Objective Physics for NEET

Abhay Kumar

Chapter 11

Oscillations and Waves - all with Video Answers

Educators

Chapter Questions

Problem 1

Which of the following expressions does not represent SHM?
(a) $A \cos \omega t$
(b) $A \sin 2 \omega t$
(c) $A \sin \omega t+B \cos \omega t$
(d) $A \sin ^{2} \omega t$

Ajay Singhal

Ajay Singhal

Numerade Educator

Problem 2

The differential equation of a particle executing simple harmonic motion along $y$ -axis is
(a) $\frac{d^{2} y}{d t^{2}}+\omega^{2} y=0$
(b) $\frac{d^{2} y}{d t^{2}}+\omega^{2} y^{2}=0$
(c) $\frac{d^{2} y}{d t^{2}}-\omega^{2} y=0$
(d) $\frac{d y}{d t}+\omega y=0$

Ajay Singhal

Numerade Educator

Problem 3

A simple harmonic motion having an amplitude $A$ and time period $T$ is represented by the equation $y=5 \sin \pi(t+4) \mathrm{m}$
Then the values of $A$ (in $\mathrm{m}$ ) and $T$ (in sec) are
(a) $A=5 ; T=2$
(b) $A=10 ; T=1$
(c) $A=5 ; T=1$
(d) $A=10 ; T=2$

Ajay Singhal

Numerade Educator

Problem 4

If the maximum velocity and acceleration of a particle executing SHM are equal in magnitude, the time period will be
(a) $1.57$ seconds
(b) $3.14$ seconds
(c) $6.28$ seconds
(d) $12.56$ seconds

Ajay Singhal

Numerade Educator

Problem 5

How long after the beginning of motion is the displacement of a harmonically oscillating point equal to one half its amplitude, if the period is 24 seconds and initial phase is zero?
(a) 12 seconds
(b) 2 seconds
(c) 4 seconds
(d) 6 seconds

Ajay Singhal

Numerade Educator

Problem 6

A particle is executing SHM with amplitude $A$ and has maximum velocity $V_{a t}$. Its speed at displacement $A / 2$ will be
(a) $(\sqrt{3}) V_{o} / 2$
(b) $V_{o} / \sqrt{2}$
(c) $V_{o}$
(d) $V_{o} / 4$

Ajay Singhal

Numerade Educator

Problem 7

A particle under the action of a SHM has a period of 3 seconds and under the effect of another it has a period 4 seconds. What will be its period under the combined action of both the SHM's in the same direction?
(a) 7 seconds
(b) 5 seconds
(c) $2.4$ seconds
(d) $0.4$ seconds

Ajay Singhal

Numerade Educator

Problem 8

The displacement $x$ of a particle in motion is given in terms of time by $x(x-4)=1-5 \cos \omega t$.
(a) The particle executes SHM.
(b) The particle executes oscillatory motion which is not SHM.
(c) The motion of the particle is neither oscillatory nor simple harmonic.
(d) The particle is not acted upon by a force when it is at $x=4$.

Ajay Singhal

Numerade Educator

Problem 9

The equation of SHM is given as $x=3 \sin 20 \pi t+4 \cos 20 \pi t$
where $x$ is in $\mathrm{cms}$ and $t$ is in seconds. The amplitude is
(a) $7 \mathrm{~cm}$
(b) $4 \mathrm{~cm}$
(c) $5 \mathrm{~cm}$
(d) $3 \mathrm{~cm}$

Ajay Singhal

Numerade Educator

Problem 10

What should be the displacement of a simple pendulum whose amplitude is $A$, at which potential energy is $\frac{1}{4}$ th of the total energy?
(a) $\frac{A}{\sqrt{2}}$
(b) $\frac{A}{2}$
(c) $\frac{A}{4}$
(d) $\frac{A}{2 \sqrt{2}}$

Ajay Singhal

Numerade Educator

Problem 11

A particle is executing SHM with amplitude $A$ and has a maximum velocity $V_{o^{\circ}}$. The displacement at which its velocity will be ( $\left.V_{a} / 2\right)$ and the velocity at displacement $A / 2$ are
(a) $\frac{A}{2}, \frac{V_{e}}{2}$
(b) $\frac{A}{3}, \frac{V_{n}}{3}$
(c) $\left(\frac{\sqrt{3}}{2}\right) A, \frac{\sqrt{3} V}{2}$
(d) $\frac{A}{\sqrt{2}}, \frac{V_{a}}{\sqrt{2}}$

Ajay Singhal

Numerade Educator

Problem 12

A particle executes simple harmonic motion between $x$ $=-A$ and $x=+A$. The time taken for it to go from 0 to $A / 2$ is $T_{1}$ and to go from $A / 2$ to $A$ is $T_{2}$. Then
(a) $T_{1}<T_{2}$
(b) $T_{1}>T_{2}$
(c) $T_{1}=T_{2}$
(d) $T_{1}=2 T_{2}$

Ajay Singhal

Numerade Educator

Problem 13

A body executes simple harmonic motion under the action of a force $F$, with a time period $(4 / 5)$ seconds. If the force is changed to $F_{2}$ it executes SHM with time period $(3 / 5)$ seconds. If both the forces $F_{1}$ and $F_{2}$ act simultaneously in the same direction on the body, its time period (in seconds) is
(a) $12 / 25$
(b) $24 / 25$
(c) $35 / 24$
(d) $25 / 12$

Ajay Singhal

Numerade Educator

Problem 14

The potential energy of a particle of mass $1 \mathrm{~kg}$ in motion along the $x$ -axis is given by $U=4(1-\cos 2 x) \mathrm{J}$, where $x$ is in metres. The period of small oscillations (in second) is
(a) $2 \pi$
(b) $\pi$
(c) $\frac{\pi}{2}$
(d) $\sqrt{2} \pi$

