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JEE Advanced > Oscillations and Waves

Explore popular questions from Oscillations and Waves for JEE Advanced. This collection covers Oscillations and Waves previous year JEE Advanced questions hand picked by experienced teachers.

Q 1.

Correct4

Incorrect-1

Two bodies {tex} M {/tex} and {tex} N {/tex} of equal masses are suspended from two separate massless springs of spring constants {tex} k _ { 1 } {/tex} and {tex} k _ { 2 } {/tex} respectively. If the two bodies oscillate vertically such that their maximum velocities are equal, the ratio of the amplitude of vibration of {tex} M {/tex} to that of {tex} N {/tex} is

A

{tex} \frac { k _ { 1 } } { k _ { 2 } } {/tex}

B

{tex} \sqrt { k _ { 1 } / k _ { 2 } } {/tex}

C

{tex} \frac { k _ { 2 } } { k _ { 1 } } {/tex}

{tex} \sqrt { k _ { 2 } / k _ { 1 } } {/tex}

Explanation

Q 2.

Correct4

Incorrect-1

A particle free to move along the {tex} x {/tex} -axis has potential energy given by {tex} U ( x ) = k \left[ 1 - \exp \left( - x ^ { 2 } \right) \right] {/tex} for {tex} - \infty \leq x \leq + \infty , {/tex} where {tex} k {/tex} is a positive constant of appropriate dimensions. Then

A

at points away from the origin, the particle is in unstable equilibrium

B

for any finite nonzero value of {tex} x , {/tex} there is a force directed away from the origin

C

if its total mechanical energy is {tex} k / 2 , {/tex} it has its minimum kinetic energy at the origin

for small displacements from {tex} x = 0 , {/tex} the motion is simple harmonic

Explanation


Q 3.

Correct4

Incorrect-1

The period of oscillation of a simple pendulum of length {tex} L {/tex} suspended from the roof of a vehicle which moves without friction down an inclined plane of inclination {tex} \alpha , {/tex} is given by

{tex} 2 \pi \sqrt { \frac { L } { g \cos \alpha } } {/tex}

B

{tex} 2 \pi \sqrt { \frac { L } { g \sin \alpha } } {/tex}

C

{tex} 2 \pi \sqrt { \frac { L } { g } } {/tex}

D

{tex} 2 \pi \sqrt { \frac { L } { g \tan \alpha } } {/tex}

Explanation


Q 4.

Correct4

Incorrect-1

A particle executes simple harmonic motion between {tex} x = - A {/tex} and {tex} x = + A . {/tex} The time taken for it to go from 0 to {tex} A / 2 {/tex} is {tex} T _ { 1 } {/tex} and to go from {tex} A / 2 {/tex} to {tex} A {/tex} is {tex} T _ { 2 } {/tex}. Then

{tex} T _ { 1 } < T _ { 2 } {/tex}

B

{tex} T _ { 1 } > T _ { 2 } {/tex}

C

{tex} T _ { 1 } = T _ { 2 } {/tex}

D

{tex} T _ { 1 } = 2 T _ { 2 } {/tex}

Explanation

Q 5.

Correct4

Incorrect-1

For a particle executing {tex}SHM{/tex} the displacement {tex} x {/tex} is given by {tex} x = A {/tex} coswt. Identify the graph which represents the variation of potential energy {tex} ( P E ) {/tex} as a function of time {tex} t {/tex} and displacement {tex} x {/tex}

{tex} \mathrm I , \mathrm { III } {/tex}

B

{tex} \mathrm { I I } , \mathrm { I V } {/tex}

C

{tex} \mathrm { I I } , \mathrm { III } {/tex}

D

{tex} \mathrm { I } , \mathrm { IV } {/tex}

Explanation

Q 6.

Correct4

Incorrect-1

A simple pendulum has time period {tex} T _ { 1 } {/tex}. The point of suspension is now moved upward according to the relation {tex} y = K t ^ { 2 } , \left( K = 1 \mathrm { m } / \mathrm { s } ^ { 2 } \right) {/tex} where {tex} y {/tex} is the vertical displacement. The time period now becomes {tex} T _ { 2 } . {/tex} The ratio of {tex} \frac { T _ { 1 } ^ { 2 } } { T _ { 2 } ^ { 2 } } {/tex} is {tex} \left( g = 10 \mathrm { m } / \mathrm { s } ^ { 2 } \right) {/tex}

A

{tex} 5 / 6 {/tex}

{tex} 6 / 5 {/tex}

C

{tex}1{/tex}

D

{tex} 4 / 5 {/tex}

Explanation


Q 7.

