a tunning fork with two
prongs ?
51. Why is the sonometer box hollow and provided with holes ?
52. The displacement of particle in S.H.M. may be given by y = a sin (ωt + φ)
show that if the time t is increased by 2π/ω, the value of y remains the same.
53. The length of simple pendulum executing SHM is increased by 21%. By
what % time period of pendulum increase ?
54. Define wave number and angular wave number and give their S.l. units.
55. Why does the sound travel faster in humid air ?

56. Use the formula v to explain, why the speed of sound in air

(a) is independent of pressure

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(b) increase with temperature

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 57. Differentiate between closed pipe and open pipe at both ends of same length
for frequency of fundamental note and harmonics.
58. Bats can ascertain distances, directions; nature and size of the obstacle
without any eyes, explain how ?
59. In a sound wave, a displacement node is a pressure antinode and vice- versa.
Explain, why ?
60. How does the frequency of a tuning fork change, when the temperature is
increased ?
61. Explain, why can we not hear an echo in a small room ?
62. What do you mean by reverberation? What is reverberation time ?
3 MARKS QUESTIONS
63. Show that for a particle in linear simple harmonic motion, the acceleration
is directly proportional to its displacement of the given instant.
64. Show that for a particle in linear simple harmonic motion, the average kinetic
energy over a period of oscillation, equals the average potential energy over
the same period.
65. Deduce an expression for the velocity of a particle executing S.H.M. when
is the particle velocity (i) Maximum (ii) minimum?
66. Draw (a) displacement time graph of a particle executing SHM with phase
angle φ equal to zero (b) velocity time graph and (c) acceleration time graph
of the particle.
67. Show that a linear combination of sine and cosine function like x(t) = a sin
ωt + b cos ωt represents a simple harmonic. Also, determine its amplitude
and phase constant.
68. Show that in a S.H.M. the phase difference between displacement and velocity
is π/2, and between displacement and acceleration is π.
69. Derive an expression for the time period of the horizontal oscillations of a
massless loaded spring.
70. Show that for small oscillations the motion of a simple pendulum is simple
harmonic. Derive an expression for its time period.
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71. Distinguish with an illustration among free, forced and resonant oscillations.

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 72. In reference to a wave motion, define the terms
(i) amplitude
(ii) time period
(iii) frequency
(iv) angular frequency
(v) wave length and wave number.
73. What do you understand by phase of a wave? How does the phase change
with time and position.
74. At what time from mean position of a body executive S.H.M. kinetic energy
and potential energy will be equal?

LONG ANSWER QUESTIONS

75. Derive expressions for the kinetic and potential energies of a simple harmonic
oscillator. Hence show that the total energy is conserved in S.H.M. in which
positions of the oscillator, is the energy wholly kinetic or wholly potential ?
76. One end of a U-tube containing mercury is connected to a suction pump and
the other end is connected to the atmosphere. A small pressure difference is
maintained between the two columns. Show that when the suction pump is
removed, the liquid in the U-tube executes S.H.M.
77. Discuss the Newton’s formula for velocity of sound in air. What correction
was applied to it by Laplace and why ?
78. What are standing waves? Desire and expression for the standing waves.
Also define the terms node and antinode and obtain their positions.
79. Discuss the formation of harmonics in a stretched string. Show that in case
of a stretched string the first four harmonics are in the ratio 1:2:3:4.
80. Give the differences between progressive and stationary waves.
81. If the pitch of the sound of a source appears to drop by 10% to a moving
person, then determine the velocity of motion of the person. Velocity of
sound = 30 ms–1.
82. Give a qualitative discussion of the different modes of vibration of an open
organ pipe.
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83. Describe the various modes of vibrations of a closed organ pipe.

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 84. What are beats? How are they produced? Briefly discuss one application
for this phenomenon.
85. Show that the speed of sound in air increases by 61 cms–1 for every 1°C rise
of temperature.
NUMERICALS
86. The time period of a body executing S.H.M is 1s. After how much time will
its displacement be of its amplitude.

87. A particle is moving with SHM in a straight line. When the distance of the
particle from the equilibrium position has values x1 and x2, the corresponding
value of velocities are u1 and u2. Show that the time period of oscillation is
given by

88. Find the period of vibrating particle (SHM), which has acceleration of
45 cm s–2, when displacement from mean position is 5 cm.
89. A 40 gm mass produces on extension of 4 cm in a vertical spring. A mass
of 200 gm is suspended at its bottom and left pulling down. Calculate the
frequency of its vibration.
90. The acceleration due to gravity on the surface of the moon is 1.7 ms–2. What
is the time period of a simple pendulum on the moon, if its time period on
the earth is 3.5 s? [g = 9.8 ms–2]
91. A particle executes simple harmonic motion of amplitude A.
(i) At what distance from the mean position is its kinetic energy equal to
its potential energy?
(ii) At what points is its speed half the maximum speed ?
92. A set of 24 tunning forks is arranged so that each gives 4 beats per second
with the previous one and the last sounds the octave of first. Find frequency
of Ist and last tunning forks.
93. The vertical motion of a huge piston in a machine is approximately S.H.M.
with a frequency of 0.5 s–1. A block of 10kg is placed on the piston. What is
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the maximum amplitude of the piston’s S.H.M. for the block and piston to
remain together ?