Ajay Singhal

Numerade Educator

Problem 15

A particle executing SHM while moving from one extremity is found at distances $x_{1}, x_{2}$ and $x_{3}$ from the centre at the end of three successive seconds. The time period of oscillation is
(a) $2 \pi / \theta$
(b) $\pi^{\prime} \theta$
(c) $\theta$
(d) $\pi / 2 \theta$
Where $\theta=\cos ^{-1}\left(\frac{x_{1}+x_{3}}{2 x_{2}}\right)$

Ajay Singhal

Numerade Educator

Problem 16

If velocity of SHM is plotted with displacement, which of the following figure should be the nearest graph
(a)
(b)
(c)
(d)

Ajay Singhal

Numerade Educator

Problem 17

The equation of motion of a particle executing simple harmonic motion is $a+16 \pi^{2} x=0 .$ In this equation, $a$ is the linear acceleration in $\mathrm{m} / \mathrm{s}^{2}$ of the particle at a displacement $x$ in metre. The time period in simple harmonic motion is
(a) $\frac{1}{4}$ second
(b) $\frac{1}{2}$ second
(c) 1 second
(d) 2 seconds

Ajay Singhal

Numerade Educator

Problem 18

The $x$ -t graph of a particle undergoing simple $x(\mathrm{~cm})$ harmonic motion is shown below. The acceleration of the particle at $t=\frac{4}{3} \mathrm{~s}$ is
(a) $\frac{\sqrt{3}}{32} \pi^{2} \mathrm{~cm} / \mathrm{s}^{2}$
(b) $\frac{-\pi^{2}}{32} \mathrm{~cm} / \mathrm{s}^{2}$
(c) $\frac{\pi^{2}}{32} \mathrm{~cm} / \mathrm{s}^{2}$
(d) $-\frac{\sqrt{3}}{32} \pi^{2} \mathrm{~cm} / \mathrm{s}^{2}$

Ajay Singhal

Numerade Educator

Problem 19

If $\langle T>$ and $\langle U>$ denote the average kinetic and the average potential energies respectively of a mass executing a simple harmonic motion, over one period, then the corresponding relation is
(a) $\langle T\rangle=-2\langle U\rangle$
(b) $\langle T\rangle=+2\langle U\rangle$
(c) $\langle T\rangle=\langle U\rangle$
(d) $\langle U\rangle=2\langle T\rangle$

Ajay Singhal

Numerade Educator

Problem 20

The maximum displacement of the particle executing SHM is $1 \mathrm{~cm}$ and the maximum acceleration is $1.57$ $\mathrm{cm} / \mathrm{s}^{2}$. Its time period is
(a) $0.25 \mathrm{~s}$
(b) $4.0 \mathrm{~s}$
(c) $1.57 \mathrm{~s}$
(d) $3.14 \mathrm{~s}$

Ajay Singhal

Numerade Educator

Problem 21

Time period of a simple pendulum is $T .$ If its length increases by $2 \%$, the new time period becomes
(a) $0.98 T$
(b) $1.02 T$
(c) $0.99 T$
(d) $1.01 T$

Ajay Singhal

Numerade Educator

Problem 22

If $x, v$ and $a$ denote the displacement, the velocity and the acceleration of a particle executing simple harmonic motion of time period $T$, then, which of the following does not change with time?
(a) $a^{2} T^{2}+4 \pi^{2} v^{2}$
(b) $\frac{a T}{x}$
(c) $a T=2 \pi v$
(d) $\frac{a T}{v}$

Ajay Singhal

Numerade Educator

Problem 24

If a simple pendulum of length $/$ has maximum angular displacement $\theta$, then the maximum kinetic energy of the bob of mass $m$ is
(a) $(1 / 2) m \sqrt{1 / g}$
(b) $(1 / 2)(m g / I)$
(c) $m g l(1-\cos \theta)$
(d) $(1 / 2) m g l \sin \theta$

Ajay Singhal

Numerade Educator

Problem 25

A simple pendulum consisting of a ball of mass $m$ tied to a string of length $l$ is made to swing on a circular arc of angle $\theta$ in a vertical plane. At the end of this arc, another ball of mass $m$ is placed at rest. The momentum translated to this ball at rest by the swinging ball is
(a) zero
(b) $m \theta \sqrt{l} / g$
(c) $m \theta / \sqrt{l} g$
(d) $\frac{m}{2} \sqrt{(l / g)}$

Ajay Singhal

Numerade Educator

Problem 26

For a particle executing simple harmonic motion, the kinetic energy $K$ is given by $K=K_{o} \cos ^{2} \omega t$. The maximum value of potential energy is
(a) $K_{e}$
(b) zero
(c) $K_{d} / 2$
(d) not obtainable

Ajay Singhal

Numerade Educator

Problem 27

A simple 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, the time period of the simple pendulum will
(a) decrease
(b) increase
(c) remain unchanged
(d) become infinite

Ajay Singhal

Numerade Educator

Problem 28

A simple pendulum is set up in a trolley which moves to the right with an acceleration $a$ on the horizontal plane. Then, the thread of the pendulum in the mean position makes an angle $\theta$ with the vertical given by
(a) $\tan ^{-1}(a / g)$ in the forward direction
(b) $\tan ^{-1}(a / g)$ in the backward direction
(c) $\tan ^{-1}(g / a)$ in the backward direction
(d) $\tan ^{-1}(g / a)$ in the forward direction

Ajay Singhal

Numerade Educator

Problem 29

A heavy brass sphere is hung from a spring and it executes vertical vibrations with period $T$. The sphere is now immersed in a non-viscous liquid with a density (1/10) th that of brass. When set into vertical vibrations with the sphere remaining inside liquid all the time, the time period will be
(a) $\sqrt{9 / 10} T$
(b) $\sqrt{10 / 9} T$
(c) $(9 / 10) T$
(d) unchanged

Ajay Singhal

Numerade Educator

Problem 30

A smooth inclined plane having angle of inclination of $30^{\circ}$ with the horizontal has a $2.5 \mathrm{~kg}$ mass held by a spring which is fixed at the upper end. If the mass is taken $2.5 \mathrm{~cm}$ up along the surface of the inclined plane, the tension in the spring reduces to zero. If the mass is now released, the angular frequency of oscillation is
(a) 7
(b) 14
(c) $0.7$
(d) $1.4$

