How many questions from Oscillations appear in JEE?

Oscillations has appeared in 117 of the last 186 JEE Main and JEE Advanced papers — about 63% of them — contributing 141 questions in total across those papers.

Is Oscillations an important chapter for JEE?

Judged by how often it is actually tested, it appears in roughly 63% of papers. Chapters above about 50% are effectively guaranteed to show up every session, so they repay thorough preparation; lower-frequency chapters are better treated as targeted revision.

Where do these Oscillations questions come from?

Every question is from an official JEE Main or JEE Advanced paper, transcribed from the original paper and tagged to this chapter. Answers follow the official answer key.

Oscillations JEE Previous Year Questions with Answers

Q1·PhysicsNumericalJEE Advanced 2026

A tank contains two immiscible liquids of densities

6\rho

and

2\rho

. The higher density liquid is filled up to a height

L/2

from the bottom. A thin rod of density

\rho

and length

L

is fully immersed and hinged at the bottom so that it can oscillate freely, as shown in the figure. If the rod is slightly disturbed from its equilibrium, the time period of small oscillations is

\frac{2\pi}{n}\sqrt{\frac{L}{g}}

, where

g

is the acceleration due to gravity. The value of

n

is:

Q2·PhysicsSingle correctJEE Main 2026

The frequency of oscillation of a mass

m

suspended by a spring is

v_1

. If the length of the spring is cut to half, the same mass oscillates with frequency

v_2

. The value of

v_2/v_1

is ________.

(A)1
(B)2
(C) $\sqrt{2}$
(D) $\sqrt{3}$

Q3·PhysicsSingle correctJEE Main 2026

A spring stretches by 2 mm when it is loaded with a mass of 200 g. From equilibrium position the mass is further pulled down by 2 mm and released. The frequency associated with the system and maximum energy in the spring are __________ Hz and __________ J, respectively. (Take g

= 10

m/s

^{2}

)

(A) $\frac{5\sqrt{50}}{\pi}$ and $8 \times 10^{-3}$
(B) $\frac{5\sqrt{50}}{\pi}$ and 8
(C) $10\sqrt{50}$ and $2 \times 10^{-3}$
(D) $\frac{5\sqrt{50}}{\pi}$ and $16 \times 10^{-3}$

Q4·PhysicsSingle correctJEE Main 2026

A particle is executing simple harmonic motion. Its amplitude is A and time period is 5 sec. The time required by it to move from x = A to

x = \frac{A}{\sqrt{2}}

is ________ sec.

(A)1/4
(B)5/4
(C)5/8
(D)3/8

Q5·PhysicsSingle correctJEE Main 2026

Match List-I with List-II. Choose the correct answer from the options given below :

List-I	List-II
A.sin² ωt	I.Periodic with time period $T = \dfrac{\pi}{\omega}$ but not simple harmonic motion (SHM)
B.sin³(2ωt)	II.Periodic with time period $T = \dfrac{2\pi}{\omega}$ but Not SHM
C.sin(ωt) + cos(πωt)	III.Periodic with time period $T = \dfrac{\pi}{\omega}$ and SHM
D.cos ωt + cos 2ωt	IV.Non-periodic

(A)A-III, B-I, C-IV, D-II
(B)A-II, B-I, C-III, D-IV
(C)A-III, B-II, C-IV, D-I
(D)A-II, B-I, C-IV, D-III

Q6·PhysicsNumericalJEE Main 2026

The velocity of a particle executing simple harmonic motion along

x

-axis is described as

v^2 = 50 - x^2

, where

x

represents displacement. If the time period of motion is

\frac{x}{7}

s, the value of

x

is ______.

Q7·PhysicsSingle correctJEE Main 2026

A uniform disc of radius

R

and mass

M

is free to oscillate about the axis

A

as shown in the figure. For small oscillations the time period is ______. (

g

is acceleration due to gravity)

(A) $2\pi\sqrt{\frac{5R}{4g}}$
(B) $2\pi\sqrt{\frac{2R}{3g}}$
(C) $2\pi\sqrt{\frac{3R}{2g}}$
(D) $2\pi\sqrt{\frac{3R}{g}}$

Q8·PhysicsSingle correctJEE Main 2026

The equation of motion of a particle is given by

x = a \sin\left(50t + \frac{\pi}{3}\right)

cm. The particle will come to rest at time

t_1

and it will have zero acceleration at time

t_2

. The

t_1

and

t_2

respectively are ________.

(A) $\frac{\pi}{300}\,s$ , $\frac{\pi}{75}\,s$
(B) $\frac{\pi}{75}\,s$ , $\frac{\pi}{300}\,s$
(C) $\frac{\pi}{300}\,s$ , $\frac{\pi}{25}\,s$
(D) $\frac{\pi}{50}\,s$ , $\frac{\pi}{100}\,s$

Q9·PhysicsNumericalJEE Main 2026

The displacement of a particle, executing simple harmonic motion with time period

T

, is expressed as

x(t)

= A sin ωt, where

A

is the amplitude. The maximum value of potential energy of this oscillator is found at

t

=

T

/2β. The value of β is __________.

Q10·PhysicsSingle correctJEE Main 2026

As shown in the figure, a spring is kept in a stretched position with some extension by holding the masses 1 kg and 0.2 kg with a separation more than spring natural length and are released. Assuming the horizontal surface to be frictionless, the angular frequency (in SI unit) of the system is :

(A)30
(B)27
(C)20
(D)5

Q11·PhysicsSingle correctJEE Main 2026

A cylindrical block of mass M and area of cross section A is floating in a liquid of density ρ and with its axis vertical. When depressed a little and released the block starts oscillating. The period of oscillation is________.

(A) $2\pi\sqrt{\dfrac{M}{\rho Ag}}$
(B) $\pi\sqrt{\dfrac{2M}{\rho Ag}}$
(C) $\pi\sqrt{\dfrac{\rho A}{Mg}}$
(D) $2\pi\sqrt{\dfrac{\rho A}{Mg}}$

Q12·PhysicsSingle correctJEE Main 2026

A spring of force constant 15 N/m is cut into two pieces. If the ratio of their length is 1:3, then the force constant of smaller piece is____N/m

(A)15
(B)20
(C)60
(D)45

Q13·PhysicsSingle correctJEE Main 2026

A simple pendulum of string length 30 cm performs 20 oscillations in 10s. The length of the string required for the pendulum to perform 40 oscillations in the same time duration is ______cm. [Assume that the mass of the pendulum remains same.]

(A)120
(B)0.75
(C)7.5
(D)15

Q14·PhysicsSingle correctJEE Main 2026

The kinetic energy of a simple harmonic oscillator is oscillating with angular frequency of 176 rad/s. The frequency of this simple harmonic oscillator is_____Hz.

\left[\text{take } \pi = \frac{22}{7}\right]

(A)14
(B)88
(C)28
(D)176

Q15·PhysicsSingle correctJEE Advanced 2025

As shown in the figures, a uniform rod OO′ of length

\ell

is hinged at the point O and held in place vertically between two walls using two massless springs of same spring constant. The springs are connected at the midpoint and at the top-end (O′) of the rod, as shown in Fig.1 and the rod is made to oscillate by a small angular displacement. The frequency of oscillation of the rod is

f_1

. On the other hand, if both the springs are connected at the midpoint of the rod, as shown in Fig. 2 and the rod is made to oscillate by a small angular displacement, then the frequency of oscillation is

f_2

. Ignoring gravity and assuming motion only in the plane of the diagram, the value of

\frac{f_1}{f_2}

is:

(A)2
(B) $\sqrt{2}$
(C) $\sqrt{\frac{5}{2}}$
(D) $\sqrt{\frac{2}{5}}$

Q16·PhysicsSingle correctJEE Advanced 2025

The center of a disk of radius r and mass m is attached to a spring of spring constant k, inside a ring of radius R > r as shown in the figure. The other end of the spring is attached on the periphery of the ring. Both the ring and the disk are in the same vertical plane. The disk can only roll along the inside periphery of the ring, without slipping. The spring can only be stretched or compressed along the periphery of the ring, following the Hooke's law. In equilibrium, the disk is at the bottom of the ring. Assuming small displacement of the disc, the time period of oscillation of center of mass of the disk is written as

