Question 1: Palette 1/141

JEE Mains 2026 MCQ

As shown in the figure, a spring is kept in a stretched position with some extension by holding the masses $1 \text{ kg}$ and $0.2 \text{ kg}$ 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

20

B

5

C

30

D

27

Question 2: Palette 2/141

JEE Mains 2026 MCQ

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

A

$2 \pi \sqrt{\frac{\rho A}{M g}}$

B

$\pi \sqrt{\frac{2 M}{\rho A g}}$

C

$2 \pi \sqrt{\frac{M}{\rho A g}}$

D

$\pi \sqrt{\frac{\rho A}{M g}}$

Question 3: Palette 3/141

JEE Mains 2026 MCQ

A spring of force constant $15 \mathrm{~N} / \mathrm{m}$ is cut into two pieces. If the ratio of their length is $1: 3$, then the force constant of smaller piece is $\_\_\_\_$ $\mathrm{N} / \mathrm{m}$.

A

20

B

45

C

60

D

15

Question 4: Palette 4/141

JEE Mains 2026 MCQ

A simple pendulum of string length 30 cm performs 20 oscillations in 10 s . 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

0.75

B

7.5

C

15

D

120

Question 5: Palette 5/141

JEE Mains 2026 MCQ

Using a simple pendulum experiment $g$ is determind by measuring its time period $T$. Which of the following plots represent the correct relation between the pendulum length $L$ and time period $T$ ?

A

B

C

D

Question 6: Palette 6/141

JEE Mains 2026 MCQ

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. [ take $\pi = \frac{22}{7}$ ]

A

88

B

14

C

28

D

176

Question 7: Palette 7/141

JEE Mains 2026 MCQ

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

$\sqrt{3}$

Question 8: Palette 8/141

JEE Mains 2026 MCQ

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 maxmimum energy in the spring are $\_\_\_\_$ Hz and $\_\_\_\_$ J, respectively.

(Take $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2$ )

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 $16 \times 10^{-3}$

Question 9: Palette 9/141

JEE Mains 2026 MCQ

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

Question 10: Palette 10/141

JEE Mains 2026 MCQ

$ \text { Match List - I with List - II. } $

$ \text { List - I } $		$ \text { List - II } $
A.	$ \sin ^2 \omega t $	I.	Periodic with time period $T=\frac{\pi}{\omega}$ but not simple harmonic motion (SHM)
B.	$ \sin ^3(2 \omega t) $	II.	Periodic with time period $T=\frac{2 \pi}{\omega}$ but Not SHM
C.	$ \sin (\omega t)+\cos (\pi \omega t) $	III.	Periodic with time period $T=\frac{\pi}{\omega}$ and SHM
D.	$ \cos \omega t+\cos 2 \omega t $	IV.	Non-periodic

Choose the correct answer from the options given below :

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

Question 11: Palette 11/141

JEE Mains 2026 MCQ

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{5 R}{4 g}} $

B

$ 2 \pi \sqrt{\frac{2 R}{3 g}} $

C

$ 2 \pi \sqrt{\frac{3 R}{2 g}} $

D

$ 2 \pi \sqrt{\frac{3 R}{g}} $

Question 12: Palette 12/141

JEE Mains 2026 MCQ

The equation of motion of a particle is given by $x = a \sin(50t + \pi/3)$ 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

$\frac{\pi}{300} \text{ s},\ \frac{\pi}{75} \text{ s}$

B

$\frac{\pi}{75} \text{ s},\ \frac{\pi}{300} \text{ s}$

C

$\frac{\pi}{300} \text{ s},\ \frac{\pi}{25} \text{ s}$

D

$\frac{\pi}{50} \text{ s},\ \frac{\pi}{100} \text{ s}$

Question 13: Palette 13/141

JEE Mains 2025 MCQ

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

$10\left[1-(2-x)^2\right]^{\frac{1}{2}} \ m/s$

B

$10\left[1-(2-x)^2\right]^{\frac{3}{2}} \ m/s$

C

$10\left[1-(2-x)^2\right] \ m/s$

D

$10\left[1-(2-x)^2\right]^2 \ m/s$

Question 14: Palette 14/141

JEE Mains 2025 MCQ

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

$\theta_1 l_2=\theta_2 l_1$

B

$\theta_1 l_1=\theta_2 l_2$

C

$\theta_1 l_2^2=\theta_2 l_1^2$

D

$\theta_1 l_1^2=\theta_2 l_2^2$

Question 15: Palette 15/141

JEE Mains 2025 MCQ

Two blocks of masses $m$ and $M,(M>m)$, are placed on a frictionless table as shown in figure. A massless spring with spring constant k is attached with the lower block. If the system is slightly displaced and released, then ( $\mu=$ coefficient of friction between the two blocks)