Ajay Singhal

Numerade Educator

Problem 31

Two pendulums of lengths $121 \mathrm{~cm}$ and $100 \mathrm{~cm}$ start vibrating. At some instant the two are in the mean position in the same phase. After how many vibrations of the shorter pendulum the two will be in phase in the mean position?
(a) 10
(b) 11
(c) 20
(d) 21

Ajay Singhal

Numerade Educator

Problem 32

A pendulum bob carries a $-$ ve charge $-q$. A positive charge $+q$ is held at the point of support. Then, the time period of the bob is
(a) greater than $2 \pi \sqrt{L / g}$
(b) less than $2 \pi \sqrt{L / g}$
(c) equal to $2 \pi \sqrt{L / g}$
(d) equal to $2 \pi \sqrt{2 L / g}$

Ajay Singhal

Numerade Educator

Problem 33

A uniform vertical field $E$ is established between two parallel plates. In this field, a small conducting sphere of mass $M$ is suspended from a string of length $l$. If the sphere is given a charge $+q$ (statcoulomb) and if lower plate is charged positively, the period of the simple pendulum is
(a) $2 \pi \sqrt{l / g}$
(b) $2 \pi \sqrt{\frac{l}{\left(g+\frac{q E}{m}\right)}}$
(c) $2 \pi \sqrt{\frac{l}{\left(g-\frac{q E}{m}\right)}}$
(d) $2 \pi \sqrt{\frac{l}{\left(\frac{q E}{m}-g\right)}}$

Ajay Singhal

Numerade Educator

Problem 34

A simple pendulum has time period $T_{1}$. The point of suspension is now moved upward according to the relation $y=k t^{2}\left(k=1 \mathrm{~m} / \mathrm{s}^{2}\right)$, where $y$ is the vertical displacement. The time period now becomes $T_{2}$. The ratio of $T_{1}^{-2} / T_{2}^{2}$ is (Take $g=10 \mathrm{~m} / \mathrm{s}^{2}$ )
(a) $6 / 5$
(b) $5 / 6$
(c) 1
(d) $4 / 5$

Ajay Singhal

Numerade Educator

Problem 35

A simple pendulum of length $L$ has an energy $E$ and amplitude $A .$ The energies of the simple pendulum
(i) when the length is doubled but with same amplitude and (ii) when the amplitude is doubled but with the same length, are respectively
(a) $2 E, 2 E$
(b) $\frac{E}{2}, \frac{E}{2}$
(c) $\frac{E}{2}, 2 E$
(d) $\frac{E}{2}, 4 E$

Ajay Singhal

Numerade Educator

Problem 36

A pendulum has period $T$ for small oscillations. An obstacle is placed directly beneath the pivot, so that only the lowest one quarter of the string can follow the pendulum bob when it swings in the left of its resting position as shown in the figure. The pendulum is released from rest at a certain point $A$. The time taken by it to return to that point is
(a) $T$
(b) $T / 2$
(c) $3 T / 4$
(d) $T / 4$

Ajay Singhal

Numerade Educator

Problem 37

Which of the following characteristics does not change due to the damping of simple harmonic motion?
(a) Angular frequency
(b) Time period
(c) Initial phase
(d) Amplitude

Ajay Singhal

Numerade Educator

Problem 38

A mass $m$ is suspended from two springs of spring constants $k_{1}$ and $k_{2}$ as shown. The time period of vertical oscillations of the mass will be
(a) $2 \pi \sqrt{\left(\frac{k_{1}+k_{2}}{m}\right)}$
(b) $2 \pi \sqrt{\frac{m}{\left(k_{1}+k_{2}\right)}}$
(c) $2 \pi \sqrt{\frac{m k_{1} k_{2}}{\left(k_{1}+k_{2}\right)}}$
(d) $2 \pi \sqrt{\frac{m\left(k_{1}+k_{2}\right)}{\left(k_{1} k_{2}\right)}}$

Ajay Singhal

Numerade Educator

Problem 39

Two masses $m_{1}$ and $m_{2}$ are suspended together by a massless spring of constant $k$. When the masses are in equilibrium, $m_{1}$ is removed without disturbing the system; the amplitude of vibration is
(a) $m_{1} g / k$
(b) $m_{2} g / k$
(c) $\frac{\left(m_{1}+m_{2}\right) g}{k}$
(d) $\frac{\left(m_{2}-m_{1}\right) g}{k}$

Ajay Singhal

Numerade Educator

Problem 40

A uniform spring has an unstretched length $l$ and $a$ force constant $k$. The spring is cut into two parts of unstretched length $l_{1}$ and $l_{2}$ such that $l_{1}=\eta l_{2}$, where $\eta$ is an integer. The corresponding force constants $k_{1}$ and $k_{2}$ are
(a) $k \eta$ and $k(\eta+1)$
(b) $\frac{k(\eta+1)}{\eta}$ and $k(\eta-1)$
(c) $\frac{k(\eta-1)}{\eta}$ and $k(\eta+1)$
(d) $\frac{k(\eta+1)}{\eta}$ and $k(\eta+1)$

Ajay Singhal

Numerade Educator

Problem 41

A particle vibrates in SHM along a straight line. Its greatest acceleration is $5 \pi^{2} \mathrm{~cm} \mathrm{~s}^{-2}$ and when its distance from the equilibrium position is $4 \mathrm{~cm}$, the velocity of the particle is $3 \pi^{2} \mathrm{~cm} \mathrm{~s}^{-2}$. The amplitude and the period of oscillation of the vibrating particle is
(a) $10 \mathrm{~cm}, 4$ seconds
(b) $5 \mathrm{~cm}, 2$ seconds
(c) $5 \mathrm{~cm}, 4$ seconds
(d) $10 \mathrm{~cm}, 2$ seconds