T = \frac{2\pi}{\omega}

. The correct expression for

\omega

is ( g is the acceleration due to gravity):

(A) $\sqrt{\frac{2}{3}\left(\frac{g}{R-r}+\frac{k}{m}\right)}$
(B) $\sqrt{\frac{2g}{3(R-r)}+\frac{k}{m}}$
(C) $\sqrt{\frac{1}{6}\left(\frac{g}{R-r}+\frac{k}{m}\right)}$
(D) $\sqrt{\frac{1}{4}\left(\frac{g}{R-r}+\frac{k}{m}\right)}$

Q17·PhysicsSingle correctJEE Main 2025

A block of mass 2 kg is attached to one end of a massless spring whose other end is fixed at a wall. The spring-mass system moves on a frictionless horizontal table. The spring’s natural length is 2 m and spring constant is 200 N/m. The block is pushed such that the length of the spring becomes 1 m and then released. At distance

x

m (

x<2

) from the wall, the speed of the block will be:

(A) $10[1-(2-x)]^{3/2}\ m/s$
(B) $10[1-(2-x)^2]^{1/2}\ m/s$
(C) $10[1-(2-x)^2]\ m/s$
(D) $10[1-(2-x)^2]^{2}\ m/s$

Q18·PhysicsSingle correctJEE Main 2025

The equation of a wave travelling on a string is

y=\sin[20\pi x+10\pi t]

, where

x

and

t

are distance and time in SI units. The minimum distance between two points having the same oscillating speed is:

(A)5.0 cm
(B)20 cm
(C)10 cm
(D)2.5 cm

Q19·PhysicsSingle correctJEE Main 2025

Two simple pendulums having lengths

l_1

and

l_2

with negligible string mass undergo angular displacements

\theta_1

and

\theta_2

from their mean positions, respectively. If the angular accelerations of both pendulums are same, then which expression is correct?

(A) $\theta_1 l_1^2=\theta_2 l_2^2$
(B) $\theta_1 l_1=\theta_2 l_2$
(C) $\theta_1 l_1=\theta_2 l_2^2$
(D) $\theta_1 l_2=\theta_2 l_1$

Q20·PhysicsSingle correctJEE Main 2025

Two blocks of masses

m

and

M

(M>m)

are placed on a frictionless table, the upper block (mass

m

) resting on the lower block (mass

M

); a massless spring of spring constant

k

is attached to the lower block, and

\mu

is the coefficient of friction between the two blocks. If the system is slightly displaced and released then: (A) The time period of small oscillation of the two blocks is

T=2\pi\sqrt{\dfrac{m+M}{k}}

. (B) The acceleration of the blocks is

a=\dfrac{kx}{M+m}

(

x=

displacement from the mean position). (C) The magnitude of the frictional force on the upper block is

\dfrac{m\mu|x|}{M+m}

. (D) The maximum amplitude of the upper block, if it does not slip, is

\dfrac{\mu(M+m)g}{k}

. (E) Maximum frictional force can be

\mu(M+m)g

. Choose the correct answer from the options given below:

(A)(A), (B), (D) Only
(B)(B), (C), (D) Only
(C)(C), (D), (E) Only
(D)(A), (B), (C) Only

Q21·PhysicsSingle correctJEE Main 2025

A particle is subjected to two simple harmonic motions as:

x_1=\sqrt{7}\sin 5t

cm and

x_2=2\sqrt{7}\sin\left(5t+\dfrac{\pi}{3}\right)

cm, where

x

is displacement and

t

is time in seconds. The maximum acceleration of the particle is

x\times10^{-2}

ms

^{-2}

. The value of

x

is:

(A)175
(B) $25\sqrt{7}$
(C) $5\sqrt{7}$
(D)125

Q22·PhysicsSingle correctJEE Main 2025

Two bodies A and B of equal mass are suspended from two massless springs of spring constant

k_1

and

k_2

, respectively. If the bodies oscillate vertically such that their amplitudes are equal, the ratio of the maximum velocity of A to the maximum velocity of B is

(A) $\sqrt{\dfrac{k_1}{k_2}}$
(B) $\dfrac{k_1}{k_2}$
(C) $\dfrac{k_2}{k_1}$
(D) $\sqrt{\dfrac{k_2}{k_1}}$

Q23·PhysicsSingle correctJEE Main 2025

Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A): Knowing initial position

x_0

and initial momentum

p_0

is enough to determine the position and momentum at any time t for a simple harmonic motion with a given angular frequency

\omega

. Reason (R): The amplitude and phase can be expressed in terms of

x_0

and

p_0

. In the light of the above statements, choose the correct answer from the options given below:

(A)Both (A) and (R) are true and (R) is NOT the correct explanation of (A)
(B)(A) is false but (R) is true
(C)(A) is true but (R) is false
(D)Both (A) and (R) are true and (R) is the correct explanation of (A)

Q24·PhysicsSingle correctJEE Main 2025

A particle is executing simple harmonic motion with time period 2s and amplitude 1 cm. If D and d are the total distance and displacement covered by the particle in 12.5 s, then

\dfrac{D}{d}

is :-

(A) $\dfrac{15}{4}$
(B)25
(C)10
(D) $\dfrac{16}{5}$

Q25·PhysicsSingle correctJEE Main 2025

A particle oscillates along the x-axis according to the law,

x(t)=x_0\sin^2\left(\dfrac{t}{2}\right)

where

x_0=1

m. The kinetic energy (K) of the particle as a function of x is correctly represented by the graph:

(A)(1)
(B)(2)
(C)(3)
(D)(4)

Q26·PhysicsSingle correctJEE Main 2025

A light hollow cube of side length 10 cm and mass 10g, is floating in water. It is pushed down and released to execute simple harmonic oscillations. The time period of oscillations is

y\pi\times10^{-2}

s, where the value of

y

is (Acceleration due to gravity,

g=10

m/s

^2

, density of water

=10^3

kg/m

^3

)

(A)2
(B)6
(C)4
(D)1

Q27·PhysicsNumericalJEE Advanced 2024

Two particles, 1 and 2, each of mass

m

, are connected by a massless spring, and are on a horizontal frictionless plane, as shown in the figure. Initially, the two particles, with their center of mass at

x_0

, are oscillating with amplitude

a

and angular frequency

\omega

. Thus, their positions at time

t

are given by

x_1(t) = (x_0 + d) + a \sin \omega t

and

x_2(t) = (x_0 - d) - a \sin \omega t

, respectively, where

d > 2a

. Particle 3 of mass

m

moves towards this system with speed

u_0 = a\omega/2

, and undergoes instantaneous elastic collision with particle 2, at time

t_0

. Finally, particles 1 and 2 acquire a center of mass speed

v_{\text{cm}}

and oscillate with amplitude

b

and the same angular frequency

\omega

. If the collision occurs at time

t_0

=

\pi/(2\omega)

, then the value of

4b^2/a^2

will be

Q28·PhysicsNumericalJEE Advanced 2024

Two particles, 1 and 2, each of mass

m

, are connected by a massless spring, and are on a horizontal frictionless plane, as shown in the figure. Initially, the two particles, with their center of mass at

x_0

, are oscillating with amplitude

a

and angular frequency

\omega

. Thus, their positions at time

t

are given by

x_1(t) = (x_0 + d) + a \sin \omega t

and

x_2(t) = (x_0 - d) - a \sin \omega t

, respectively, where

d > 2a

. Particle 3 of mass

m

moves towards this system with speed

u_0 = a\omega/2

, and undergoes instantaneous elastic collision with particle 2, at time

t_0

. Finally, particles 1 and 2 acquire a center of mass speed

v_{\text{cm}}

and oscillate with amplitude

b

and the same angular frequency

\omega

. If the collision occurs at time

t_0

= 0, the value of

v_{\text{cm}}/(a\omega)

will be

Q29·PhysicsSingle correctJEE Advanced 2024

Two beads, each with charge

q

and mass

m

, are on a horizontal, frictionless, non-conducting, circular hoop of radius

R

. One of the beads is glued to the hoop at some point, while the other one performs small oscillations about its equilibrium position along the hoop. The square of the angular frequency of the small oscillations is given by [

\varepsilon_{0}

is the permittivity of free space.]