Correct4

Incorrect-1

The {tex} x - t {/tex} graph of a particle undergoing simple harmonic motion is shown below. The acceleration of the particle at {tex} t = \frac 4 3 \mathrm { s } {/tex} is

A

{tex} \frac { \sqrt { 3 } } { 32 } p ^ { 2 } \mathrm { cm } / \mathrm { s } ^ { 2 } {/tex}

B

{tex} \frac { - \pi ^ { 2 } } { 32 } \mathrm { cm } / \mathrm { s } ^ { 2 } {/tex}

C

{tex} \frac { \pi ^ { 2 } } { 32 } \mathrm { cm } / \mathrm { s } ^ { 2 } {/tex}

{tex} - \frac { \sqrt { 3 } } { 32 } \pi ^ { 2 } \mathrm { cm } / \mathrm { s } ^ { 2 } {/tex}

Explanation


Q 8.

Correct4

Incorrect-1

A uniform rod of length {tex} L {/tex} and mass {tex} M {/tex} is pivoted at the centre. Its two ends are attached to two springs of equal spring constants {tex} k . {/tex} The springs are fixed to rigid supports as shown in the
figure, and the rod is free to oscillate in the horizontal plane. The rod is gently pushed through a small angle {tex} \theta {/tex} in one direction and released. The frequency of oscillation is

A

{tex} \frac { 1 } { 2 \pi } \sqrt { \frac { 2 k } { M } } {/tex}

B

{tex} \frac { 1 } { 2 \pi } \sqrt { \frac { k } { M } } {/tex}

{tex} \frac { 1 } { 2 \pi } \sqrt { \frac { 6 k } { M } } {/tex}

D

{tex} \frac { 1 } { 2 \pi } \sqrt { \frac { 24 k } { M } } {/tex}

Explanation



Q 9.

Correct4

Incorrect-1

The mass {tex} M {/tex} shown in the figure oscillates in simple harmonic motion with amplitude {tex} A . {/tex} The amplitude of the point {tex} P {/tex} is

A

{tex}\frac{k_1 A}{k_2}{/tex}

B

{tex}\frac{k_2 A}{k_1}{/tex}

C

{tex}\frac{k_1 A}{k_1+k_2}{/tex}

{tex}\frac{k_2 A}{k_1+k_2}{/tex}

Explanation


Q 10.

Correct4

Incorrect-1

A point mass is subjected to two simultaneous sinusoidal displacements in {tex} x {/tex} -direction, {tex} x _ { 1 } ( t ) = A \sin \omega t {/tex} and {tex} x _ { 2 } ( t ) = {/tex} {tex} A \sin \left( \omega t + \frac { 2 \pi } { 3 } \right) {/tex}. Adding a third sinusoidal displacement {tex} x _ { 3 } ( t ) = B \sin ( \omega t + \phi ) {/tex} brings the mass to a complete rest. The values of {tex} B {/tex} and {tex} \phi {/tex} are

A

{tex} \sqrt { 2 } A , \frac { 3 \mathrm { p } } { 4 } {/tex}

{tex} A , \frac { 4 p } { 3 } {/tex}

C

{tex} \sqrt { 3 } A , \frac { 5 \mathrm { p } } { 6 } {/tex}

D

{tex} A , \frac { \mathrm { p } } { 3 } {/tex}

Explanation


Q 11.

Correct4

Incorrect-1

A small block is connected to one end of a massless spring of un-stretched length 4.9 m. The other end of the spring (see the figure) is fixed, The system lies on a horizontal frictionless surface. The block is stretched by 0.2 m and released from rest at t=0. It then executes simple harmonic motion With angular frequency {tex}\omega=\pi/3{/tex} rad/s. Simultaneously at t=0, a small pebble is projected with speed {tex}v{/tex} form point P at an angle of {tex}45^\circ{/tex} as shown in the figure. Point {tex}P{/tex} is at a horizontal distance of 10 m from {tex}O {/tex}. If the pebble hits the block at t = 1 s, the value of {tex}v{/tex} is (take{tex} g= 10 m/s^2{/tex})

{tex} \sqrt { 50 } \mathrm { m } / \mathrm { s } {/tex}

B

{tex} \sqrt { 51 } \mathrm { m } / \mathrm { s } {/tex}

C

{tex} \sqrt { 52 } \mathrm { m } / \mathrm { s } {/tex}

D

{tex} \sqrt { 53 } \mathrm { m } / \mathrm { s } {/tex}

Explanation

Q 12.