Ajay Singhal

Numerade Educator

Problem 42

In the arrangement, spring constant $k$ has value $2 \mathrm{~N}$ $\mathrm{m}^{-1}$, mass $M=3 \mathrm{~kg}$ and mass $m=1 \mathrm{~kg}$. Mass $M$ is in contact with a smooth surface. The coefficient of friction between two blocks is $0.1$. The time period of SHM executed by the system is
(a) $\pi \sqrt{6}$
(b) $\pi \sqrt{2}$
(c) $2 \sqrt{2} \pi$
(d) $2 \pi$

Ajay Singhal

Numerade Educator

Problem 43

A particle of mass $m$ is attached to three identical springs $A, B$ and $C$ each of force constant $k$ as shown in figure. If the particle of mass $m$ is pushed slightly against the spring $A$ and released, then the time period of oscillation is
(a) $2 \pi \sqrt{\frac{2 m}{k}}$
(b) $2 \pi \sqrt{\frac{m}{2 k}}$
(c) $2 \pi \sqrt{\frac{m}{k}}$
(d) $2 \pi \sqrt{\frac{m}{3 k}}$

Ajay Singhal

Numerade Educator

Problem 44

A uniform circular disc of mass $12 \mathrm{~kg}$ is held by two identical springs as shown in the figure. When the disc is pressed down slightly and released, it executes SHM with a time period of 2 seconds. The force constant of each spring is
(a) $236 \mathrm{~N} / \mathrm{m}$
(b) $118.3 \mathrm{~N} / \mathrm{m}$
(c) $59.15 \mathrm{~N} / \mathrm{m}$
(d) None of these

Ajay Singhal

Numerade Educator

Problem 45

To make the frequency double of an oscillator, we have to
(a) double the mass
(b) half the mass
(c) quadruple the mass
(d) reduce the mass to one-fourth

Ajay Singhal

Numerade Educator

Problem 46

Two identical springs of constant $k$ are connected in series and parallel as shown in the figure. A mass $M$ is suspended from them. The ratio of their frequencies of vertical oscillations will be
(a) $2: 1$
(b) $1: 2$
(c) $1: 4$
(d) $4: 1$

Ajay Singhal

Numerade Educator

Problem 47

When a body of mass $1.0 \mathrm{~kg}$ is suspended from a certain light spring hanging vertically, its length increases by $5 \mathrm{~cm}$. By suspending $2.0 \mathrm{~kg}$ block to the spring and if the block is pulled through $10 \mathrm{~cm}$ and released, the maximum velocity of it (in $\mathrm{m} / \mathrm{s}$ ) is
(a) $0.5$
(b) 1
(c) 2
(d) 4

Ajay Singhal

Numerade Educator

Problem 48

A block of mass $m$ compresses a spring of stiffness $k$ through a distance $I / 2$ as shown in the figure. If the block is not fixed to the spring, the period of motion of the block is
(a) $2 \pi \sqrt{m / k}$
(b) $(\pi+4) \sqrt{m / k}$
(c) $(1+\pi) \sqrt{m / k}$
(d) None of these

Ajay Singhal

Numerade Educator

Problem 49

Two identical particles each of mass $m$ are interconnected by a light spring of stiffness $k$, the time period for small oscillation is equal to
(a) $2 \pi \sqrt{\frac{m}{k}}$
(b) $\pi \sqrt{\frac{m}{k}}$
(c) $2 \pi \sqrt{\frac{m}{2 k}}$
(d) $\pi \sqrt{\frac{2 m}{k}}$

Ajay Singhal

Numerade Educator

Problem 50

In the given diagram, $S_{1}$ and $S_{2}$ are identical springs. The frequency of oscillation of the mass $m$ is $f$. If one of the springs is removed, the frequency will be
(a) $\bar{f}$
(b) $2 f$
(c) $\sqrt{2} f$
(d) $f / \sqrt{2}$

Ajay Singhal

Numerade Educator

Problem 51

An object on a spring vibrates in simple harmonic motion at a frequency of $4.0 \mathrm{~Hz}$ and an amplitude of $8.0 \mathrm{~cm}$. If the mass of the object is $0.20 \mathrm{~kg}$, the spring constant is
(a) $40 \mathrm{~N} / \mathrm{m}$
(b) $87 \mathrm{~N} / \mathrm{m}$
(c) $126 \mathrm{~N} / \mathrm{m}$
(d) $160 \mathrm{~N} / \mathrm{m}$

Ajay Singhal

Numerade Educator

Problem 52

An electric motor of mass $40 \mathrm{~kg}$ is mounted on four vertical springs each having spring constant of 4000 $\mathrm{N} / \mathrm{m}$. The period with which the motor vibrates vertically is
(a) $0.314 \mathrm{~s}$
(b) $3.14 \mathrm{~s}$
(c) $0.628 \mathrm{~s}$
(d) $0.157 \mathrm{~s}$

Ajay Singhal

Numerade Educator

Problem 53

Two SHM's are respectively represented by $y=a \sin$ $(\omega t-k x)$ and $y=b \cos (\omega t-k x)$. The phase difference between the two is
(a) $\pi / 2$
(b) $\pi / 4$
(c) $\pi / 6$
(d) $3 \pi / 4$

Ajay Singhal

Numerade Educator

Problem 54

Two particles $P$ and $Q$ describe SHM of same amplitude $a$ and frequency $\mathrm{v}$ along the same straight line. The maximum distance between two particle is $\sqrt{2} a$. The initial phase difference between the particles is
(a) zero
(b) $\pi / 2$
(c) $\pi / 6$
(d) $\pi / 3$

Ajay Singhal

Numerade Educator

Problem 55

A particle is subjected to two mutually perpendicular simple harmonic motions such that its $x$ and $y$ coordinates are given by $x=2 \sin \omega t$
$y=2 \sin \left(\omega t+\frac{\pi}{4}\right)$
The path of the particle will be
(a) an ellipse
(b) a straight line
(c) a parabola
(d) a circle

Ajay Singhal

Numerade Educator

Problem 56

Two simple harmonic motions with same frequency act on a particle at right angles, i.e., along $x$ and $y$ -axis. If the two amplitudes are equal and the phase difference is $\pi / 2$ the resultant motion will be
(a) a straight line inclined at $45^{\circ}$ to the $x$ -axis.
(b) an ellipse with the major axis along the $x$ -axis.
(c) an ellipse with the major axis along the $y$ -axis.
(d) a circle.