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 94. At what temperature will the speed of sound be double its value
at 273°K?
95. 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 spring, when displaced and
released, oscillates with a period of 0.60 s. What is the weight of the body ?
96. If the pitch of the sound of a source appears to drop by 10% to a moving
person, then determine the velocity of motion of the person. Velocity of
sound = 330 ms–1.
97. A body of mass m suspended from a spring executes SHM. Calculate ratio
of K.E. and P.E. of body when it is at a displacement half of its amplitude
from mean position.
98. A string of mass 2.5 kg is under a tension of 200N. The length of the stretched
string is 20m. If a transverse jerk is struck at one end of the string, how long
does the disturbance take to reach the other end ?
99. Which of the following function of time represent (a) periodic and (b) non-
periodic motion? Give the period for each case of periodic motion. [w is any
positive constant].
(i) sin ωt + cos ωt
(ii) sin ωt + sin 2ωt + sin 4 ωt
(iii) e–ωt
(iv) log (ωt)

100. The equation of a plane progressive wave is given by the equation y = 10

sin 2π (t – 0.005x) where y and x are in cm and t in seconds. Calculate the
amplitude, frequency, wave length and velocity of the wave.
101. A tuning fork arrangement (pair) produces 4 beats s–1 with one fork of
frequency 288 cps. A little wax is placed on the unknown fork and it then
produces 2 beats s–1. What is the frequency of the unknown fork ?
102. A pipe 20 cm long is closed at one end, which harmonic mode of the pipe
is resonantly excited by a 430 Hz source? Will this same source can be in
resonance with the pipe, if both ends are open? Speed of sound = 340 ms–1.
103. The length of a wire between the two ends of a sonometer is 105 cm. Where
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should the two bridges be placed so that the fundamental frequencies of the
three segments are in the ratio of 1 : 3 : 15 ?

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 104. The transverse displacement of a string (clamped at its two ends) is
given by

where x, y are in m and t is in s. The length of the string is 1.5 m and its mass
is 3.0 ×10–2 kg. Answer the following.
(a) Does the function represent a travelling or a stationary wave?
(b) Interpret the wave as a superposition of two waves travelling in opposite
directions. What are the wavelength frequency and speed of propagation
of each wave ?
(c) Determine the tension in the string.
105. A wire stretched between two rigid supports vibrates in its fundamental
mode with a frequency 45 Hz. The mass of the wire is 3.5 × 10–2 kg and its
linear density is 4.0 × 10–2 kg m–1. What is (a) the speed of transverse wave
on the string and (b) the tension in the string ?
106. A steel rod 100 cm long is clamped at its middle. The fundamental frequency
of longitudinal vibrations of the rod as given to be 2.53 kHz. What is the
speed of sound in steel ?
107. A progressive wave of frequency 500 Hz is travelling with velocity 360 m/s.
How far apart are two points 60° out of phase ?
108. An observer moves towards a stationary source of sound with a velocity one
fifth of velocity of sound. What is the % increase in apparent frequency ?

ASSERTION - REASON BASED QUESTIONS

Direction:- Read the assertion and reason carefully to mark the correct
option out of the options given below :
(a) If both assertion and reason are true and the reason is the correct explanation
of the assertion.
(b) If both assertion and reason are true but reason is not the correct explanation
of the assertion.
(c) If assertion is true but reason is false.
(d) If the assertion and reason both are false.
(e) If assertion is false but reason is true.
1. Assertion : All oscillatory motions are necessarily periodic motion but all
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periodic motions are not oscillatory.

Reason: Simple pendulum is an example of oscillatory motion.
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 2. Assertion: Simple harmonic motion is a uniform motion.
Reason: Simple harmonic motion is the projection of uniform circular
motion.
3. Assertion: Acceleration is proportional to the displacement. This condition
is not sufficient for motion in simple harmonic.
Reason: In simple harmonic motion direction of displacement is also
considered.
4. Assertion: Sine and cosine functions are periodic functions.
Reason: Sinusoidal functions repeats it values after a definite interval of
time.
5. Assertion: The graph between velocity and displacement for a harmonic
oscillator is a parabola.
Reason: Velocity does not change uniformly with displacement in harmonic
motion.
6. Assertion: When a simple pendulum is made to oscillate on the surface of
moon, its time period Increases.
Reason: Moon is much smaller as compared to earth.
7. Assertion: Resonance is special case of forced vibration in which the
natural frequency of vibration of the body is the same as the impressed
frequency of external periodic force and the amplitude of forced vibration
is maximum.
Reason: The amplitude of forced vibrations of a bod increases with an
increase in the frequency of the externally impressed periodic force.
8. Assertion: The graph of total energy of a particle in SHM w.r.t. position
is a straight line with zero slope.
Reason: Total energy of particle in SHM remains constant throughout its
motion.
9. Assertion: The percentage change in time period is 1.5%, if the length of
simple pendulum increases by 3%.
Reason: Time period is directly proportional to length of pendulum.
10. Assertion: The frequency of a second pendulum in an elevator moving up
with an acceleration half the acceleration due to gravity is 0.612 Hz .
Reason: The frequency of a second pendulum does not depend upon
acceleration due to gravity.
11. Assertion: Damped oscillation indicates loss of energy.
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Reason: The energy loss in damped oscillation may be due to friction, air
resistance etc.

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 HINTS AND ANSWERS
1. (b) Both assertion and reason are correct but reason is not the correct
explanation of assertion.
2. (e) simple harmonic motion, v = ω a 2 − y 2 as y changes,velocity v will also
change. So simple hanllonic motion is not uniform motion. But simple
harmonic motion may be dehned as the projection of uni fonll circular
motion along one of the diameter of the circle.
3. (a) In SHM, the acceleratio n is always in a direction opposite to that of the
displacement i.e., proportional to (–y).
4. (a) A periodic function is one whose value repeats after a dehnite interval
of time. sinq and cosq are periodic functions because they repeat itself
after 2p interval of time.