(A) $q^{2} / \left(4\pi\varepsilon_{0}R^{3}m\right)$
(B) $q^{2} / \left(32\pi\varepsilon_{0}R^{3}m\right)$
(C) $q^{2} / \left(8\pi\varepsilon_{0}R^{3}m\right)$
(D) $q^{2} / \left(16\pi\varepsilon_{0}R^{3}m\right)$

Q30·PhysicsSingle correctJEE Advanced 2024

A block of mass 5 kg moves along the x-direction subject to the force

F = (-20x + 10)

N, with the value of

x

in metre. At time

t = 0

s, it is at rest at position

x = 1

m. The position and momentum of the block at

t = (\pi/4)

s are

(A) $-0.5$ m, 5 kg m/s
(B)0.5 m, 0 kg m/s
(C)0.5 m, $-5$ kg m/s
(D) $-1$ m, 5 kg m/s

Q31·PhysicsNumericalJEE Main 2024

A particle of mass 0.50 kg executes simple harmonic motion under force

F=-50\,(\text{N/m})\,x

. The time period of oscillation is

\dfrac{x}{35}

s. The value of x is _______. (Given

\pi=\dfrac{22}{7}

)

Q32·PhysicsNumericalJEE Main 2024

The position, velocity and acceleration of a particle executing simple harmonic motion are found to have magnitudes of 4 m,

2\ \mathrm{ms^{-1}}

and

16\ \mathrm{ms^{-2}}

at a certain instant. The amplitude of the motion is

\sqrt{x}

m where

x

is ______.

Q33·PhysicsNumericalJEE Main 2024

An object of mass 0.2 kg executes simple harmonic motion along the x axis with frequency of

\dfrac{25}{\pi}

Hz. At the position x = 0.04 m the object has kinetic energy 0.5 J and potential energy 0.4 J. The amplitude of oscillation is ________ cm.

Q34·PhysicsNumericalJEE Main 2024

A particle is doing simple harmonic motion of amplitude 0.06 m and time period 3.14 s. The maximum velocity of the particle is _______ cm/s.

Q35·PhysicsNumericalJEE Main 2024

An elastic spring under tension of

3\,N

has a length

a

. Its length is

b

under tension

2\,N

. For its length

(3a-2b)

, the value of tension will be ___ N.

Q36·PhysicsNumericalJEE Main 2024

The displacement of a particle executing SHM is given by

x=10\sin\left(\omega t+\tfrac{\pi}{3}\right)

m. The time period of motion is 3.14 s. The velocity of the particle at

t=0

is ______ m/s.

Q37·PhysicsSingle correctJEE Main 2024

In a simple harmonic motion, the total mechanical energy of given system is E. If mass of the oscillating particle P is doubled then the new energy of the system for same amplitude is

(A) $\tfrac{E}{\sqrt{2}}$
(B)E
(C) $E\sqrt{2}$
(D) $2E$

Q38·PhysicsNumericalJEE Main 2024

A mass m is suspended from a spring of negligible mass and the system oscillates with a frequency

f_1

. The frequency of oscillations if a mass 9m is suspended from the same spring is

f_2

. The value of

\dfrac{f_1}{f_2}

is __________.

Q39·PhysicsNumericalJEE Main 2024

A particle performs simple harmonic motion with amplitude A. Its speed is increased to three times at an instant when its displacement is

\dfrac{2A}{3}

. The new amplitude of motion is

\dfrac{nA}{3}

. The value of n is ______.

Q40·PhysicsNumericalJEE Main 2024

The time period of simple harmonic motion of mass M in the given figure is

\pi\sqrt{\dfrac{\alpha M}{5K}}

, where the value of

\alpha

is ______.

Q41·PhysicsNumericalJEE Main 2024

A simple harmonic oscillator has an amplitude A and time period

6\pi

second. Assuming the oscillation starts from its mean position, the time required by it to travel from

x=A

to

x=\dfrac{\sqrt{3}}{2}A

will be

\dfrac{\pi}{x}

s, where

x=

___.

Q42·PhysicsNumericalJEE Main 2024

When the displacement of a simple harmonic oscillator is one third of its amplitude, the ratio of total energy to the kinetic energy is

\dfrac{x}{8}

, where

x=

______ .

Q43·PhysicsNumericalJEE Main 2024

A particle executes simple harmonic motion with an amplitude of 4 cm. At the mean position, velocity of the particle is 10 cm/s. The distance of the particle from the mean position when its speed becomes 5 cm/s is

\sqrt{\alpha}

cm, where

\alpha=

__________.

Q44·PhysicsSingle correctJEE Main 2024

A ball suspended by a thread swings in a vertical plane so that its magnitude of acceleration in the extreme position and lowest position are equal. The angle

\theta

of thread deflection in the extreme position will be :

(A) $\tan^{-1}(\sqrt2)$
(B) $2\tan^{-1}\left(\dfrac12\right)$
(C) $\tan^{-1}\left(\dfrac12\right)$
(D) $2\tan^{-1}\left(\dfrac{1}{\sqrt5}\right)$

Q45·PhysicsNumericalJEE Advanced 2023

Two point-like objects of masses 20 gm and 30 gm are fixed at the two ends of a rigid massless rod of length 10 cm. This system is suspended vertically from a rigid ceiling using a thin wire attached to its center of mass, as shown in the figure. The resulting torsional pendulum undergoes small oscillations. The torsional constant of the wire is

1.2 \times 10^{-8}

Nm

rad^{-1}

. The angular frequency of the oscillations in

n \times 10^{-3}

rad

s^{-1}

. The value of n is ______

Q46·PhysicsSingle correctJEE Main 2023

In a linear simple harmonic motion (SHM) (A) Restoring force is directly proportional to the displacement. (B) The acceleration and displacement are opposite in direction. (C) The velocity is maximum at mean position. (D) The acceleration is minimum at extreme points. Choose the correct answer from the options given below :

(A)(A), (B) and (C) only
(B)(C) and (D) only
(C)(A), (B) and (D) only
(D)(A), (C) and (D) only

Q47·PhysicsSingle correctJEE Main 2023

A particle executes SHM of amplitude A. The distance from the mean position when its kinetic energy becomes equal to its potential energy is:

(A) $\sqrt{2}A$
(B) $2A$
(C) $\dfrac{1}{\sqrt{2}}A$
(D) $\dfrac{1}{2}A$

Q48·PhysicsSingle correctJEE Main 2023

Which graph represents the difference between total energy and potential energy of a particle executing SHM Vs its distance from mean position:

(A)(1)
(B)(2)
(C)(3)
(D)(4)

Q49·PhysicsNumericalJEE Main 2023

At a given point of time the value of displacement of a simple harmonic oscillator is given as

y=A\cos(30^\circ)

. If amplitude is

40\,cm

and kinetic energy at that time is

200\,J

, the value of force constant is

1.0\times 10^{x}\,Nm^{-1}

. The value of

x

is _____.