Ajay Singhal

Numerade Educator

Problem 57

If two SHMs are represented by equations $y_{1}=10 \sin \left(3 \pi t+\frac{\pi}{4}\right)$
and $y_{2}=5[\sin (3 \pi t)+\sqrt{3} \cos (3 \pi t)]$, the ratio of their
amplitudes is
(a) $2: 1$
(b) $1: 2$
(c) $1: 1$
(d) $1: \sqrt{2}$

Ajay Singhal

Numerade Educator

Problem 58

Which of the following combinations of Lissajous' figure will be like infinite $(\infty)$ ?
(a) $x=a \sin \omega t, y=b \sin \omega t$
(b) $x=a \sin 2 \omega t, y=b \sin \omega t$
(c) $x=a \sin \omega t, y=b \sin 2 \omega t$
(d) $x=a \sin 2 \omega t, y=b \sin 2 \omega t$

Ajay Singhal

Numerade Educator

Problem 59

A particle is subjected simultaneously to two SHM's, one along the $x$ -axis and the other along the $y$ -axis. The two vibrations are in phase and have unequal amplitudes. The particle will execute
(a) straight line motion
(b) circular motion
(c) elliptic motion
(d) parabolic motion

Ajay Singhal

Numerade Educator

Problem 60

Which of the following statements is correct?
(a) Both sound and light waves in air are longitudinal.
(b) Both sound and light waves in air are transverse.
(c) Sound waves in air are transverse while light longitudinal.
(d) Sound waves in air are longitudinal while light waves transverse.

Ajay Singhal

Numerade Educator

Problem 61

Of the following properties of a wave, the one that is independent of the other is its
(a) amplitude
(b) velocity
(c) wavelength
(d) frequency

Ajay Singhal

Numerade Educator

Problem 62

Which of the following properties of a wave does not change with a change in medium?
(a) Frequency
(b) Wavelength
(c) Velocity
(d) Amplitude

Ajay Singhal

Numerade Educator

Problem 64

The particles of a medium vibrate about their mean positions whenever a wave travels through that medium. The phase difference between the vibrations of two such particles
(a) varies with time
(b) varies with distance separating them
(c) varies with time as well as distance
(d) is always zero

Ajay Singhal

Numerade Educator

Problem 65

The angle between particle velocity and wave velocity in a transverse wave is
(a) zero
(b) $\pi / 4$
(c) $\pi / 2$
(d) $\pi$

Ajay Singhal

Numerade Educator

Problem 66

A heavy rope is suspended from a rigid support. A wave pulse is set up at the lower end; then
(a) the pulse will travel with uniform speed.
(b) the pulse will travel with increasing speed.
(c) the pulse will travel with decreasing speed.
(d) the pulse cannot travel through the rope.

Ajay Singhal

Numerade Educator

Problem 67

Define Mach Number.
(a) It is the ratio of the stress to strain.
(b) It is the ratio of the strain to stress.
(c) It is the ratio of the velocity of an object to the velocity of sound.
(d) It is the ratio of the velocity of sound to the velocity of an object.

Ajay Singhal

Numerade Educator

Problem 68

Which one of the following does not represent a travelling wave?
(a) $y=y_{m} f(x-v t)$
(b) $y=y_{m} \sin k(x+v t)$
(c) $y=y_{m} \log (x-v t)$
(d) $y=f\left(x^{2}-v t^{2}\right)$

Ajay Singhal

Numerade Educator

Problem 69

Which of the following equations represents a wave?
(a) $y=a(\omega t-k x)$
(b) $y=a \sin \omega t$
(c) $y=a \cos k x$
(d) $y=a \sin (a t-b x+c)$

Ajay Singhal

Numerade Educator

Problem 70

A plane sound wave is travelling in a medium. With reference to a frame $A$, its equation is $y=a \cos (\omega t-k x)$. With reference to a frame $B$, moving with a constant velocity $v$ in the direction of propagation of the wave, equation of the wave will be
(a) $y=a \cos [(\omega+k v) t-k x]$
(b) $y=-a \cos [(\omega-k v) t-k x]$
(c) $y=-a \cos [(\omega-k v) t-k x]$
(d) $y=a \cos [(\omega+k v) t+k x]$

Ajay Singhal

Numerade Educator

Problem 71

In a sine wave, the position of different particles at time $t$ $=0$ is shown in the figure. The equation for this wave if it is travelling along positive $x$ -axis can be
(a) $y=A \sin (\omega t-k x)$
(b) $y=A \sin (k x-\omega t)$
(c) $y=A \cos (\omega t-k x)$
(d) $y=A \cos (k x-\omega t)$

Ajay Singhal

Numerade Educator

Problem 72

A transverse wave is travelling in a string. Equation of the wave
(a) is not equal to the shape of the string at an instant $t$.
(b) is general equation for displacement of a particle of the string.
(c) must be sinusoidal equation.
(d) is an equation for displacement of the particle of one end only.

Ajay Singhal

Numerade Educator

Problem 73

A wave equation which gives the displacement along $Y$ -direction is given $y=10^{4} \sin (60 t+2 x)$
where, $x$ and $y$ are in metre and $t$ in sec. Among the following choose the correct statement
(a) It represents a wave propagating along positive $x$ -axis with a velocity of $30 \mathrm{~m} / \mathrm{s}$.
(b) It represents a wave propagating along negative $x$ -axis with a velocity of $120 \mathrm{~m} / \mathrm{s}$.
(c) It represents a wave propagating along negative $x$ -axis with a velocity of $30 \mathrm{~m} / \mathrm{s}$.
(d) It represents a wave propagating along negative $x$ -axis with a velocity of $10^{4} \mathrm{~m} / \mathrm{s}$.