 2  2
O O

sin curve cos curve

It is also true that moon is smaller than the earth, but this statement is
not explaining the assertion.
5. (e) In SHM, v =
ω a 2 − y2 or v2 = w2y2.
v2 y2
Dividing both sides by w2a2 , 2 2 + =
1. This is the equation of an
ωa a2
ellipse. Hence the graph between v and y is an ellipse not a parabola.
1
6. (b) T = 2π . On moon, g is much smaller compared to g on earth.
g
Therefore, T increases.
7. (c) Amplitude of oscillation for a forced, damped oscillator is
F0 / m
A= , where b is constant related to the strength
(ω 2
)
− ω02 + ( bω / m )
2

of the resistive force, ω0 = k / m is natural frequency of undamped

oscillator (b = 0).
When the Frequency of driving force ω ≈ ω0 , then amplitude A is
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very larger.
For w < w or w > w, the amplitude decrease.
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 8. (a) The total energy of S.H.M. = Kinetic energy of particle +
potential energy of particle.
The variation of total energy of the particle in SHM with time is shown
in a graph.

9. (c) Time period of simple pendulum of length is,

l ∆T 1 ∆l
T =2π ⇒T∝ l ⇒ =
g T 2 l

∆T 1
∴ = × 3 = 1.5%
T 2

10. (c) Frequency of second pendulum n = (1 / 2)s–1. When elevator is
movmg upwards with acceleration g/2, the effective acceleration due
to gravity is
g = g + a = g +g / 2 = 3g / 2.
1 g
=
As n so n 2 ∝ g.
2π l
n2 g 3g / 2 3 n1 3
∴ 12 = 1 = = or = =1.225
n g g 2 n 2
2

or, nl = 1.225n = 1.225 × (l / 2) = 0.612s–1.

11. (b) Energy of damped oscillator at an any instant t is given by

1
E = E0e–bt/m [where E0 = kx2 = maximum energy]
2
Due to damping forces the amplitude of oscillator will go on decreasing
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with time whose energy is expressed by above equation.

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 CASE STUDY BASED QUESTIONS

Simple Pendulum
An ideal simple pendulum consists of a heavy point mass body (bob)
suspend d by a weightless, inextensible and perfectly flexible string from
a rigid support about which it is free to oscillate.
But in reality neither point mass nor weightless string exist, so we can
never construct a simple pendulum strictly according to the definition.
Suppose simple pendulum of length l is displaced through a small
angle q from it's mean (vertical) position. Consider m as of the bob is m
and linear displacement from mean position is x.

Answer the following questions :-

1. The period of a simple pendulum is doubled, when
(a) Its length is doubled
(b) The mass of the bob is doubled
(c) Its length is made four times
(d) The mass of the bob and the length of the pendulum are doubled

2. The period of oscillation of a simple pendulum of constant length at

earth surface is T. Its period inside a mine is
(a) Greater than T
(b) Less than T
(c) Equal to T
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(d) Cannot be compared

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 3. A pendulum suspended from the ceiling of a train has a period T,
when the train is at rest. When the train is accelerating with a uniform
acceleration a, the period of oscillation will
(a) Increase (b) Decrease
(c) Remain unaffected (d) Become infinite

4. Which of the following statements is not true? In the case of a simple

pendulum for small amplitudes the period of oscillation is
(a) Directly proportional to square root of the length of the pendulum
(b) Inversely proportional to the square root of the acceleration due
to gravity
(c) Dependent on the mass, size and material of the bob
(d) Independent of the amplitude

5. The time period of a second's pendulum is 2 sec. The spherical bob

which is empty from inside has a mass of 50 gm. This is now replaced
by another solid bob of same radius but having different mass of
100 gm. The new time period will be
(a) 4 sec (b) l sec (c) 2sec (d) 8sec

HINTS AND ANSWERS

l
1. (c) T =2π ⇒T∝ l
g
2. (a) Inside the mine g decreases
Hence from T = 2π l ; T increases
g

3. (b) Initially time period was T = 2π l .

g
When train acceleration, the effective
value of g becomes (g 2
)
+ a 2 which
is greater than g.
Hence, new time period, becomes
less than the initial time period.
4. (c)
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l
5. (c) T = 2π (independent of mass)
g

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 WAVES
ASSERTION - REASON BASED QUESTIONS

Direction:- Read the assertion and reason carefully to mark the correct
option out of the options given below :
(a) If both assertion and reason are true and the reason is the correct explanation
of the assertion.
(b) If both assertion and reason are true but reason is not the correct explanation
of the assertion.
(c) If assertion is true but reason is false.
(d) If the assertion and reason both are false.
(e) If assertion is false but reason is true.

1. Assertion : Two persons on the surface of moon cannot talk to each other.
Reason: There is no atmosphere on moon.
2. Assertion: Transverse waves are not produced in liquids and gases.
Reason: Light waves are transverse waves.
3. Assertion: Sound waves cannot propagate through vacuum but light waves
can.
Reason: Sound waves cannot be polarised but light waves can be polarised.
4. Assertion: The velocity of sound increases with increase in humidity.
Reason: Velocity of sound does not depend upon the medium.
5. Assertion: Ocean waves hitting a beach are always found to be nearly
normal to the shore.
Reason: Ocean waves are longitudinal waves.
6. Assertion: Compression and rarefaction involve changes in density and
pressure.
Reason: When particles are compressed, density of medium increases and
when they are rarefied, density of medium decreases.
7. Assertion: Transverse waves travel through air in an organ pipe.
Reason: Air possesses only volume elasticity.
8. Assertion: Sound would travel faster on a hot summer day than on a cold
winter day.
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Reason: Velocity of sound is directly proportional to the square of its

absolute temperature.