Q50·PhysicsSingle correctJEE Main 2023

A particle is executing Simple Harmonic Motion (SHM). The ratio of potential energy and kinetic energy of the particle when its displacement is half of its amplitude will be:

(A)1 : 1
(B)2 : 1
(C)1 : 4
(D)1 : 3

Q51·PhysicsSingle correctJEE Main 2023

The variation of kinetic energy (KE) of a particle executing simple harmonic motion with the displacement (x) starting from mean position to extreme position (A) is given by

(A)(1)
(B)(2)
(C)(3)
(D)(4)

Q52·PhysicsSingle correctJEE Main 2023

A particle executes S.H.M. of amplitude A along x-axis. At

t=0

, the position of the particle is

x=\dfrac{A}{2}

and it moves along positive x-axis. The displacement of particle in time t i.e.

x=A\sin(\omega t+\delta)

, then the value of

\delta

will be:

(A) $\dfrac{\pi}{6}$
(B) $\dfrac{\pi}{3}$
(C) $\dfrac{\pi}{4}$
(D) $\dfrac{\pi}{2}$

Q53·PhysicsNumericalJEE Main 2023

A rectangular block of mass

5

kg attached to a horizontal spiral spring executes simple harmonic motion of amplitude

1

m and time period

3.14

s. The maximum force exerted by the spring on the block is _______ N.

Q54·PhysicsSingle correctJEE Main 2023

For a periodic motion represented by the equation

Y=\sin\omega t+\cos\omega t

, the amplitude of the motion is:

(A) $0.5$
(B) $\sqrt{2}$
(C) $1$
(D) $2$

Q55·PhysicsSingle correctJEE Main 2023

For a particle P revolving round the centre O with radius of circular path

r

and angular velocity

\omega

, as shown in the figure, the projection of OP on the x-axis at time t is

(A) $x(t)=r\cos\left(\omega t+\dfrac{\pi}{6}\right)$
(B) $x(t)=r\cos(\omega t)$
(C) $x(t)=r\sin\left(\omega t+\dfrac{\pi}{6}\right)$
(D) $x(t)=r\cos\left(\omega t-\dfrac{\pi}{6}\right)$

Q56·PhysicsNumericalJEE Main 2023

A simple pendulum with length 100 cm and bob of mass 250 g is executing S.H.M. of amplitude 10 cm. The maximum tension in the string is found to be

\dfrac{x}{40}

N. The value of

x

is

Q57·PhysicsSingle correctJEE Main 2023

A small block of mass

100\,g

is tied to a spring of spring constant

7.5\,N/m

and length

20\,cm

. The other end of spring is fixed at a particular point

A

. If the block moves in a circular path on a smooth horizontal surface with constant angular velocity

5\,rad/s

about point

A

, then tension in the spring is:

(A) $1.5\,N$
(B) $0.75\,N$
(C) $0.25\,N$
(D) $0.50\,N$

Q58·PhysicsSingle correctJEE Main 2023

A mass

m

is attached to two springs as shown in figure. The spring constants of two springs are

K_1

and

K_2

. For the frictionless surface, the time period of oscillation of mass is:

(A) $\dfrac{1}{2\pi}\sqrt{\dfrac{K_1+K_2}{m}}$
(B) $\dfrac{1}{2\pi}\sqrt{\dfrac{K_1-K_2}{m}}$
(C) $2\pi\sqrt{\dfrac{m}{K_1+K_2}}$
(D) $2\pi\sqrt{\dfrac{m}{K_1-K_2}}$

Q59·PhysicsSingle correctJEE Main 2023

Choose the correct length (L) versus square of time period (T

^2

) graph for a simple pendulum executing simple harmonic motion.

(A)(1)
(B)(2)
(C)(3)
(D)(4)

Q60·PhysicsNumericalJEE Main 2023

The amplitude of a particle executing SHM is

3

cm. The displacement at which its kinetic energy will be

25\%

more than the potential energy is _______ cm.

Q61·PhysicsNumericalJEE Main 2023

A block is fastened to a horizontal spring. The block is pulled to a distance

x=10

cm from its equilibrium position (at

x=0

) on a frictionless surface from rest. The energy of the block at

x=5

cm is

0.25

J. The spring constant of the spring is _________ Nm

^{-1}

.

Q62·PhysicsNumericalJEE Main 2023

In the figure given below, a block of mass

M = 490

g placed on a frictionless table is connected with two springs having same spring constant (

K = 2\,\text{N m}^{-1}

). If the block is horizontally displaced through '

X

' m then the number of complete oscillations it will make in

14\pi

seconds will be __________.

Q63·PhysicsSingle correctJEE Main 2023

The maximum potential energy of a block executing simple harmonic motion is

25

J.

A

is amplitude of oscillation. At

A/2

, the kinetic energy of the block is :

(A) $18.75$ J
(B) $9.75$ J
(C) $37.5$ J
(D) $12.5$ J

Q64·PhysicsNumericalJEE Main 2023

The velocity of a particle executing SHM varies with displacement (

x

) as

4v^{2}=50-x^{2}

. The time period of oscillation is

\dfrac{x}{7}

s. The value of

x

is _______. (Take

\pi=\dfrac{22}{7}

)

Q65·PhysicsSingle correctJEE Main 2023

For a simple harmonic motion in a mass-spring system on a frictionless surface, the angular frequency is

\omega_1

when the mass of the block is 1 kg and

\omega_2

when the mass is 2 kg. The ratio

\dfrac{\omega_2}{\omega_1}

is:

(A) $\dfrac{1}{\sqrt{2}}$
(B) $\sqrt{2}$
(C) $2$
(D) $\dfrac{1}{2}$

Q66·PhysicsNumericalJEE Main 2023

The general displacement of a simple harmonic oscillator is

x=A\sin\omega t

. Let T be its time period. The slope of its potential energy (U) - time (t) curve will be maximum when

t=\dfrac{T}{\beta}

. The value of

\beta

is

Q67·PhysicsNumericalJEE Main 2023

A particle of mass

250\,g

executes a simple harmonic motion under a periodic force

F=(-25x)N

. The particle attains a maximum speed of

4\,m/s

during its oscillation. The amplitude of the motion is _____ cm.

Q68·PhysicsSingle correctJEE Main 2023

T is the time period of a simple pendulum on the earth's surface. Its time period becomes

xT

when taken to a height R (equal to earth's radius) above the earth's surface. Then, the value of

x

will be:

(A) $4$
(B) $2$
(C) $\dfrac{1}{4}$
(D) $\dfrac{1}{2}$

Q69·PhysicsSingle correctJEE Main 2023

A particle executes simple harmonic motion between

x = -A

and

x = +A

. If time taken by particle to go from

x = 0

to

\frac{A}{2}

is

2

s; then time taken by particle in going from

x = \frac{A}{2}

to A is

(A) $4$ S
(B) $1.5$ S
(C) $2$ S
(D) $3$ S

Q70·PhysicsNumericalJEE Main 2023

A mass

m

attached to free end of a spring executes SHM with a period of

1

s. If the mass is increased by

3

kg the period of oscillation increases by one second, the value of mass

m

is _________ kg.

Q71·PhysicsNumericalJEE Main 2023

A block of mass 2 kg is attached with two identical springs of spring constant 20 N/m each. The block is placed on a frictionless surface and the ends of the springs are attached to rigid supports (see figure). When the mass is displaced from its equilibrium position, it executes a simple harmonic motion. The time period of oscillation is

\tfrac{\pi}{\sqrt{x}}

in SI unit. The value of

x

is _______ .

Q72·PhysicsIntegerJEE Advanced 2022

On a frictionless horizontal plane, a bob of mass m = 0.1 kg is attached to a spring with natural length l

_0

=0.1 m. The spring constant is k

_1

= 0.009Nm

^{-1}

when the length of the spring l>l

_0

and is k

_2

= 0.016Nm

^{-1}

when l < l

_0

. Initially the bob is released from l=0.15m . Assume that Hooke's law remains valid throughout the motion. If the time period of the full oscillation is T=(n π) s, then the integer closest to n is _______

Q73·PhysicsSingle correctJEE Main 2022

The time period of oscillation of a simple pendulum of length L suspended from the roof of a vehicle, which moves without friction down an inclined plane of inclination

\alpha

, is given by :

(A) $2\pi\sqrt{L/(g\cos\alpha)}$
(B) $2\pi\sqrt{L/(g\sin\alpha)}$
(C) $2\pi\sqrt{L/g}$
(D) $2\pi\sqrt{L/(g\tan\alpha)}$

Q74·PhysicsNumericalJEE Main 2022

The metallic bob of simple pendulum has the relative density 5. The time period of this pendulum is 10 s. If the metallic bob is immersed in water, then the new time period becomes

5\sqrt{x}

s. The value of x will be ________.