Correct4

Incorrect-1

A cylindrical tube open at both ends, has a fundamental frequency {tex}'f'{/tex}in air. The tube is dipped vertically in air. The tube is dipped vertically in water so that half of it is in water. The fundamental frequency of the air column in now

A

{tex} \frac { f } { 2 } {/tex}

B

{tex} \frac { 3 f } { 4 } {/tex}

{tex} f {/tex}

D

{tex}2f{/tex}

Explanation

Q 13.

Correct4

Incorrect-1

A wave represented by the equation {tex} y = a \cos ( k x - \omega t ) {/tex} is superposed with another wave to form a stationary wave such that point {tex} x = 0 {/tex} is a node. The equation for the other wave is

A

{tex} a \sin ( k x + \omega t ) {/tex}

B

{tex} - a \cos ( k x - \omega t ) {/tex}

{tex} - a \cos ( k x + \omega t ) {/tex}

D

{tex} - a \sin ( k x - \omega t ) {/tex}

Explanation

Q 14.

Correct4

Incorrect-1

An object of specific gravity {tex} \rho {/tex} is hung from a thin steel wire. The fundamental frequency for transverse standing waves in the wire is {tex} 300 \mathrm { Hz } {/tex}. The object is immersed in water so that one half of its volume is submerged. The new fundamental frequency in {tex} \mathrm { Hz } {/tex} is

{tex} 300 \left( \frac { 2 \rho - 1 } { 2 \rho } \right) ^ { 1 / 2 } {/tex}

B

{tex} 300 \left( \frac { 2 \rho } { 2 \rho - 1 } \right) ^ { 1 / 2 } {/tex}

C

{tex} 300 \left( \frac { 2 \rho } { 2 \rho - 1 } \right) {/tex}

D

{tex} 300 \left( \frac { 2 \rho - 1 } { 2 \rho } \right) {/tex}

Explanation


Q 15.

Correct4

Incorrect-1

A wave disturbance in a medium is described by {tex} y ( x , t ) = 0.02 \cos \left( 50 \pi t + \frac { \pi } { 2 } \right) \cos ( 10 \pi x ) {/tex} where {tex} x {/tex} and {tex} y {/tex} are in metre and {tex} t {/tex} is in second

A

A node occurs at {tex} x = 0.15 \mathrm { m } {/tex}

B

An antinode occurs at {tex} x = 0.3 \mathrm { m } {/tex}

The speed wave is {tex} 5 \mathrm { ms } ^ { - 1 } {/tex}

D

The wave length is {tex} 0.3 \mathrm { m } {/tex}

Explanation

Q 16.

Correct4

Incorrect-1

The extension in a string, obeying Hooke's law, is {tex} x {/tex}. The speed of sound in the stretched string is {tex} v {/tex}. If the extension in the string is increased to {tex} 1.5 x , {/tex} the speed of sound will be

{tex} 1.22 v {/tex}

B

{tex} 0.61 v {/tex}

C

{tex} 1.50 v {/tex}

D

{tex} 0.75 v {/tex}

Explanation

Q 17.

Correct4

Incorrect-1

An open pipe is suddenly closed at one end with the result that the frequency of third harmonic of the closed pipe is found to be higher by {tex} 100 \mathrm { Hz } {/tex} than the fundamental frequency of the open pipe. The fundamental frequency of the open pipe is

{tex} 200 \mathrm { Hz } {/tex}

B

{tex} 300 \mathrm { Hz } {/tex}

C

{tex} 240 \mathrm { Hz } {/tex}

D

{tex} 480 \mathrm { Hz } {/tex}

Explanation



Q 18.

Correct4

Incorrect-1

A travelling wave in a stretched string is described by the equation {tex} y = A \sin ( k x - \omega t ) {/tex} The maximum particle velocity is

{tex} A \omega {/tex}

B

{tex} \omega / k {/tex}

C

{tex} d \omega / d k {/tex}

D

{tex} x / t {/tex}

Explanation

Q 19.