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 9. Assertion: The basic of Laplace correction was that, exchange of heat
between the region of compression and rarefaction in air is not possible.
Reason: Air is a bad conductor of heat and velocity of sound in air is large.
10. Assertion: Particle velocity and wave velocity both are independent of
time.
Reason: For the propagation of wave motion, the medium must have the
properties of elasticity and inertia.
11. Assertion: When we start filling an empty bucket with water, the pitch of
sound produced goes on decreasing.
Reason: The frequency of man voice is usually higher than that of woman.
12. Assertion: A tuning fork is made of an alloy of steel, nickel and chromium.
Reason: The alloy of steel, nickel and chromium is called elinvar.

HINTS AND ANSWERS

1. (a) Sound waves require material medium to travel. As there is no atmosphere
(vacuum) on the surface of moon, therefore the sound waves cannot reach
from one person to another.
2. (b) Transverse waves travel in the form of crests and troughs involving
change in shape of the medium. As liquids and gases do not possess the
elasticity of shape, therefore, transverse waves cannot be produced in
liquid and gases. Also, light wave is one example of transverse wave.
3. (b) Sound waves cannot propagate through vacuum because sound waves
are mechanical waves. Light waves can propagate through vacuum
because light waves are electromagnetic waves. Since sound waves are
longitudinal waves, the particles move in the direction of propagation,
therefore these waves cannot be polarised.
K γρ
4. (c) Velocity of sound in gas medium is=v = is ratio of its principal
ρ ρ
heat capacities ( Cp / C v ) . For moist air r is less than that for dry air and
γ is slightly greater.
∴ velocity of sound increases with increase in humidity.
5. (c) Ocean waves are transverse waves travelling in concentric circles of
ever-increasing radius. When they hit the shore, their radius of curvature
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is so large that they can be treated as plane waves. Hence, they hit the
shore nearly normal to the shore.

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 6. (a) A compression is a region of medium in which particles come closer i.e.,
distance between the particles becomes less than the normal distance
between them. Thus, there is a temporary decrease in volume and a
consequent increase in density of medium. Similarly in rarefaction,
particle get farther apart and a consequent decrease in density.
7. (e) Since transverse wave can propagate through medium which possess
elasticity of shape. Air posses only volume elasticity therefore transverse
wave cannot propagate through air.
8. (c) The velocity of sound in a gas is directly proportional to the square root
 γRT 
of its absolute temperature  as v =  . Since temperature of a hot
 M 
day is more than cold winter day, therefore sound would travel faster
on a hot summer day than on a cold winter day.
9. (c) According to Laplace, the changes in pressure and volume of a gas, when
sound waves propagated through it, are not isothermal, but adiabatic.
A gas is a bad conductor of heat. It does not allow the free exchange of
heat between compressed layer, rarefied layer and surrounding.
10. (e) The velocity of every oscillating particle of the medium is different of
its different positions in one oscillation but the velocity of wave motion
is always constant i.e., particle velocity vary with respect to time, while
the wave velocity is independent of time.
Also for wave propagation medium must have the properties of elasticity
and inertia.
11. (d) A bucket can be treated as a pipe closed at one end. The frequency of
v
the note produced l = , here L equal to depth of water level from the
4L
open end. As the bucket is filled with water L decreases, hence frequency
increases. Therefore, frequency or pitch of sound produced goes on
increasing. Also, the frequency of woman voice is usually higher than
that of man.
12. (b) A tuning fork is made of a material for which elasticity does not change.
Since the alloy of nickel, steel and chromium (elinvar) has constant
elasticity, therefore it is used for the preparation of tuning fork.
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 CASE STUDY BASED QUESTIONS
I. Doppler's Effect or Doppler Shift is the change in frequency of a wave
in relation to an observer who is moving relative to the wave source. It is
named after the Austrian physicist Christian Doppler, who described the
phenomenon in 1842.
Whenever there is a relative motion between a source of sound and the
observer (listener), the frequency of sound heard by the observer is different
from the actual frequency of sound emitted by the source. The frequency
observed by the observer is called the apparent frequency. It may be less
than or greater than the actual frequency emitted by the sound source. The
difference depends on the relative motion between the source and observer.
A common example of Doppler shift is the change of pitch heard when
a vehicle sounding a horn approaches and recedes from an observer.
Compared to the emitted frequency, the received frequency is higher during
the approach, identical at the instant of passing by, and lower during the
recession.
Answer the following questions :-

1. Doppler shift in frequency does not depend upon
(a) The frequency of the wave produced
(b) The velocity of the source
(c) The velocity of the observer
(d) Distance from the source to the listener
2. A source of sound of frequency 450 cycles/sec is moving towards
a stationary observer with 34 m/sec speed. If the speed of sound is
340 m/sec, then the apparent frequency will be
(a) 410 cycles/sec (b) 500 cycles/sec
(c) 550 cycles/sec (d) 450 cycles/sec
3. The wavelength is 120 cm when the source is stationary. If the source
is moving with relative velocity of 60 m/sec towards the observer,
then the wavelength of the sound wave reaching to the observer will
be (velocity of sound = 330 m/s)
(a) 98 cm (b) 140 cm (c) 120 cm (d) 144 cm
4. The frequency of a whistle of an engine is 600 cycles/sec is moving
with the speed of 30 m/sec towards an observer. The apparent frequency
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will be (velocity of sound = 330 m/s)