Ajay Singhal

Numerade Educator

Problem 74

If the equation of a progressive wave is given by $y=4 \sin \pi\left[\frac{t}{5}-\frac{x}{9}+\frac{\pi}{6}\right]$
Then, which of the following is correct?
(a) $v=5 \mathrm{~cm} / \mathrm{s}$
(b) $\lambda=18 \mathrm{~m}$
(c) $A=0.04 \mathrm{~cm}$
(d) $f=50 \mathrm{~Hz}$

Ajay Singhal

Numerade Educator

Problem 75

The equation for the displacement of a stretched string is given by $y=4 \sin 2 \pi\left[\frac{t}{0.02}-\frac{x}{100}\right]$where, $y$ and $x$ are in $\mathrm{cm}$ and $t$ in second. The (i) frequency (ii) velocity of the wave (iii) maximum particle velocity are
(a) $50 \mathrm{~Hz}, 50 \mathrm{~m} / \mathrm{s}, 20 \pi \mathrm{m} / \mathrm{s}$
(b) $50 \mathrm{~Hz}, 20 \mathrm{~m} / \mathrm{s}, 50 \mathrm{~m} / \mathrm{s}$
(c) $50 \mathrm{~Hz}, 50 \mathrm{~m} / \mathrm{s}, 2 \pi \mathrm{m} / \mathrm{s}$
(d) $50 \mathrm{~Hz}, 50 \mathrm{~m} / \mathrm{s}, 4 \pi \mathrm{m} / \mathrm{s}$

Ajay Singhal

Numerade Educator

Problem 76

A wave is represented by the equation $y=7 \sin \left(7 \pi t-0.04 x+\frac{\pi}{3}\right)$
where, $x$ is in metres and $t$ in seconds. The speed of the waves is
(a) $(175 \pi) \mathrm{m} / \mathrm{s}$
(b) $(49 \pi) \mathrm{m} / \mathrm{s}$
(c) $(49 / \pi) \mathrm{m} / \mathrm{s}$
(d) $(0.28 \pi) \mathrm{m} / \mathrm{s}$

Ajay Singhal

Numerade Educator

Problem 77

The equation for a wave propagating with a velocity of $330 \mathrm{~m} / \mathrm{s}$ and having a frequency of $110 \mathrm{~Hz}$ and amplitude $0.05 \mathrm{~m}$ is
(a) $y=0.05 \sin 2 \pi\left[110 t+\frac{x}{3}\right]$
(b) $y=0.05 \sin 2 \pi\left[110 t-\frac{x}{3}\right]$
(c) $y=0.05 \sin 2 \pi\left[110 t \pm \frac{x}{3}\right]$
(d) $y=0.05 \sin [110 t-330 x]$

Ajay Singhal

Numerade Educator

Problem 78

The amplitude of a wave disturbance propagating in the positive $Y$ -direction is given by $y=\frac{1}{1+x^{2}}$ at $t=0$ and $y=\frac{1}{\left[1+(x-1)^{2}\right]}$ at $t=2 \mathrm{~s}$
where, $x$ and $y$ are in $\mathrm{m}$. If the shape of the wave disturbance does not change during the propagation, what is the velocity of the wave?
(a) $1 \mathrm{~m} / \mathrm{s}$
(b) $1.5 \mathrm{~m} / \mathrm{s}$
(c) $0.5 \mathrm{~m} / \mathrm{s}$
(d) $2 \mathrm{~m} / \mathrm{s}$

Ajay Singhal

Numerade Educator

Problem 79

A transverse wave is described by the equation $y=y_{o} \sin 2 \pi\left(f t-\frac{x}{\lambda}\right) .$ The maximum particle velocity
is equal to four times the wave velocity, if
(a) $\lambda=\frac{\pi y_{o}}{4}$
(b) $\lambda=\frac{\pi y_{o}}{2}$
(c) $\lambda=\pi y_{o}$
(d) $\lambda=2 \pi y_{o}$

Ajay Singhal

Numerade Educator

Problem 80

A simple harmonic wave is represented by the relation $y(x, t)=a_{o} \sin 2 x\left(v t-\frac{x}{\lambda}\right)$
If the maximum particle velocity is three times the wave velocity, the wavelength $\lambda$ of the wave is
(a) $\pi a_{a} / 3$
(b) $2 \pi a_{o} / 3$
(c) $\pi a_{o}$
(d) $\pi a_{e} / 2$

Ajay Singhal

Numerade Educator

Problem 81

The equation $y=a \sin 2 \pi\left(\frac{t}{T}-\frac{x}{\lambda}\right)$ of a simple harmonic wave gives us
(a) the displacement of all particles of the medium at a particular instant of time only.
(b) the displacement of a single particle at any time.
(c) the displacement of all the particles of the medium at a particular instant of time as well as the displacement of a single particle at any time.
(d) the behaviour of the medium as a whole.

Ajay Singhal

Numerade Educator

Problem 82

The diagram below shows the propagation of a wave. Which points are in phase?
(a) $A B$
(b) $B C$
(c) $B D$
(d) $E B$

Ajay Singhal

Numerade Educator

Problem 83

A wave equation is $y=10^{-4} \sin (60 t+2 x)$
where, $x$ and $y$ are in metres and $t$ is in second. Which of the following statements is correct?
(a) The wave travels with a velocity of $300 \mathrm{~m} / \mathrm{s}$ in the negative direction of the $x$ -axis.
(b) Its wavelength is $\pi$ metre.
(c) Its frequency is $50 \pi$ hertz.
(d) All of these

Ajay Singhal

Numerade Educator

Problem 84

A simple harmonic progressive wave is represented by the equation $y=8 \sin 2 \pi(0.1 x-2 t)$
where, $x$ and $y$ are in $\mathrm{cm}$ and $t$ is in seconds. At any instant the phase difference between two particles separated by $2.0 \mathrm{~cm}$ in the $X$ -direction is
(a) $18^{\circ}$
(b) $54^{\circ}$
(c) $36^{\circ}$
(d) $72^{\circ}$