A. The time period of small oscillation of the two blocks is $T=2 \pi \sqrt{\frac{(m+M)}{k}}$

B. The acceleration of the blocks is $a=-\frac{k x}{M+m}$ ( $x=$ displacement of the blocks from the mean position)

C. The magnitude of the frictional force on the upper block is $\frac{m \mu|x|}{M+m}$

D. The maximum amplitude of the upper block, if it does not slip, is $\frac{\mu(M+m) g}{k}$

E. Maximum frictional force can be $\mu(\mathrm{M}+\mathrm{m}) \mathrm{g}$.

Choose the correct answer from the options given below :

A

B, C, D Only

B

C, D, E Only

C

A, B, D Only

D

A, B, C Only

Question 16: Palette 16/141

JEE Mains 2025 MCQ

A particle is subjected to two simple harmonic motions as : $ x_1=\sqrt{7} \sin 5 \mathrm{tcm} $ and $x_2=2 \sqrt{7} \sin \left(5 t+\frac{\pi}{3}\right) \mathrm{cm}$ where $x$ is displacement and $t$ is time in seconds. The maximum acceleration of the particle is $x \times 10^{-2} \mathrm{~ms}^{-2}$. The value of $x$ is :

A

$5 \sqrt{7}$

B

125

C

$25 \sqrt{7}$

D

175

Question 17: Palette 17/141

JEE Mains 2025 MCQ

Two bodies A and B of equal mass are suspended from two massless springs of spring constant k₁ and k₂, 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{\frac{k_2}{k_1}} $

B

$ \sqrt{\frac{k_1}{k_2}} $

C

$ \frac{k_2}{k_1} $

D

$ \frac{k_1}{k_2} $

Question 18: Palette 18/141

JEE Mains 2025 MCQ

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

Assertion (A) : Time period of a simple pendulum is longer at the top of a mountain than that at the base of the mountain.

Reason (R) : Time period of a simple pendulum decreases with increasing value of acceleration due to gravity and vice-versa.

In the light of the above statements, choose the most appropriate answer from the options given below :

A

Both (A) and (R) are true but (R) is not the correct explanation of (A).

B

(A) is true but (R) is false.

C

Both (A) and (R) are true and (R) is the correct explanation of (A).

D

(A) is false but (R) is true.

Question 19: Palette 19/141

JEE Mains 2025 MCQ

Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R).

Assertion (A) : Knowing initial position $\mathrm{x}_0$ 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 $\mathrm{X}_0$ an $\mathrm{p}_0$.

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

A

(A) is true but (R) is false

B

Both (A) and (R) are true but (R) is NOT the correct explanation of (A)

C

(A) is false but (R) is true

D

Both (A) and (R) are true and (R) is the correct explanation of (A)

Question 20: Palette 20/141

JEE Mains 2025 MCQ

A particle oscillates along the $x$-axis according to the law, $x(\mathrm{t})=x_0 \sin ^2\left(\frac{\mathrm{t}}{2}\right)$ where $x_0=1 \mathrm{~m}$. The kinetic energy $(\mathrm{K})$ of the particle as a function of $x$ is correctly represented by the graph

A

B

C

D

Question 21: Palette 21/141

JEE Mains 2025 MCQ

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

A

$10$

B

$\frac{16}{5}$

C

$25$

D

$\frac{15}{4}$

Question 22: Palette 22/141

JEE Mains 2025 MCQ

A light hollow cube of side length 10 cm and mass 10 g , is floating in water. It is pushed down and released to execute simple harmonic oscillations. The time period of oscillations is $y \pi \times 10^{-2} \mathrm{~s}$, where the value of $y$ is (Acceleration due to gravity, $g=10 \mathrm{~m} / \mathrm{s}^2$, density of water $=10^3 \mathrm{~kg} / \mathrm{m}^3$ )

A

2

B

4

C

1

D

6

Question 23: Palette 23/141

JEE Mains 2025 MCQ

Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R).