(a) 600 cps (b) 660 cps (c) 990 cps (d) 330 cps

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 5. A source of sound emits waves with frequency f Hz and speed
V m/sec. Two observers move away from this source in opposite
directions each with a speed 0.2 V relative to the source. The ratio
of frequencies heard by the two observers will be
(a) 3 : 2 (b) 2 : 3 (c) 1 : 1 (d) 4 : 10

II. Standing Wave in a Organ Pipe

Organ pipes are the musical instrument which are used for producing
musical sound by blowing air into the pipe. Longitudinal stationary
waves are Formed on account of superimposition of incident and reflected
longitudinal waves.
Equation of standing wave y = 2a cos
2πvt 2πx
sin
λ λ
v
Frequency or vibration n =
λ

Answer the following questions :-

1. A tube closed at one end and containing air is excited. It produces
the fundamental note of frequency 512 Hz. If the same tube is open
at both the ends the fundamental frequency that can be produced is
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(a) 1024 Hz (b) 512 Hz (c) 256 Hz (d) 128 Hz

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 2. A closed pipe and an open pipe have their first overtones identical
in frequency. Their lengths are in the ratio
(a) 1 : 2 (b) 2 : 3 (c) 3 : 4 (d) 4 : 5
3. The first overtone in a closed pipe has a frequency
(a) Same as the fundamental frequency of an open tube of same length
(b) Twice the fundamental frequency of an open tube of same length
(c) Same as that of the first overtone of an open tube of same length
(d) None of the above
4. An empty vessel is partially filled with water, then the frequency of
vibration of air column in the vessel
(a) Remains same (b) Decreases
(c) Increases (d) First increases then decreases
5. It is desired to increase the fundamental resonance frequency in a
tube which is closed at one end. This can be achieved by
(a) Replacing the air in the tube by hydrogen gas
(b) Increasing the length of the tube
(c) Decreasing the length of the tube

HINTS AND ANSWERS

I. 1. (d)
 v   340 
2.=
(b) n ' n=
  450  =  500 cycles / sec.
 v − vO   340 − 34 

 v   v − vs 
3. (a) n' = n  ⇒ λ' = λ 
 v − vs   v 
 330 − 60 
= ⇒ λ ' 120
=   98 cm.
 330 
 v   330 
4.=
(b) n ' n=
 =
 600   660 cps.
 v − vS   300 
5. (c) Both listeners, hears the same frequencies.
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 II. 1. (a) Fundamental frequency of open pipe is double that of the closed
pipe.
2. (c) If is given that
First over tone of closed pipe = First over tone of open pipe
 v   v 
⇒ 3  = 2 ;
 4l1   2l2 
where l and l are the lengths of closed and open organ pipes hence
l1 3
=
l2 4

3v
3. (d) First overtone for closed pipe =
4l
v
Fundamental frequency for open pipe =
2v 2l
First overtone for open pipe = .
2l
v 1
4. (c) For closed pipe in general=n (2N − 1) ⇒ n ∝
4l l
i.e. if length of air column decreases frequency increases.
5.
v
(a,c,d) Fundamental frequency for closed pipe n =
4l
γRT 1
where v = ⇒v∝
M M

 M H < M air ⇒ v H > vair

2 2

Hence, fundamental frequency with H will be more as compared

to air. So option (a) is correct.
1
Aslo n ∝ , hence if l decrease n increases so option (c) is
correct. l
It is well known that (n) = 2(n) hence option (d) is correct.
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 SOLUTIONS
ANSWERS OF ONE MARK QUESTIONS
1. No effect on time period when amplitude of pendulum is increased or
decreased.
2. The spring constant of a spring is the change in the force it exerts, divided
by the change in deflection of the spring. (K = f/x)

3. At x = a/ ., KE = PE =
4. P.E. or K.E. completes two vibrations in a time during which S.H.M.
completes one vibration or the frequency of R.E. or K.E is double than that
of S.H.M.
5. The frequency of total energy of particle is S.H.M. is zero because it remains
constant.
6. Length of the seconds pendulum proportional to (acceleration due to gravity)
7. Increased
1
8. As T α , T will increase.
g
9. In the y-z plane or in plane perpendicular to x-axis.
10. It is the angle covered per unit time or it is the quantity obtained by
multiplying frequency by a factor of 2π.
ω = 2πn, S.I. unit is rad s–1.

11. Intensity = amplitude2 ∝

Required ratio = y/x

12. No, the resultant of Tension in the string and weight of bob is not always
towards the mean position.
13. T =2π/ω
14. Swinging through small angles.
15. No, it is a circular and periodic motion but not SHM.
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16. In SHM, The velocity leads the displacement by a phase π/2 radians and
acceleration leads the velocity by a phase π/2 radians.

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 17. The component of weight (mg sin θ).

18. times, as
19. A harmonic wave function is a periodic function whose functional form is
sine or cosine.
20. S.H.M.
21. Both amplitude and energy of the particle can be maximum only in the case
of resonance, for resonance to occur ω1 = ω2.
22. Properties of elasticity and inertia.
23. When the sound wave travel through air adiabatic changes take place in the
medium.
24. Sound travel faster in iron or solids because iron or solid is highly elastic as
compared to water (liquids) or air (gases).
25. When the displacement of bob from the mean position is so small that
sin θ ≈ θ.
26.