Ajay Singhal

Numerade Educator

Problem 85

A travelling wave in a gas along the positive $X$ -direction has an amplitude of $2 \mathrm{~cm}$, velocity $45 \mathrm{~m} / \mathrm{s}$ and frequency $75 \mathrm{~Hz}$. Particle acceleration after an interval of 3 seconds at a distance of $135 \mathrm{~cm}$ from the origin is
(a) $0.44 \times 10^{2} \mathrm{~cm} / \mathrm{s}^{2}$
(b) $4.4 \times 10^{5} \mathrm{~cm} / \mathrm{s}^{2}$
(c) $4.4 \times 10^{3} \mathrm{~cm} / \mathrm{s}^{2}$
(d) $44 \times 10^{5} \mathrm{~cm} / \mathrm{s}^{2}$

Ajay Singhal

Numerade Educator

Problem 86

The speed of a wave in a certain medium is $960 \mathrm{~m} / \mathrm{s}$. If 3600 waves pass over a certain point of the medium in 1 minute the wavelength is
(a) 2 metre
(b) 8 metre
(c) 4 metre
(d) 16 metre

Ajay Singhal

Numerade Educator

Problem 87

The phase difference between two points separated by $0.8 \mathrm{~m}$ in a wave of frequency $120 \mathrm{~Hz}$ is $0.5 \pi$. The wave velocity is
(a) $144 \mathrm{~m} / \mathrm{s}$
(b) $256 \mathrm{~m} / \mathrm{s}$
(c) $384 \mathrm{~m} / \mathrm{s}$
(d) $720 \mathrm{~m} / \mathrm{s}$

Ajay Singhal

Numerade Educator

Problem 88

The graph between wave number $(\bar{v})$ and angular frequency $(\omega)$ is
(a)
(b)
Wave no. (\overline{v} )
(c)
(d)

Ajay Singhal

Numerade Educator

Problem 89

The amplitude of a wave represented by displacement equation $y=\frac{1}{\sqrt{a}} \sin \omega t \pm \frac{1}{\sqrt{b}} \cos \omega t$ will be
(a) $\frac{\sqrt{a+b}}{a b}$
(b) $\frac{\sqrt{a}+\sqrt{b}}{a b}$
(c) $\frac{\sqrt{a}-\sqrt{b}}{a b}$
(d) $\frac{a+b}{a b}$

Ajay Singhal

Numerade Educator

Problem 90

In a plane progressive harmonic wave particle speed is always less than the wave speed if
(a) amplitude of wave $<\lambda / 2 \pi$
(b) amplitude of wave $>\lambda / 2 \pi$
(c) amplitude of wave $<\lambda$
(d) amplitude of wave $>\lambda \pi$

Ajay Singhal

Numerade Educator

Problem 91

When a wave travels in a medium, the particle displacement is given by $y=a \sin 2 \pi(b t-c x)$, where $a, b$ and $c$ are constants. The maximum particle velocity will be twice the wave velocity if
(a) $c=\frac{1}{\pi a}$
(b) $c=\pi a$
(c) $b=a c$
(d) $b=\frac{1}{a c}$

Ajay Singhal

Numerade Educator

Problem 92

A sine wave has an amplitude $A$ and wavelength $\lambda$ The ratio of particle velocity and the wave velocity is $(2 \pi A=\lambda)$
(a) $\leq 1$
(b) $=1$
(c) $\geq 1$
(d) None of these

Ajay Singhal

Numerade Educator

Problem 93

Two points on a travelling wave having frequency 500 $\mathrm{Hz}$ and velocity $300 \mathrm{~m} / \mathrm{s}$ are $60^{\circ}$ out of phase, then the minimum distance between the two points is
(a) $0.2$
(b) $0.1$
(c) $0.5$
(d) $0.4$

Ajay Singhal

Numerade Educator

Problem 94

The equation of a wave travelling on a string is $y=4 \sin \left[\frac{\pi}{2}\left(8 t-\frac{x}{8}\right)\right]$, where $x, y$ are in $\mathrm{cm}$ and $t$ in
second. The velocity of the wave is
(a) $64 \mathrm{~cm} / \mathrm{s}$, in $-X$ -direction
(b) $32 \mathrm{~cm} / \mathrm{s}$, in $-X$ -direction
(c) $32 \mathrm{~cm} / \mathrm{s}$, in $+X$ -direction
(d) $64 \mathrm{~cm} / \mathrm{s}$, in $+X$ -direction

Ajay Singhal

Numerade Educator

Problem 96

A wave travelling along positive $x$ -axis is given by $y=A \sin (\omega t-k x)$. If it is reflected from rigid boundary such that $80 \%$ amplitude is reflected, then equation of reflected wave is
(a) $y=A \sin (\omega t+k x)$
(b) $y=-0.8 A \sin (\omega t+k x)$
(c) $y=0.8 A \sin (\omega t+k x)$
(d) $y=A \sin (\omega t+0.8 \mathrm{kx})$

Ajay Singhal

Numerade Educator

Problem 97

A transverse sinusoidal wave moves along a string in the positive $X$ -direction at a speed of $10 \mathrm{~cm} / \mathrm{s}$. The wavelength of the wave is $0.5 \mathrm{~m}$ and its amplitude is $10 \mathrm{~cm}$. At a particular time $t$, the snap-shot of the wave is shown in the figure. The velocity of point $P$ when its displacement is $5 \mathrm{~cm}$, is
(a) $\frac{\sqrt{3} \pi}{50} \hat{j} \mathrm{~m} / \mathrm{s}$
(b) $-\frac{\sqrt{3} \pi}{50} \hat{j} \mathrm{~m} / \mathrm{s}$
(c) $\frac{\sqrt{3} \pi}{50} \hat{i} \mathrm{~m} / \mathrm{s}$
(d) $-\frac{\sqrt{3} \pi}{50} \hat{i} \mathrm{~m} / \mathrm{s}$

Ajay Singhal

Numerade Educator

Problem 98

A transverse wave is travelling along a string from left to right. The adjoining figure represents the shape of the string a given instant. At this instant, among the following, choose the wrong statement
(a) Points $D, E$ and $F$ have upwards positive velocity
(b) Points $A, B$ and $H$ have downwards negative velocity
(c) Point $C$ and $G$ have zero velocity
(d) Points $A$ and $E$ have minimum velocity