Q75·PhysicsNumericalJEE Main 2022

The potential energy of a particle of mass 4 kg in motion along the x-axis is given by

U = 4(1 - \cos 4x)

J. The time period of the particle for small oscillation

(\sin\theta \simeq \theta)

is

\left(\dfrac{\pi}{K}\right)

s. The value of K is ........

Q76·PhysicsSingle correctJEE Main 2022

Two identical positive charges Q each are fixed at a distance of ‘2a’ apart from each other. Another point charge q

_{0}

with mass ‘m’ is placed at midpoint between two fixed charges. For a small displacement along the line joining the fixed charges, the charge q

_{0}

executes SHM. The time period of oscillation of charge q

_{0}

will be :

(A) $\sqrt{\dfrac{4\pi^{3}\varepsilon_{0}ma^{3}}{q_{0}Q}}$
(B) $\sqrt{\dfrac{q_{0}Q}{4\pi^{3}\varepsilon_{0}ma^{3}}}$
(C) $\sqrt{\dfrac{2\pi^{2}\varepsilon_{0}ma^{3}}{q_{0}Q}}$
(D) $\sqrt{\dfrac{8\pi^{3}\varepsilon_{0}ma^{3}}{q_{0}Q}}$

Q77·PhysicsNumericalJEE Main 2022

A mass 0.9 kg, attached to a horizontal spring, executes SHM with an amplitude

A_1

. When this mass passes through its mean position, then a smaller mass of 124 g is placed over it and both masses move together with amplitude

A_2

. If the ratio

\frac{A_1}{A_2}

is

\frac{\alpha}{\alpha-1}

, then the value of

\alpha

will be ____ .

Q78·PhysicsNumericalJEE Main 2022

As per given figures, two springs of spring constants K and 2K are connected to mass m. If the period of oscillation in figure (a) is 3s, then the period of oscillation in figure (b) will be

\sqrt{x}

s. The value of x is__________ .

Q79·PhysicsSingle correctJEE Main 2022

When a particle executes simple Harmonic motion, the nature of graph of velocity as function of displacement will be :

(A)Circular
(B)Ellipitical
(C)Sinusoidal
(D)Straight line

Q80·PhysicsSingle correctJEE Main 2022

In figure (A), mass '2 m' is fixed on mass 'm' which is attached to two springs of spring constant k. In figure (B), mass 'm' is attached to two spring of spring constant 'k' and '2k'. If mass 'm' in (A) and (B) are displaced by distance 'x' horizontally and then released, then time period

T_1

and

T_2

corresponding to (A) and (B) respectively follow the relation.

(A) $\frac{T_1}{T_2} = \frac{3}{\sqrt{2}}$
(B) $\frac{T_1}{T_2} = \sqrt{\frac{3}{2}}$
(C) $\frac{T_1}{T_2} = \sqrt{\frac{2}{3}}$
(D) $\frac{T_1}{T_2} = \frac{\sqrt{2}}{3}$

Q81·PhysicsNumericalJEE Main 2022

A body is performing simple harmonic with an amplitude of 10 cm. The velocity of the body was tripled by air Jet when it is at 5 cm from its mean position. The new amplitude of vibration is

\sqrt{x}

cm. The value of x is___________.

Q82·PhysicsSingle correctJEE Main 2022

The motion of a simple pendulum excuting S.H.M. is represented by following equation.

Y = A \sin(\pi t + \phi)

, where time is measured in second. The length of pendulum is :

(A)97.23 cm
(B)25.3 cm
(C)99.4 cm
(D)406.1 cm

Q83·PhysicsNumericalJEE Main 2022

A particle executes simple harmonic motion. Its amplitude is 8 cm and time period is 6s. The time it will take to travel from its position of maximum displacement to the point corresponding to half of its amplitude, is _________ s.

Q84·PhysicsSingle correctJEE Main 2022

The equation of a particle executing simple harmonic motion is given by

x=\sin\pi\left(t+\dfrac{1}{3}\right)

m . At t = 1s, the speed of particle will be (Given :

\pi=3.14

)

(A) $0$ cm s $^{-1}$
(B) $157$ cm s $^{-1}$
(C) $272$ cm s $^{-1}$
(D) $314$ cm s $^{-1}$

Q85·PhysicsSingle correctJEE Main 2022

The displacement of simple harmonic oscillator after 3 seconds starting from its mean position is equal to half of its amplitude. The time period of harmonic motion is :

(A)6 s
(B)8 s
(C)12s
(D)36 s

Q86·PhysicsSingle correctJEE Main 2022

Time period of a simple pendulum in a stationary lift is 'T'. If the lift accelerates with

\frac{g}{6}

vertically upwards then the time period will be : (where g = acceleration due to gravity)

(A) $\sqrt{\frac{6}{5}}T$
(B) $\sqrt{\frac{5}{6}}T$
(C) $\sqrt{\frac{6}{7}}T$
(D) $\sqrt{\frac{7}{6}}T$

Q87·PhysicsSingle correctJEE Main 2021

A mass of 5 kg is connected to a spring. The potential energy curve of the simple harmonic motion executed by the system is shown in the figure. A simple pendulum of length 4 m has the same period of oscillation as the spring system. What is the value of acceleration due to gravity on the planet where these experiments are performed?

(A)10 m/s $^{2}$
(B)5 m/s $^{2}$
(C)4 m/s $^{2}$
(D)9.8 m/s $^{2}$

Q88·PhysicsSingle correctJEE Main 2021

For a body executing S.H.M. : (a) Potential energy is always equal to its K.E. (b) Average potential and kinetic energy over any given time interval are always equal. (c) Sum of the kinetic and potential energy at any point of time is constant. (d) Average K.E. in one time period is equal to average potential energy in one time period. Choose the most appropriate option from the options given below :

(A)(c) and (d)
(B)only (c)
(C)(b) and (c)
(D)only (b)

Q89·PhysicsSingle correctJEE Main 2021

A bob of mass 'm' suspended by a thread of length

l

undergoes simple harmonic oscillations with time period T. If the bob is immersed in a liquid that has density

\frac{1}{4}

times that of the bob and the length of the thread is increased by

1/3^{rd}

of the original length, then the time period of the simple harmonic oscillations will be :-

(A)T
(B) $\frac{3}{2}$ T
(C) $\frac{3}{4}$ T
(D) $\frac{4}{3}$ T

Q90·PhysicsNumericalJEE Main 2021

A particle of mass 1 kg is hanging from a spring of force constant 100 Nm

^{-1}

. The mass is pulled slightly downward and released so that it executes free simple harmonic motion with time period T. The time when the kinetic energy and potential energy of the system will become equal, is

\frac{T}{x}

.The value of x is ____.

Q91·PhysicsSingle correctJEE Main 2021

The variation of displacement with time of a particle executing free simple harmonic motion is shown in the figure. The potential energy U(x) versus time (t) plot of the particle is correctly shown in figure :

(A)(1)
(B)(2)
(C)(3)
(D)(4)

Q92·PhysicsNumericalJEE Main 2021

Two simple harmonic motion, are represented by the equations

y_1 = 10 \sin\left(3\pi t + \frac{\pi}{3}\right)

y_2 = 5(\sin 3\pi t + \sqrt{3}\cos 3\pi t)

Ratio of amplitude of

y_1

to

y_2

= x : 1. The value of x is _________ .

Q93·PhysicsSingle correctJEE Main 2021

If the length of the pendulum in pendulum clock increases by 0.1%, then the error in time per day is:

(A)86.4 s
(B)4.32 s
(C)43.2 s
(D)8.64 s

Q94·PhysicsNumericalJEE Main 2021

Two simple harmonic motions are represented by the equations

x_1 = 5 \sin\left(2\pi t + \frac{\pi}{4}\right)

and

x_2 = 5\sqrt{2}\,(\sin 2\pi t + \cos 2\pi t)

. The amplitude of second motion is ............. times the amplitude in first motion.