27. The frequency of external periodic force is different from the natural
frequency of the oscillator in case of forced oscillation but in resonance two
frequencies are equal.
28. The maximum displacement of oscillating particle on either side of its mean
position is called its amplitude.
29. A periodic motion repeats after a definite time interval T.
So, y(t) = y(t + T) = y(t + 2T) etc.
30. On a hot day, the velocity of sound will be more since (frequency proportional
to velocity) the frequency of sound increases and hence its pitch increases.
31. On reflection from a denser medium, a wave suffers a sudden phase reversal.
32. If sounds are produced by different musical instruments simultaneously, then
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all these sounds are heard at the same time.

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 33. Explosion at the bottom of lake or sea create enormous increase in pressure
of medium (water). A shock wave is thus a longitudinal wave travelling at
a speed which is greater than that of ordinary wave.
34. When a wave passes through different media, velocity and wavelength change
but frequency does not change.
35. Wave velocity is constant for a given medium and is given by V = nλ. But
particle velocity changes harmonically with time and it is maximum at mean
position and zero at extreme position.
36. The frequency of vibration depends on the length of the air column and not
on reflecting media, hence frequency does not change.
37. Doppler effect will not be observed, if the source of sound moves towards
the listener with a velocity greater than the velocity of sound. Same is also
true if listener moves with velocity greater than the velocity of sound towards
the source of sound.
38. Wave length of sound changes.
39. The number of sound waves received by the listener changes.
40. Yes, the sound waves will produce 56 beats every second. But due to
persistence of hearing, we would not be able to hear these beats.
41. A medium in which speed of wave motion is independent of frequency of
wave is called non-dispersive medium. For sound, air is non dispersive
medium.
ANSWERS OF TWO MARKS QUESTIONS

42. Condition (i) is not sufficient, because direction of acceleration is not

mentioned. In SHM, the acceleration is always in a direction opposite to
that of the displacement.

43. Although length of the spring does not appear in the expression for the time
period, yet the time period depends on the length of the spring. It is because,
force constant of the spring depends on the length of the spring.

44. The time period of the liquid in a U-tube executing S.H.M. does not depend
upon density of the liquid, therefore time period will be same, when the
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mercury is filled up to the same height in place of water in the U-tube.

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 45. We have,

So, when a hard spring is loaded with a mass m. The extension I will be
lesser w.r.t. delicate one. So frequency of the oscillation of the hard spring
will be more and if time period is asked it will be lesser.
46. Restoring force in case of simple pendulum is given by

So force constant itself proportional to m as the value of k is substituted in
the formula, m is cancelled out.
47. The pendulum is in a state of weightlessness i.e. g = 0. The frequency of
pendulum

48. Amax = ω2a = A0, Umax = ωa = v0

⇒ .

49. At a given temperature, the velocity of sound is independent of pressure, so

velocity of sound in tube will remain 330 ms–1.

50. Two prongs of a tunning fork set each other in resonant vitorations and help
to maintain the vibrations for a longer time.

51. When the stem of the a tunning fork gently pressed against the top of
sonometer box, the air enclosed in box also vibrates and increases the
intensity of sound. The holes bring the inside air incontact with the outside
air and check the effect of elastic fatigue.
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 52. The displacement at any time t is

y = a sin (ωt + φ)

displacement at any time (t + 2π/ω) will be

y = a sin [ω (t + 2π/ω) + φ] = [sin {ωt + φ) + 2π}]

y = a sin (ωt + φ) [ sin (2π + φ) = sin φ]

Hence, the displacement at time t and (t + 2π/ω) are same.

53. When a number of waves travel through the same region at the same time,
each wave travels independently as if all other waves were absent.

This characteristic of wave is known as independent behaviour of waves.

For example we can distinguish different sounds in a full orchestra.

54. Wave number is the number of waves present in a unit distance of medium.
S.I. unit of k is rad m–1.
Angular wave number or propagation constant is 2π/λ. It represents phase
change per unit path difference and denoted by k = 2π/λ. S.I. unit of k is
rad m–1.
55. Because the density of water vapour is less than that of the dry air hence
density of air decreases with the increase of water vapours or humidity and
velocity of sound inversely proportional to square root of density.

56. Given,

(a) Let V be the volume of 1 mole of air, then

or
for 1 mole of air PV = RT

or

⇒ ....(i)
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So at constant temperature v is constant as γ, R and M are constant.

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 (b) From equation (i) we know that so with the increase in
temperature velocity of sound increases.
57. (i) In a pipe open at both ends, the frequency of fundamental note produced
is twice as that produced by a closed pipe of same length.
(ii) An open pipe produces all the harmonics, while in a closed pipe, the
even harmonics are absent,
58. Bats emit ultrasonic waves of very small wavelength (high frequencies) and
so high speed. The reflected waves from an obstacle in their path give them
idea about the distance, direction, nature and size of the obstacle.
59. At the point, where a compression and a rarefaction meet, the displacement is
minimum and it is called displacement node. At this point, pressure difference
is maximum i.e. at the same point it is a pressure antinode. On the other
hand, at the mid point of compression or a rarefaction, the displacement
variation is maximum i.e. such a point is pressure node, as pressure variation
is minimum at such point.
60. As the temperature increases, the length of the prong of the tunning fork
increases. This increases the wavelength of the stationary waves set up in

the tunning fork. As frequency, so frequency of the tunning fork

decreases.
61. For an echo of a simple sound to be heard, the minimum distance between
the speaker and the walls should be 17 m, so in any room having length less
than 17 m, our ears can not distinguish between sound received directly and
sound received after reflection.
62. The phenomenon of persistence or prolongation of sound after the source
has stopped emitting sound is called reverberation. The time for which the
sound persists until it becomes inaudible is called the reverberation time.
SOLUTION / HINTS OF NUMERICALS

86. y = r sin ωt = r sin

Here and T = 1s
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 87. When

When

As

or ....(i)

and or ....(ii)
Subtracting (ii) from (i), we get

or ω=

T=

88. Here y = 5 cm and acceleration a = 45 cm s–2

We know a = ω2y

or rad s–1

and T =
89. Here mg = 40 g = 40 × 980 dyne ; l = 4 cm.
say k is the force constant of spring, then
mg = kl or k = mg/l

k= dyne cm–1
when the spring is loaded with mass m = 200 g

v=
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= 1.113 s–1.