Assertion (A) : A simple pendulum is taken to a planet of mass and radius, 4 times and 2 times, respectively, than the Earth. The time period of the pendulum remains same on earth and the planet.

Reason (R) : The mass of the pendulum remains unchanged at Earth and the other planet.

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 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 but (R) is NOT the correct explanation of (A)

Question 24: Palette 24/141

JEE Mains 2024 MCQ

A simple pendulum doing small oscillations at a place $R$ height above earth surface has time period of $T_1=4 \mathrm{~s}$. $\mathrm{T}_2$ would be it's time period if it is brought to a point which is at a height $2 \mathrm{R}$ from earth surface. Choose the correct relation [$\mathrm{R}=$ radius of earth] :

A

$3 \mathrm{~T}_1=2 \mathrm{~T}_2$

B

$\mathrm{T}_1=\mathrm{T}_2$

C

$2 \mathrm{~T}_1=3 \mathrm{~T}_2$

D

$2 \mathrm{~T}_1=\mathrm{T}_2$

Question 25: Palette 25/141

JEE Mains 2024 MCQ

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

A

$E / \sqrt{2}$

B

$2 E$

C

$E \sqrt{2}$

D

$E$

Question 26: Palette 26/141

JEE Mains 2024 MCQ

A simple pendulum of length $1 \mathrm{~m}$ has a wooden bob of mass $1 \mathrm{~kg}$. It is struck by a bullet of mass $10^{-2} \mathrm{~kg}$ moving with a speed of $2 \times 10^2 \mathrm{~ms}^{-1}$. The bullet gets embedded into the bob. The height to which the bob rises before swinging back is. (use $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2$ )

A

$0.20 \mathrm{~m}$

B

$0.40 \mathrm{~m}$

C

$0.30 \mathrm{~m}$

D

$0.35 \mathrm{~m}$

Question 27: Palette 27/141

JEE Mains 2024 MCQ

The bob of a pendulum was released from a horizontal position. The length of the pendulum is $10 \mathrm{~m}$. If it dissipates $10 \%$ of its initial energy against air resistance, the speed with which the bob arrives at the lowest point is:

[Use, $\mathrm{g}: 10 \mathrm{~ms}^{-2}$]

A

$5 \sqrt{6} \mathrm{~ms}^{-1}$

B

$5 \sqrt{5} \mathrm{~ms}^{-1}$

C

$2 \sqrt{5} \mathrm{~ms}^{-1}$

D

$6 \sqrt{5} \mathrm{~ms}^{-1}$

Question 28: Palette 28/141

JEE Mains 2023 MCQ

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

${\text {(A), (B) and (D) only }}$

B

(C) and (D) only

C

(A), (B) and (C) Only

D

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

Question 29: Palette 29/141

JEE Mains 2023 MCQ

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

A

$\frac{1}{\sqrt{2}} A$

B

$\frac{1}{2} A$

C

$2 \mathrm{~A}$

D

$\sqrt{2 A}$

Question 30: Palette 30/141

JEE Mains 2023 MCQ

Which graph represents the difference between total energy and potential energy of a particle executing SHM vs it's distance from mean position ?

A

B

C

D

Question 31: Palette 31/141

JEE Mains 2023 MCQ

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

1 : 4

C

2 : 1

D

1 : 3

Question 32: Palette 32/141

JEE Mains 2023 MCQ

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

B

C

D

Question 33: Palette 33/141

JEE Mains 2023 MCQ

A particle executes S.H.M. of amplitude A along x-axis. At t = 0, the position of the particle is $x=\frac{A}{2}$ and it moves along positive x-axis. The displacement of particle in time t is $x = A\sin (wt + \delta )$, then the value of $\delta$ will be

A

$\frac{\pi}{2}$

B

$\frac{\pi}{3}$

C

$\frac{\pi}{4}$

D

$\frac{\pi}{6}$

Question 34: Palette 34/141

JEE Mains 2023 MCQ

For particle P revolving round the centre O with radius of circular path $\mathrm{r}$ and angular velocity $\omega$, as shown in below figure, the projection of OP on the $x$-axis at time $t$ is