Q95·PhysicsSingle correctJEE Main 2021

A particle starts executing simple harmonic motion (SHM) of amplitude 'a' and total energy E. At any instant, its kinetic energy is

\frac{3E}{4}

then its displacement 'y' is given by :

(A)y = $\frac{a}{2}$
(B)y = $\frac{a}{\sqrt{2}}$
(C)y = $\frac{a\sqrt{3}}{2}$
(D)y = a

Q96·PhysicsSingle correctJEE Main 2021

An object of mass 0.5 kg executing simple harmonic motion. Its amplitude is 5 cm and time period (T) is 0.2 s. What will be the potential energy of the object at an instant t =

\frac{T}{4}

s starting from mean position. Assume that the initial phase of the oscillation is zero.

(A)6.2 × 10 $^{-3}$ J
(B)0.62 J
(C)6.2 × 10 $^{3}$ J
(D)1.2 × 10 $^{3}$ J

Q97·PhysicsNumericalJEE Main 2021

A particle executes simple harmonic motion represented by displacement function as x(t) = A sin(ωt + φ) If the position and velocity of the particle at t = 0s are 2cm and 2ω cm s

^{-1}

respectively, then its amplitude is x√2 cm where the value of x is __________.

Q98·PhysicsSingle correctJEE Main 2021

In a simple harmonic oscillation, what fraction of total mechanical energy is in the form of kinetic energy, when the particle is midway between mean and extreme position.

(A) $\frac{1}{2}$
(B) $\frac{3}{4}$
(C) $\frac{1}{3}$
(D) $\frac{1}{4}$

Q99·PhysicsNumericalJEE Main 2021

In the reported figure, two bodies A and B of masses 200g and 800g are attached with the system of springs. Springs are kept in a stretched position with some extension when the system is released. The horizontal surface is assumed to be frictionless. The angular frequency will be _______ rad /s when k = 20 N / m.

Q100·PhysicsSingle correctJEE Main 2021

T_0

is the time period of a simple pendulum at a place. If the length of the pendulum is reduced to

\frac{1}{16}

times of its initial value, the modified time period is :

(A) $T_0$
(B) $8\pi\,T_0$
(C) $4\,T_0$
(D) $\frac{1}{4}T_0$

Q101·PhysicsSingle correctJEE Main 2021

The motion of a mass on a spring, with spring constant K is as shown in figure. The equation of motion is given by

x(t)=A\sin\omega t+B\cos\omega t

with

\omega=\sqrt{\frac{K}{m}}

. Suppose that at time t = 0, the position of mass is

x(0)

and velocity

\upsilon(0)

, then its displacement can also be represented as

x(t)=C\cos(\omega t-\phi)

, where C and

\phi

are :

(A) $C=\sqrt{\frac{\upsilon(0)^2}{\omega^2}+x(0)^2},\ \phi=\tan^{-1}\left(\frac{x(0)\omega}{\upsilon(0)}\right)$
(B) $C=\sqrt{\frac{2\upsilon(0)^2}{\omega^2}+x(0)^2},\ \phi=\tan^{-1}\left(\frac{x(0)\omega}{2\upsilon(0)}\right)$
(C) $C=\sqrt{\frac{\upsilon(0)^2}{\omega^2}+x(0)^2},\ \phi=\tan^{-1}\left(\frac{\upsilon(0)}{x(0)\omega}\right)$
(D) $C=\sqrt{\frac{2\upsilon(0)^2}{\omega^2}+x(0)^2},\ \phi=\tan^{-1}\left(\frac{\upsilon(0)}{x(0)\omega}\right)$

Q102·PhysicsSingle correctJEE Main 2021

A particle is making simple harmonic motion along the X-axis. If at a distances

x_1

and

x_2

from the mean position the velocities of the particle are

\upsilon_1

and

\upsilon_2

respectively. The time period of its oscillation is given as :

(A) $T = 2\pi \sqrt{ \dfrac{x_2^{2} - x_1^{2}}{\upsilon_1^{2} - \upsilon_2^{2}} }$
(B) $T = 2\pi \sqrt{ \dfrac{x_2^{2} + x_1^{2}}{\upsilon_1^{2} - \upsilon_2^{2}} }$
(C) $T = 2\pi \sqrt{ \dfrac{x_2^{2} - x_1^{2}}{\upsilon_1^{2} + \upsilon_2^{2}} }$
(D) $T = 2\pi \sqrt{ \dfrac{x_2^{2} + x_1^{2}}{\upsilon_1^{2} + \upsilon_2^{2}} }$

Q103·PhysicsNumericalJEE Main 2021

A particle performs simple harmonic motion with a period of 2 second. The time taken by the particle to cover a displacement equal to half of its amplitude from the mean position is

\dfrac{1}{a}

s. The value of 'a' to the nearest integer is _______ .

Q104·PhysicsSingle correctJEE Main 2021

The function of time representing a simple harmonic motion with a period of

\frac{\pi}{\omega}

is :

(A)sin(ωt) + cos (ωt)
(B)cos(ωt) + cos (2ωt) + cos (3ωt)
(C) $\sin^{2}(\omega t)$
(D) $3\cos\left(\frac{\pi}{4}-2\omega t\right)$

Q105·PhysicsNumericalJEE Main 2021

Consider two identical springs each of spring constant k and negligible mass compared to the mass M as shown. Fig.1 shows one of them and Fig.2 shows their series combination. The ratios of time period of oscillation of the two SHM is

\frac{\mathrm{T}_b}{\mathrm{T}_a} = \sqrt{\mathrm{x}}

, where value of x is _______. (Round off to the Nearest Integer)

Q106·PhysicsSingle correctJEE Main 2021

For what value of displacement the kinetic energy and potential energy of a simple harmonic oscillation become equal ?

(A) $x = 0$
(B) $x = \pm A$
(C) $x = \pm \frac{A}{\sqrt{2}}$
(D) $x = \frac{A}{2}$

Q107·PhysicsSingle correctJEE Main 2021

Two particles A and B of equal masses are suspended from two massless springs of spring constants K

_{1}

and K

_{2}

respectively.If the maximum velocities during oscillations are equal, the ratio of the amplitude of A and B is

(A) $\frac{K_{2}}{K_{1}}$
(B) $\frac{K_{1}}{K_{2}}$
(C) $\sqrt{\frac{K_{1}}{K_{2}}}$
(D) $\sqrt{\frac{K_{2}}{K_{1}}}$

Q108·PhysicsSingle correctJEE Main 2021

Time period of a simple pendulum is T inside a lift when the lift is stationary. If the lift moves upwards with an acceleration g/2, the time period of pendulum will be :

(A) $\sqrt{3}\,T$
(B) $\frac{T}{\sqrt{3}}$
(C) $\sqrt{\frac{3}{2}}\,T$
(D) $\sqrt{\frac{2}{3}}\,T$

Q109·PhysicsSingle correctJEE Main 2021

Amplitude of a mass-spring system, which is executing simple harmonic motion decreases with time. If mass = 500g, Decay constant = 20 g/s then how much time is required for the amplitude of the system to drop to half of its initial value ? (ln 2 = 0.693)

(A)34.65 s
(B)17.32 s
(C)0.034 s
(D)15.01 s

Q110·PhysicsNumericalJEE Main 2021

Time period of a simple pendulum is T. The time taken to complete

\frac{5}{8}

oscillations starting from mean position is

\frac{\alpha}{\beta}

T. The value of

\alpha

is _____________.