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 90. Here on earth, T = 3.5 s; g = 9.8 ms–2

For simple pendulum

....(i)
on moon, g′ = 1.7 ms–2 and if T′ is time period

then ....(ii)
Dividing eqn. (ii) by eqn. (i), we get

or

91. (i)

(ii)
92. Let frequency of Ist tunning fork = x
frequency of IInd tunning fork = x + 4
frequency of IIIrd tunning fork = x + 2 (4)
frequency of IVth tunning fork = x + 3 (4)
Let frequency of 24th tunning fork = x + 23 (4)
octave means, (twice in freq.)
freq. of 24th = 2 × freq. of Ist = 2x
2x = x + 23 (4) ⇒ x = 92
freq. of 24th = 2 × 92 = 184 H3.
93. Given, v = 0.5 s–1, g = 9.8 ms–1

amax at the extreme position i.e., r = y

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amax = 4π2v2r and amax = g to remain in contact.

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 or

94. Say v1 in the velocity of sound at T1 = 273°K and v2 = 2v1 at temperature T2

Now

or T2 = 4 × 273 = 1092°K.
95. Here m = 50 kg, l = 0.2 m

we know mg = kl or Nm–1

T = 0.60 s and M is the mass of the body, then using

T= kg

Weight of body Mg = 22.34 × 9.8 = 218.93 N.

96. Apparent freq.

or

v = 330 ms–1

v0 = 330 – 297 = 33 m/s.

97. KE

at
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KE =

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 PE =

98. Given T = 200 N, length of string l = 20 m

total mass of the string = 2.5 kg
mass per unit length of the string

kg m–1

Now v = ms–1
Hence time taken by the transverse wave to reach other end

t =

99. (i) sin ωt + cos ωt =

It is simple harmonic function with period

(ii) sin ωt + sin 2ωt + sin 4ωt is a periodic but not simple harmonic function.

Its time period is .

(iii) e–ωt is exponential function, which never repeat itself. Hence it is non-
periodic function.
(iv) log ωt is also non-periodic function.
100. Here y =

y = ....(i)
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 The equation of a travelling wave is given by

y = ....(ii)
Comparing the equation (i) and (ii), we have
α = 10 cm, λ = 200 cm and v = 200 ms–1

Now v =

101. Unknown freq. = Known freq. I Beat freq.

= 288 ± 4 = 292 or 284 Hz
On putting wax, freq. decreases, beat freq. is also descrease to 2
unknown freq. = 292 Hz (higher one)
102. The frequency of nth mode of vibration of a pipe closed at one end is given
by

vn =
river v = 340 ms–1, L = 20 cm = 0.2 m; vn = 430 Hz

430 =

Therefore, first mode of vibration of the pipe is excited, for open pipe since
n must be an integer, the same source can not be in resonance with the pipe
with both ends open.

103. Total length of the wire, L = 105 cm

v1 : v2 : v3 = 1 : 3 : 15

Let L1, L2 and L3 be the length of the three parts. As v ∞

L1 : L2 : L3 =
Sum of the ratios = 15 + 5 + 1 = 21

L1 = cm; L2 = cm;
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 L3 = cm
Hence the bridges should be placed at 75 cm and (75 + 25) = 100 cm from
one end.

104. ....(i)

(a) The displacement which involves harmonic functions of x and t separately

represents a stationary wave and the displacement, which is harmonic
function of the form (vt ± x), represents a travelling wave. Hence, the
equation given above represents a stationary wave.

(b) When a wave pulse travelling along x-axis is

superimposed by the reflected pulse.

from the other end, a stationary wave is formed and

is given by

y = ....(ii)
Comparing the eqs. (i) and (ii), we have

or λ = 3m

= 120π or v = 60λ = 60 × 3 = 180 ms–1

Now frequency γ = Hz
(c) Velocity of transverse wave in a string is given by

Here m= kgm–1
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Also v = 180 ms–1

T = v2 m = (180)2 × 2 × 10–2 = 648N.

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 105. Frequency of fundamental mode, v = 45Hz
Mass of wire M = 3.5 × 10–2 kg; mass per unit length, m = 4.0 × 10–2 kgm1

Length of wire L =

(a) For fundamental mode L = or λ = 2L = 0.875 × 2 = 1.75 m

velocity v = vλ = 45 × 1.75 = 78.75 ms–1
(b) The velocity of transverse wave

v =

106. Given : u = 2.53 kHz = 2.53 × 103 Hz

(L) Length of steel rod = 100 cm = 1 m.
when the steel rod clamped at its middle executes longitudinal vibrations of
its fundamental frequency, then

L= or λ = 2L = 2 × 1 = 2 m
The speed of sound in steel
v = nλ = 2.53 × 103 × 2 = 5.06 × 103 ms–1.

107. ∆φ = 60° = 60 rad.

v = m

As ∆φ =

∆x =
= 0.12 m.