Correct4

Incorrect-1

A train moves towards a stationary observer with speed {tex}34 \mathrm { m } / \mathrm { s } {/tex}. The train sounds a whistle and its frequency registered by the observer is {tex} f _ { 1 } {/tex}. If the train's speed is reduced to {tex}17 \mathrm { m } / \mathrm { s } {/tex}, the frequency registered is {tex} f _ { 2 } {/tex}. If the speed of sound is {tex} 340 \mathrm { m } / \mathrm { s } {/tex}, then the ratio {tex} f _ { 1 } / f _ { 2 } {/tex} is

A

{tex} 18 / 19 {/tex}

B

{tex} 1 / 2 {/tex}

C

{tex} 2 {/tex}

{tex} 19 / 18 {/tex}

Explanation

Q 20.

Correct4

Incorrect-1

Two vibrating strings of the same material but lengths {tex} L {/tex} and {tex} 2 L {/tex} have radii {tex} 2 r {/tex} and {tex} r {/tex} respectively. They are stretched under the same tension. Both the strings vibrate in their fundamental nodes, the one of length {tex} L {/tex} with frequency {tex} v _ { 1 } {/tex} and the other with frequency {tex} v _ { 2 } . {/tex} The raio {tex} v _ { 1 } / v _ { 2 } {/tex} is given by

A

2

B

4

C

8

1

Explanation

Q 21.

Correct4

Incorrect-1

Two monatomic ideal gases 1 and 2 of molecular masses {tex} m _ { 1 } {/tex} and {tex} m _ { 2 } {/tex} respectively are enclosed in separate containers kept at the same temperature. The ratio of the speed of sound in gas 1 to that in gas 2 is given by

A

{tex} \sqrt { \frac { m _ { 1 } } { m _ { 2 } } } {/tex}

{tex} \sqrt { \frac { m _ { 2 } } { m _ { 1 } } } {/tex}

C

{tex} \frac { m _ { 1 } } { m _ { 2 } } {/tex}

D

{tex} \frac { m _ { 2 } } { m _ { 1 } } {/tex}

Explanation

Q 22.

Correct4

Incorrect-1

Two pulses in a stretched string whose centers are initially {tex} 8 \mathrm { cm } {/tex} apart are moving towards each other as shown in the figure. The speed of each pulse is {tex} 2 \mathrm { cm } / \mathrm { s } {/tex}. After {tex}2{/tex} seconds, the total energy of the pulses will be

A

zero

purely kinetic

C

purely potential

D

partly kinetic and partly potential

Explanation

Q 23.

Correct4

Incorrect-1

The ends of a stretched wire of length {tex} L {/tex} are fixed at {tex} x = 0 {/tex} and {tex} x = L . {/tex} In one experiment, the displacement of the wire is {tex} y _ { 1 } = A \sin ( \pi x / L ) \sin \omega t {/tex} and energy is {tex} E _ { 1 } {/tex} and in another experiment its displacement is {tex} y _ { 2 } = A \sin ( 2 \pi x / L ) \sin 2 \omega t {/tex} and energy is {tex} E _ { 2 } . {/tex} Then

A

{tex} E _ { 2 } = E _ { 1 } {/tex}

B

{tex} E _ { 2 } = 2 E _ { 1 } {/tex}

{tex} E _ { 2 } = 4 E_1 {/tex}

D

{tex} E _ { 2 } = 16 E _ { 1 } {/tex}

Explanation

Q 24.

Correct4

Incorrect-1

A siren placed at a railway platform is emitting sound of frequency {tex} 5 \mathrm { kHz } {/tex}. A passenger sitting in a moving train {tex} A {/tex} records a frequency of {tex} 5.5 \mathrm { kHz } {/tex} while the train approaches the siren. During his return journey in a different train {tex} B {/tex} he records a frequency of {tex} 6.0 \mathrm { kHz } {/tex} while approaching the same siren. The ratio of the velocity of train {tex} B {/tex} to that train {tex} A {/tex} is

A

{tex} 242 / 252 {/tex}

{tex}2{/tex}

C

{tex} 5 / 6 {/tex}

D

{tex} 11 / 6 {/tex}

Explanation

Q 25.

Correct4

Incorrect-1

A police car moving at {tex} 22 \mathrm { m } / \mathrm { s } {/tex}, chases a motorcyclist. The police man sounds his horn at {tex} 176 \mathrm { Hz } {/tex}, while both of them move towards a stationary siren of frequency {tex} 165 \mathrm { Hz } {/tex}. Calculate the speed of the motorcycle, if it is given that he does not observes any beats.

A

{tex} 33 \mathrm { m } / \mathrm { s } {/tex}

{tex} 22 \mathrm { m } / \mathrm { s } {/tex}

C

zero

D

{tex} 11 \mathrm { m } / \mathrm { s } {/tex}

Explanation