Q111·PhysicsNumericalJEE Main 2021

A particle excutes S.H.M with amplitude 'a' and time period T. The displacement of the particle when its speed is half of maximum speed is

\frac{\sqrt{x}\,a}{2}

. The value of x is ______________

Q112·PhysicsSingle correctJEE Main 2021

A particle executes S.H.M., the graph of velocity as a function of displacement is:

(A)a circle
(B)a parabola
(C)an ellipse
(D)a helix

Q113·PhysicsSingle correctJEE Main 2021

If two similar springs each of spring constant K

_{1}

are joined in series, the new spring constant and time period would be changed by a factor :

(A) $\frac{1}{2}$ , $\sqrt{2}$
(B) $\frac{1}{4}$ , $2\sqrt{2}$
(C) $\frac{1}{2}$ , $2\sqrt{2}$
(D) $\frac{1}{4}$ , $\sqrt{2}$

Q114·PhysicsSingle correctJEE Main 2021

Given below are two statements : Statement (1) :- A second's pendulum has a time period of 1 second. Statement (2) :- It takes precisely one second to move between the two extreme positions. In the light of the above statements, choose the correct answer from the options give below.

(A)Both Statement I and Statement II are false
(B)Statement I is true but Statement II is false
(C)Statement I is false but Statement II is true
(D)Both Statement I and Statement II is true

Q115·PhysicsSingle correctJEE Main 2021

The point A moves with a uniform speed along the circumference of a circle of radius 0.36m and cover 30

^{o}

in 0.1s. The perpendicular projection 'P' form 'A' on the diameter MN represents the simple harmonic motion of 'P'. The restoration force per unit mass when P touches M will be: A 0.1s 0.36m30

^{o}

MN OP

(A)100 N
(B)50 N
(C)9.87 N
(D)0.49 N

Q116·PhysicsSingle correctJEE Main 2021

If the time period of a two meter long simple pendulum is 2 s, the acceleration due to gravity at the place where pendulum is executing S.H.M. is :

(A) $2\pi^2 \mathrm{ms}^{-2}$
(B) $16 \mathrm{m/s}^2$
(C) $9.8 \mathrm{ms}^{-2}$
(D) $\pi^2 \mathrm{ms}^{-2}$

Q117·PhysicsSingle correctJEE Main 2021

Y = A sin(ωt + φ0) is the time – displacement equation of a SHM, At t = 0 the displacement of A the particle is Y = and it is moving along negative x-direction. Then the initial phase angle φ0 2 will be.

(A) $\dfrac{\pi}{6}$
(B) $\dfrac{\pi}{3}$
(C) $\dfrac{2\pi}{3}$
(D) $\dfrac{5\pi}{6}$

Q118·PhysicsSingle correctJEE Main 2021

Two identical spring of spring constant '2K' are attached to a block of mass m and to fixed support (see figure). When the mass is displaced from equilibrium position on either side, it executes simple harmonic motion. Then time period of oscillations of this system is: 2km2k mmmm

(A) $\pi\sqrt{\dfrac{m}{k}}$
(B) $\pi\sqrt{\dfrac{m}{2k}}$
(C) $2\pi\sqrt{\dfrac{m}{k}}$
(D) $2\pi\sqrt{\dfrac{m}{2k}}$

Q119·PhysicsSingle correctJEE Main 2021

In the given figure, a mass M is attached to a horizontal spring which is fixed on one side to a rigid support. The spring constant of the spring is k. The mass oscillates on a frictionless surface with time period T and amplitude A. When the mass is in equilibrium position, as shown in the figure, another mass m is gently fixed upon it. The new amplitude of oscillation will be -

(A) $A\sqrt{\frac{M}{M+m}}$
(B) $A\sqrt{\frac{M}{M-m}}$
(C) $A\sqrt{\frac{M-m}{M}}$
(D) $A\sqrt{\frac{M+m}{M}}$

Q120·PhysicsSingle correctJEE Main 2021

When a particle executes SHM, the nature of graphical representation of velocity as a function of displacement is :

(A)elliptical
(B)parabolic
(C)straight line
(D)circular

Q121·PhysicsNumericalJEE Advanced 2020

One end of a spring of negligible unstretched length and spring constant k is fixed at the origin (0,0). A point particle of mass m carrying a positive charge q is attached at its other end. The entire system is kept on a smooth horizontal surface. When a point dipole

\vec{p}

pointing towards the charge q is fixed at the origin, the spring gets stretched to a length

\ell

and attains a new equilibrium position (see figure below). If the point mass is now displaced slightly by

\Delta\ell \ll \ell

from its equilibrium position and released, it is found to oscillate at frequency

\frac{1}{\delta}\sqrt{\frac{k}{m}}

. The value of δ is ______.

Q122·PhysicsSingle correctJEE Main 2020

An object of mass m is suspended at the end of a massless wire of length L and area of cross – selection, A. Young modulus of the material of the wire is Y. If the mass is pulled down slightly its frequency of oscillation along the vertical direction is:

(A) $f=\dfrac{1}{2\pi}\sqrt{\dfrac{mL}{YA}}$
(B) $f=\dfrac{1}{2\pi}\sqrt{\dfrac{YA}{mL}}$
(C) $f=\dfrac{1}{2\pi}\sqrt{\dfrac{mA}{YL}}$
(D) $f=\dfrac{1}{2\pi}\sqrt{\dfrac{YL}{mA}}$

Q123·PhysicsSingle correctJEE Main 2020

When a particle of mass m is attached to a vertical spring of spring constant k and released, its motion is described by

y(t) = y_{0}\sin^{2}\omega t

, where 'y' is measured from the lower end of unstretched spring. Then ω is:

(A) $\frac{1}{2}\sqrt{\frac{g}{y_{0}}}$
(B) $\sqrt{\frac{g}{y_{0}}}$
(C) $\sqrt{\frac{g}{2y_{0}}}$
(D) $\sqrt{\frac{2g}{y_{0}}}$

Q124·PhysicsSingle correctJEE Main 2020

A ring is hung on a nail. It can oscillate, without slipping or sliding (i) in its plane with a time period T

_1

and, T(ii) back and forth in a direction perpendicular to its plane, with a period T

_2

. The ratio

\dfrac{T_1}{T_2}

will be:

(A) $\dfrac{2}{\sqrt{3}}$
(B) $\dfrac{3}{\sqrt{3}}$
(C) $\dfrac{\sqrt{2}}{3}$
(D) $\dfrac{2}{3}$

Q125·PhysicsSingle correctJEE Main 2020

A block of mass m attached to a massless spring is performing oscillatory motion of amplitude 'A' on a frictionless horizontal plane. If half of the mass of the block breaks off when it is passing through its equilibrium point, the amplitude of oscillation for the remaining system become

fA

. The value of f is:

(A) $\dfrac{1}{2}$
(B) $\dfrac{1}{\sqrt{2}}$
(C) $\sqrt{2}$
(D)1

Q126·PhysicsSingle correctJEE Main 2019

A spring whose unstretched length is

\ell

has a force constant k. The spring is cut into two pieces of unstretched lengths

\ell_{1}

and

\ell_{2}

where,

\ell_{1} = n\ell_{2}

and n is an integer. The ratio k

_{1}

/k

_{2}

of the corresponding force constants, k

_{1}

and k

_{2}

will be:

(A)n
(B) $1/n^{2}$
(C) $n^{2}$
(D) $1/n$

Q127·PhysicsSingle correctJEE Main 2019

A simple pendulum of length L is placed between the plates of a parallel plate capacitor having electric field E, as shown in figure. Its bob has mass m and charge q. the time period of the pendulum is given by

(A) $2π\sqrt{\dfrac{L}{\sqrt{g^{2} + \left(\dfrac{qE}{m}\right)^{2}}}}$
(B) $2π\sqrt{\dfrac{L}{\left(g + \dfrac{qE}{m}\right)}}$
(C) $2π\sqrt{\dfrac{L}{\left(g - \dfrac{qE}{m}\right)}}$
(D) $2π\sqrt{\dfrac{L}{\sqrt{g^{2} - \dfrac{q^{2}E^{2}}{m^{2}}}}}$

Q128·PhysicsSingle correctJEE Main 2019

The displacement of a damped harmonic oscillator is given by

x(t) = e^{-0.1t}\cos(10\pi t + \varphi)