A

$x(t)=\operatorname{cos}\left(\omega t-\frac{\pi}{6} \omega\right)$

B

$x(t)=\operatorname{cos}(\omega t)$

C

$x(t)=r \cos \left(\omega t+\frac{\pi}{6}\right)$

D

$x(t)=r \sin \left(\omega t+\frac{\pi}{6}\right)$

Question 35: Palette 35/141

JEE Mains 2023 MCQ

A mass $m$ is attached to two strings as shown in figure. The spring constants of two springs are $\mathrm{K}_{1}$ and $\mathrm{K}_{2}$. For the frictionless surface, the time period of oscillation of mass $m$ is :

A

$2\pi \sqrt {{m \over {{K_1} + {K_2}}}} $

B

$2\pi \sqrt {{m \over {{K_1} - {K_2}}}} $

C

${1 \over {2\pi }}\sqrt {{{{K_1} + {K_2}} \over m}} $

D

${1 \over {2\pi }}\sqrt {{{{K_1} - {K_2}} \over m}} $

Question 36: Palette 36/141

JEE Mains 2023 MCQ

Choose the correct length (L) versus square of the time period ($\mathrm{T}^{2}$) graph for a simple pendulum executing simple harmonic motion.

A

B

C

D

Question 37: Palette 37/141

JEE Mains 2023 MCQ

The maximum potential energy of a block executing simple harmonic motion is $25 \mathrm{~J}$. A is amplitude of oscillation. At $\mathrm{A / 2}$, the kinetic energy of the block is

A

9.75 J

B

37.5 J

C

18.75 J

D

12.5 J

Question 38: Palette 38/141

JEE Mains 2023 MCQ

For a simple harmonic motion in a mass spring system shown, the surface is frictionless. When the mass of the block is $1 \mathrm{~kg}$, the angular frequency is $\omega_{1}$. When the mass block is $2 \mathrm{~kg}$ the angular frequency is $\omega_{2}$. The ratio $\omega_{2} / \omega_{1}$ is

A

$1 / \sqrt{2}$

B

$1 / 2$

C

2

D

$\sqrt{2}$

Question 39: Palette 39/141

JEE Mains 2023 MCQ

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

3 s

D

2 s

Question 40: Palette 40/141

JEE Mains 2023 MCQ

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

$\frac{1}{2}$

C

2

D

$\frac{1}{4}$

Question 41: Palette 41/141

JEE Mains 2022 MCQ

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

$2 \pi \sqrt{\mathrm{L} /(\mathrm{g} \cos \alpha)}$

B

$2 \pi \sqrt{\mathrm{L} /(\mathrm{g} \sin \alpha)}$

C

$2 \pi \sqrt{\mathrm{L} / \mathrm{g}}$

D

$2 \pi \sqrt{\mathrm{L} /(\mathrm{g} \tan \alpha)}$

Question 42: Palette 42/141

JEE Mains 2022 MCQ

Assume there are two identical simple pendulum clocks. Clock - 1 is placed on the earth and Clock - 2 is placed on a space station located at a height h above the earth surface. Clock - 1 and Clock - 2 operate at time periods 4 s and 6 s respectively. Then the value of h is -

(consider radius of earth $R_{E}=6400 \mathrm{~km}$ and $\mathrm{g}$ on earth $10 \mathrm{~m} / \mathrm{s}^{2}$ )

A

1200 km

B

1600 km

C

3200 km

D

4800 km

Question 43: Palette 43/141

JEE Mains 2022 MCQ

When a particle executes Simple Hormonic Motion, the nature of graph of velocity as a function of displacement will be :

A

Circular

B

Elliptical

C

Sinusoidal

D

Straight line

Question 44: Palette 44/141

JEE Mains 2022 MCQ

In figure $(\mathrm{A})$, mass '$2 \mathrm{~m}^{\text {' }}$ is fixed on mass '$\mathrm{m}$ ' which is attached to two springs of spring constant $\mathrm{k}$. In figure (B), mass '$\mathrm{m}$' is attached to two springs of spring constant '$\mathrm{k}$' and '$2 \mathrm{k}^{\prime}$. If mass '$\mathrm{m}$' in (A) and in (B) are displaced by distance '$x^{\prime}$ horizontally and then released, then time period $\mathrm{T}_{1}$ and $\mathrm{T}_{2}$ corresponding to $(\mathrm{A})$ and (B) respectively follow the relation.