Ajay Singhal

Numerade Educator

Problem 99

For the wave shown in figure given below, the frequency and wavelength, if its speed is $320 \mathrm{~m} / \mathrm{s}$, are
(a) $8 \mathrm{~cm}, 400 \mathrm{~Hz}$
(b) $80 \mathrm{~cm}, 40 \mathrm{~Hz}$
(c) $8 \mathrm{~cm}, 4000 \mathrm{~Hz}$
(d) $40 \mathrm{~cm}, 8000 \mathrm{~Hz}$

Ajay Singhal

Numerade Educator

Problem 100

The distance between two points differing in phase by $60^{\circ}$ on a wave having a wave velocity $360 \mathrm{~m} / \mathrm{s}$ and frequency $500 \mathrm{~Hz}$ is
(a) $0.72$ metre
(b) $0.18$ metre
(c) $0.12$ metre
(d) $0.36$ metre

Ajay Singhal

Numerade Educator

Problem 101

A simple harmonic wave train of amplitude $2 \mathrm{~cm}$ and time period $0.01$ sec is travelling with a velocity of 10 $\mathrm{m} / \mathrm{s}$ in the positive $X$ -direction. The displacement of the particle from the mean position, the particle velocity and particle acceleration at $x=150 \mathrm{~cm}$ from the origin and at $t=3$ seconds are
(a) $0,0,0$
(b) $0,400 \pi \mathrm{cm} / \mathrm{s}, 0$
(c) $0,0,400 \pi \mathrm{cm} / \mathrm{s}^{2}$
(d) $400 \pi \mathrm{cm}, 0,0$

Ajay Singhal

Numerade Educator

Problem 103

Two particles $P$ and $Q$ describe SHM of same amplitude $a$, frequency $\mathrm{v}$ along the same straight line. The maximum distance between the two particles is $a \sqrt{2}$. The initial phase difference between the particles is
(a) zero
(b) $\pi / 2$
(c) $\pi / 6$
(d) $\pi / 3$

Ajay Singhal

Numerade Educator

Problem 104

Three waves of equal frequencies having amplitudes $10 \mu \mathrm{m}, 4 \mu \mathrm{m}$, and $7 \mu \mathrm{m}$, arrive at a given point with successive phase difference of $\pi / 2$. The amplitude of the resulting wave (in $\mu \mathrm{m}$ ) is given by
(a) 7
(b) 6
(c) 5
(d) 4

Ajay Singhal

Numerade Educator

Problem 105

A string of length $L$ is stretched by $L / 20$ and the speed of transverse waves along it is $v$. The speed of wave when it is stretched by $L / 10$ will be (assume that Hooke's law is applicable)
(a) $2 v$
(b) $v / \sqrt{2}$
(c) $\sqrt{2} v$
(d) $4 v$

Ajay Singhal

Numerade Educator

Problem 106

A travelling wave is partly reflected and partly transmitted from a rigid boundary. Let $a_{i}, a_{r}$ and $a_{t}$ be the amplitudes of incident wave, reflected wave and transmitted wave and $I, I$ and $I$ be the corresponding intensities. Then, choose the correct alternative
(a) $\frac{I_{i}}{I_{r}}=\left(\frac{a_{i}}{a_{r}}\right)^{2}$
(b) $\frac{I_{t}}{I_{t}}=\left(\frac{a_{i}}{a_{i}}\right)^{2}$
(c) $\frac{I_{r}}{I_{t}}=\left(\frac{a_{r}}{a_{i}}\right)^{2}$
(d) All of these

Ajay Singhal

Numerade Educator

Problem 107

A line source emits a cylindrical wave. If the medium absorbs no energy the amplitude will vary with distance $r$ from the source as proportional to
(a) $r^{-1}$
(b) $r^{-2}$
(c) $r^{-1 / 2}$
(d) $\underline{r^{1 / 2}}$

Ajay Singhal

Numerade Educator

Problem 108

A stone thrown into still water, creates a circular wave pattern moving radially outwards. If $r$ is the distance measured from the centre of the pattern, the amplitude of the wave varies as
(a) $r^{-1 / 2}$
(b) $r^{-1}$
(c) $r^{-2}$
(d) $r^{-3 / 2}$

Ajay Singhal

Numerade Educator

Problem 109

Two waves are given by $y_{1}=\cos (4 t-2 x)$ and $y_{2}=\sin \left(4 t-2 x+\frac{\pi}{4}\right)$. The phase difference between the two waves is
(a) $\frac{\pi}{4}$
(b) $-\frac{\pi}{4}$
(c) $\frac{3 \pi}{4}$
(d) $\frac{\pi}{2}$

Ajay Singhal

Numerade Educator

Problem 110

The amplitude of a wave is given by
$A=\frac{c}{(a+b-c)}$
Resonance will occur when
(a) $b=-c / 2$
(b) $b=-a / 2$
(c) $b=0, a=c$
(d) None of these

Ajay Singhal

Numerade Educator

Problem 111

The phase difference between two waves, represented by $y_{1}=10^{-6} \sin [100 t+(x / 50)+0.5] \mathrm{m}$
$y_{2}=10^{-6} \cos [100 t+(x / 50)] \mathrm{m}$
where $x$ is expressed in metres and $t$ is expressed in seconds, is approximately
(a) $1.07 \mathrm{rad}$
(b) $2.07 \mathrm{rad}$
(c) $0.5 \mathrm{rad}$
(d) $1.5 \mathrm{rad}$

Ajay Singhal

Numerade Educator

Problem 112

Two waves of frequencies $20 \mathrm{~Hz}$ and $30 \mathrm{~Hz}$ travel out from a common point. How will they differ in phase at the end of $0.75$ second?
(a) $15 \pi$
(b) $\pi$
(c) $7 \pi$
(d) $2 \pi$

Ajay Singhal

Numerade Educator