. Here t is in seconds. The time taken for its amplitude of vibration to drop to half of its initial value is close to:

(A)13 s
(B)27 S
(C)4 s
(D)7 s

Q129·PhysicsSingle correctJEE Main 2019

A simple pendulum oscillating in air has period T. The bob of the pendulum is completely immersed in a non-viscous liquid. The density of the liquid is

\dfrac{1}{16}

th of the material of the bob. If the bob is inside liquid all the time, its period of oscillation in this liquid is:

(A) $2T\sqrt{\dfrac{1}{10}}$
(B) $2T\sqrt{\dfrac{1}{14}}$
(C) $4T\sqrt{\dfrac{1}{15}}$
(D) $4T\sqrt{\dfrac{1}{14}}$

Q130·PhysicsSingle correctJEE Main 2019

A damped harmonic oscillator has a frequency of 5 oscillations per second. The amplitude drops to half its value for every 10 oscillations. The time it will take to drop to

\frac{1}{1000}

of the original amplitude is close to:

(A)10s
(B)100s
(C)50s
(D)20s

Q131·PhysicsSingle correctJEE Main 2019

A simple harmonic motion is represented by :

y = 5(\sin 3\pi t + \sqrt{3} \cos 3\pi t)

cm The amplitude and time period of the motion are :

(A)10 cm, $\frac{2}{3}$ s
(B)10 cm, $\frac{3}{2}$ s
(C)5 cm, $\frac{3}{2}$ s
(D)5 cm, $\frac{2}{3}$ s

Q132·PhysicsSingle correctJEE Main 2019

Two light identical springs of spring constant k are attached horizontally at the two ends of a uniform horizontal rod AB of length l and mass m. the rod is pivoted at its centre 'O' and can rotate freely in horizontal plane. The other ends of the two springs are fixed to rigid supports as shown in figure. The rod is gently pushed through a small angle and released. The frequency of resulting oscillation is:

(A) $\frac{1}{2\pi}\sqrt{\frac{3k}{m}}$
(B) $\frac{1}{2\pi}\sqrt{\frac{2k}{m}}$
(C) $\frac{1}{2\pi}\sqrt{\frac{6k}{m}}$
(D) $\frac{1}{2\pi}\sqrt{\frac{k}{m}}$

Q133·PhysicsSingle correctJEE Main 2019

A pendulum is executing simple harmonic motion and its maximum kinetic energy is K

_1

. If the length of the pendulum is doubled and it performs simple harmonic motion with the same amplitude as in the first case, its maximum kinetic energy is K

_2

then:

(A)K $_2$ = 2K $_1$
(B)K $_2$ = $\frac{K_1}{2}$
(C)K $_2$ = $\frac{K_1}{4}$
(D)K $_2$ = K $_1$

Q134·PhysicsSingle correctJEE Main 2019

A particle is executing simple harmonic motion (SHM) of amplitude A, along the x-axis, about x = 0. When its potential energy (PE) equals kinetic energy (KE), the position of the particle will be

(A) $\dfrac{A}{2}$
(B) $\dfrac{A}{2\sqrt{2}}$
(C) $\dfrac{A}{\sqrt{2}}$
(D)A

Q135·PhysicsSingle correctJEE Main 2019

A rod of mass ‘M’ and length ‘2L’ is suspended at its middle by a wire. It exhibits torsional oscillations; If two masses each of ‘m’ are attached at distance ‘L/2’ from its centre on both sides, it reduces the oscillation frequency by 20%. The value of ratio m/M is close to:

(A)0.77
(B)0.57
(C)0.37
(D)0.17

Q136·PhysicsSingle correctJEE Main 2019

A block of mass m, lying on a smooth horizontal surface, is attached to a spring (of negligible mass) of spring constant k. The other end of the spring is fixed, as shown in the figure. The block is initially at rest in a equilibrium position. If now the block is pulled with a constant force F, the maximum speed of the block is

(A) $\dfrac{2F}{\sqrt{mk}}$
(B) $\dfrac{F}{\pi\sqrt{mk}}$
(C) $\dfrac{\pi F}{\sqrt{mk}}$
(D) $\dfrac{F}{\sqrt{mk}}$

Q137·PhysicsSingle correctJEE Main 2019

Two masses

m

and

\dfrac{m}{2}

are connected at the two ends of a massless rigid rod of length

\ell

. The rod is suspended by a thin wire of torsional constant

k

at the centre of mass of the rod-mass system (see figure). Because of torsional constant

k

, the restoring torque is

\tau = k\theta

for angular displacement

\theta

. If the rod is rotated by

\theta_0

and released, the tension in it when it passes through its mean position will be

(A) $\dfrac{3k\theta_0^2}{\ell}$
(B) $\dfrac{2k\theta_0^2}{\ell}$
(C) $\dfrac{k\theta_0^2}{\ell}$
(D) $\dfrac{k\theta_0^2}{2\ell}$

Q138·PhysicsNumericalJEE Advanced 2018

A spring-block system is resting on a frictionless floor as shown in the figure. The spring constant is

2.0\ N\,m^{-1}

and the mass of the block is

2.0\ kg

. Ignore the mass of the spring. Initially the spring is in an unstretched condition. Another block of mass

1.0\ kg

moving with a speed of

2.0\ m\ s^{-1}

collides elastically with the first block. The collision is such that the

2.0\ kg

block does not hit the wall. The distance, in metres, between the two blocks when the spring returns to its unstretched position for the first time is __________.

Q139·PhysicsMultiple correctJEE Advanced 2016

A block with mass M is connected by a massless spring with stiffness constant k to a rigid wall and moves without friction on a horizontal surface. The block oscillates with small amplitude A about an equilibrium position

x_{0}

. Consider two cases: (i) when the block is at

x_{0}

; and (ii) when the block is at

x = x_{0} + A

. In both the cases, a particle with mass m (< M) is softly placed on the block after which they stick to each other. Which of the following statement(s) is (are) true about the motion after the mass m is placed on the mass M?

(A)The amplitude of oscillation in the first case changes by a factor of $\sqrt{\frac{M}{m+M}}$ , whereas in the second case it remains unchanged
(B)The final time period of oscillation in both the cases is same
(C)The total energy decreases in both the cases
(D)The instantaneous speed at $x_{0}$ of the combined masses decreases in both the cases

Q140·PhysicsMultiple correctJEE Advanced 2015

Two independent harmonic oscillators of equal mass are oscillating about the origin with angular frequencies

\omega_{1}

and

\omega_{2}

and have total energies

E_{1}

and

E_{2}

, respectively. The variations of their momenta

p

with positions

x

are shown in the figures. If

\dfrac{a}{b} = n^{2}

and

\dfrac{a}{R} = n

, then the correct equation(s) is(are)

(A) $E_{1}\omega_{1} = E_{2}\omega_{2}$
(B) $\dfrac{\omega_{2}}{\omega_{1}} = n^{2}$
(C) $\omega_{1}\omega_{2} = n^{2}$
(D) $\dfrac{E_{1}}{\omega_{1}} = \dfrac{E_{2}}{\omega_{2}}$

Q141·PhysicsMultiple correctJEE Advanced 2013

A particle of mass

m

is attached to one end of a mass-less spring of force constant

k

, lying on a frictionless horizontal plane. The other end of the spring is fixed. The particle starts moving horizontally from its equilibrium position at time

t = 0

with an initial velocity

u_{0}

. When the speed of the particle is

0.5u_{0}

. It collides elastically with a rigid wall. After this collision,

(A)the speed of the particle when it returns to its equilibrium position is $u_{0}$ .
(B)the time at which the particle passes through the equilibrium position for the first time is $t = \pi\sqrt{\frac{m}{k}}$ .
(C)the time at which the maximum compression of the spring occurs is $t = \frac{4\pi}{3}\sqrt{\frac{m}{k}}$ .
(D)the time at which the particle passes through the equilibrium position for the second time is $t = \frac{5\pi}{3}\sqrt{\frac{m}{k}}$ .

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