A

$ \frac{\mathrm{T}_{1}}{\mathrm{~T}_{2}}=\frac{3}{\sqrt{2}} $

B

$ \frac{\mathrm{T}_{1}}{\mathrm{~T}_{2}}=\sqrt{\frac{3}{2}} $

C

$ \frac{\mathrm{T}_{1}}{\mathrm{~T}_{2}}=\sqrt{\frac{2}{3}} $

D

$ \frac{T_{1}}{T_{2}}=\frac{\sqrt{2}}{3} $

Question 45: Palette 45/141

JEE Mains 2022 MCQ

The motion of a simple pendulum executing S.H.M. is represented by the 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

Question 46: Palette 46/141

JEE Mains 2022 MCQ

Motion of a particle in x-y plane is described by a set of following equations $x = 4\sin \left( {{\pi \over 2} - \omega t} \right)\,m$ and $y = 4\sin (\omega t)\,m$. The path of the particle will be :

A

circular

B

helical

C

parabolic

D

elliptical

Question 47: Palette 47/141

JEE Mains 2022 MCQ

The equation of a particle executing simple harmonic motion is given by $x = \sin \pi \left( {t + {1 \over 3}} \right)m$. 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

Question 48: Palette 48/141

JEE Mains 2022 MCQ

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

12 s

D

36 s

Question 49: Palette 49/141

JEE Mains 2022 MCQ

Time period of a simple pendulum in a stationary lift is 'T'. If the lift accelerates with ${g \over 6}$ vertically upwards then the time period will be :

(Where g = acceleration due to gravity)

A

$\sqrt {{6 \over 5}} T$

B

$\sqrt {{5 \over 6}} T$

C

$\sqrt {{6 \over 7}} T$

D

$\sqrt {{7 \over 6}} T$

Question 50: Palette 50/141

JEE Mains 2022 MCQ

Two massless springs with spring constants 2 k and 9 k, carry 50 g and 100 g masses at their free ends. These two masses oscillate vertically such that their maximum velocities are equal. Then, the ratio of their respective amplitudes will be :

A

1 : 2

B

3 : 2

C

3 : 1

D

2 : 3

Question 51: Palette 51/141

JEE Mains 2021 MCQ

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/s²

B

5 m/s²

C

4 m/s²

D

9.8 m/s²

Question 52: Palette 52/141

JEE Mains 2021 MCQ

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 ${1 \over 4}$ 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

${3 \over 2}$T

C

${3 \over 4}$T

D

${4 \over 3}$T

Question 53: Palette 53/141

JEE Mains 2021 MCQ

For a body executing S.H.M. :

(1) Potential energy is always equal to its K.E.

(2) Average potential and kinetic energy over any given time interval are always equal.

(3) Sum of the kinetic and potential energy at any point of time is constant.

(4) 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

(3) and (4)

B

only (3)

C

(2) and (3)

D

only (2)

Question 54: Palette 54/141

JEE Mains 2021 MCQ

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

B

C

D

Question 55: Palette 55/141

JEE Mains 2021 MCQ

An object of mass 0.5 kg is executing simple harmonic motion. It amplitude is 5 cm and time period (T) is 0.2 s. What will be the potential energy of the object at an instant $t = {T \over 4}s$ starting from mean position. Assume that the initial phase of the oscillation is zero.

A

0.62 J

B

6.2 $\times$ 10^$-$3 J

C

1.2 $\times$ 10³ J

D

6.2 $\times$ 10³ J

Question 56: Palette 56/141

JEE Mains 2021 MCQ

A particle starts executing simple harmonic motion (SHM) of amplitude 'a' and total energy E. At any instant, its kinetic energy is ${{3E} \over 4}$ then its displacement 'y' is given by :

A

y = a

B

$y = {a \over {\sqrt 2 }}$

C

$y = {{a\sqrt 3 } \over 2}$

D

$y = {a \over 2}$

Question 57: Palette 57/141

JEE Mains 2021 MCQ

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

${1 \over 2}$

B

${3 \over 4}$

C

${1 \over 3}$

D

${1 \over 4}$

Question 58: Palette 58/141

JEE Mains 2021 MCQ

T₀ is the time period of a simple pendulum at a place. if the length of the pendulum is reduced to ${1 \over {16}}$ times of its initial value, the modified time period is :