108. v0 =
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Apparent freq. =

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 =

% change =

OBJECTIVE QUESTIONS
109. The periodic time of a body executing simple harmonic motion is 3s. After
how much interval from t = 0, its displacement will be half of its amplitude?
1 1 1 1 1 1 1 1
(a) s s s s (b) s s s s
8 6 4 3 8 6 4 3
1 1 1 1 1 1 1 1
s(c) s s s s s (d)s s
8 6 4 3 8 6 4 3
110. Two equations of two SHM y = a Sin (ωt–α) and y = a Cos (ωt–α). The
phase difference between the two is
(a) 0° (b) α°
(c) 90° (d) 180°
111. If a simple pendulum oscillates with an amplitude of 50 mm and time period
of 2s, its maximum velocity is
(a) 0.10 m/s (b) 0.15 m/s
(c) 0.8 m/s (d) 0.26 m/s
112. The equation of simple harmonic motion y = a sin (2π t + α) then its phase
at time t is
(a) 2πn t (b) α
(c) 2π t + α (d) 2π t
113. The equation of simple harmonic motion y = a sin (2π t + α) then its phase
at time t = 0s is
(a) 2πn t (b) α
(c) 2π t + α (d) 2π t
114. A particle is oscillating according to the equation x = 7 cos (0.5π t), where t
is in second. The point moves from the position of equilibrium to maximum
displacement in time
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(a) 4s (b) 2s
(c) 1s (d) 0.5s

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 115. The instantaneous displacement of a simple pendulum oscillator is given
by x = A cos  ωt + π  . If speed will be maximum at time
 4 π π π 2π
π π π 2π
(a) (b)
4ω 2ω ω ω 4ω 2ω ω ω
π π π 2π π π π 2π
(c) (d)
4ω 2ω ω ω 4ω 2ω ω ω
116. The velocity of particle in SHM at displacement y from mean position is
(a) w (a 2 + y 2 ) w (a 2 −wy 2 ) (a w+(b)
2 2 2
) 2 −wy 2 )(a 2 − y 2 )
y (a w 2 (a 2 − y 2 )

(c) wy (d) w 2 a 2 + y2

117. A particle is executing SHM with a period of T seconds and amplitude a

meter. The shortest time it takes to reach a point a2 m from its mean position
in seconds is
T T T
(a) T (b)
4 8 16
T T T
(c) T T T (d)
4 8 16 4 8 16
118. Displacement between maximum potential energy position and maximum
kinetic energy position for a particle executing SHM is
(a) –a (b) +a
a
(c) ±a (d) ±
4
119. If tension in the string is increased from 1 KN to 4 KN, other factors
remaining unchanged, the frequency of the second harmonic will
(a) be halved (b) main changed
(c) be doubled (d) becomes four times
120. An open organ pipe and a closed organ pipe have the frequency of their first
overtone identical. What is the ratio of their lengths?
1 4 3 1 4 3
(a) (b)
2 3 4 2 3 4
1 4 3
(c) (d) 1
2 3 4
121. The fundamental frequency of a stretched string is v0. If the length is reduced
by 35% and tension increased by 69% the fundamental frequency will be
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(a) 0.2 v0 (b) 0.5 v0

(c) 2.0 v0 (d) 1.6 v0

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 122. Two waves of same frequency traveling in the same medium in opposite
direction when super imposed give rise to
(a) beats (b) harmonics
(c) standing waves (d) resonance
 π
123. Equation of a progressive wave is given by y = 0.2 cos π (0.04 t + 0.02 x) − 6  The
 
distance is expressed in cm and time in second. What will be the minimum
distance between two particles having the phase difference of π/2?
(a) 4 cm (b) 8 cm
(c) 25 cm (d) 12.5 cm
124. For two systems to be in resonance, which of the following properties should
be equal?
(a) Wavelength (b) Frequency
(c) Amplitude (d) Wave velocity
125. Fundamental frequency of a sonometer wire is n. If the length, diameter
and tension are doubled, the new fundamental frequency will be
n n
(a) n (b) 2n
2 2 2
n n n n
(c)
2n 2n (d)
2 2 2 2 2 2
126. The frequency of an open organ pipe is v. If half part of organ pipe is dipped
in water then its frequency is
ν ν
(a) v (b)
4 2
(c)
ν ν (d) O
4 2
127. Two tuning forks when sounded together given one beat every 0.2 s. What
is the difference of frequencies?
(a) 0.2 (b) 2
(c) 5 (d) 10
128. Angle between wave velocity and particle velocity of a longitudinal
wave is
(a) 90° (b) 60°
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(c) 0° (d) 120°

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 Answer : (Objective Type Questions)

109. (c) 110. (c) 111. (b) 112. (c) 113. (b) 114. (c)

115. (a) 116. (b) 117. (c) 118. (c) 119. (c) 120. (c)

121. (c) 122. (c) 123. (c) 124. (b) 125. (d) 126. (a)

127. (c) 118. (c)

HINTS :

a 1
=
109. y a =
sin wt as y =we get t s=
(Given T 3s)
2 4
5 2π
111. V=
max a=
w ×
100 2
m/s

T
114. wt = 0.5 π t ⇒ w = 0.5 π ⇒ T= 4s req. time = = 1s
4
1
=
117. y a=
sin wt y
2
119. να T
1 γP
120. For open pipe, frequency of I overtone, ν1 =
L1 P
3 γP
For closed organ pipe, frequency of I overtone, ν 2 =
4L 2 P

1 T 1 T + 69% of T
121. ν 0 = Frequency in new cond. ν = 2(65% of L) M
2L M
λ
123. Req. distance = 4

****
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