A

4 T₀

B

${1 \over {4}}$ T₀

C

T₀

D

8$\pi$ T₀

Question 59: Palette 59/141

JEE Mains 2021 MCQ

A particle is making simple harmonic motion along the X-axis. If at a distances x₁ and x₂ from the mean position the velocities of the particle are v₁ and v₂ respectively. The time period of its oscillation is given as :

A

$T = 2\pi \sqrt {{{x_2^2 + x_1^2} \over {v_1^2 - v_2^2}}} $

B

$T = 2\pi \sqrt {{{x_2^2 + x_1^2} \over {v_1^2 + v_2^2}}} $

C

$T = 2\pi \sqrt {{{x_2^2 - x_1^2} \over {v_1^2 + v_2^2}}} $

D

$T = 2\pi \sqrt {{{x_2^2 - x_1^2} \over {v_1^2 - v_2^2}}} $

Question 60: Palette 60/141

JEE Mains 2021 MCQ

The function of time representing a simple harmonic motion with a period of ${\pi \over \omega }$ is :

A

cos($\omega$t) + cos(2$\omega$t) + cos(3$\omega$t)

B

sin²($\omega$t)

C

sin($\omega$t) + cos($\omega$t)

D

3cos$\left( {{\pi \over 4} - 2\omega t} \right)$

Question 61: Palette 61/141

JEE Mains 2021 MCQ

The time period of a simple pendulum is given by $T = 2\pi \sqrt {{l \over g}} $. The measured value of the length of pendulum is 10 cm known to a 1mm accuracy. The time for 200 oscillations of the pendulum is found to be 100 second using a clock of 1s resolution. The percentage accuracy in the determination of 'g' using this pendulum is 'x'. The value of 'x' to be nearest integer is :-

A

2%

B

3%

C

5%

D

4%

Question 62: Palette 62/141

JEE Mains 2021 MCQ

Two particles A and B of equal masses are suspended from two massless springs of spring constants K₁ and K₂ respectively. If the maximum velocities during oscillations are equal, the ratio of the amplitude of A and B is

A

${{{K_1}} \over {{K_2}}}$

B

$\sqrt {{{{K_1}} \over {{K_2}}}} $

C

${{{K_2}} \over {{K_1}}}$

D

$\sqrt {{{{K_2}} \over {{K_1}}}} $

Question 63: Palette 63/141

JEE Mains 2021 MCQ

A block of mass 1 kg attached to a spring is made to oscillate with an initial amplitude of 12 cm. After 2 minutes the amplitude decreases to 6 cm. Determine the value of the damping constant for this motion . (take ln 2 = 0.693)

A

0.69 $\times$ 10² kg s^$-$1

B

3.3 $\times$ 10² kg s^$-$1

C

1.16 $\times$ 10^$-$2 kg s^$-$1

D

5.7 $\times$ 10^$-$3 kg s^$-$1

Question 64: Palette 64/141

JEE Mains 2021 MCQ

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

A

x = ${A \over 2}$

B

x = $\pm$ A

C

x = $\pm$ ${A \over {\sqrt 2 }}$

D

x = 0

Question 65: Palette 65/141

JEE Mains 2021 MCQ

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 drop to half of its initial value? (ln 2 = 0.693)

A

17.32 s

B

34.65 s

C

0.034 s

D

15.01 s

Question 66: Palette 66/141

JEE Mains 2021 MCQ

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

$\sqrt {{2 \over 3}} T$

C

${T \over {\sqrt 3 }}$

D

$\sqrt {{3 \over 2}} T$

Question 67: Palette 67/141

JEE Mains 2021 MCQ

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

A

a parabola

B

a helix

C

an ellipse

D

a circle

Question 68: Palette 68/141

JEE Mains 2021 MCQ

Given below are two statements :

Statement I : A second's pendulum has a time period of 1 second.

Statement II : 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 given below :

A

Both Statement I and Statement II are false

B

Statement I is false but Statement II is true

C

Both Statement I and Statement II are true

D

Statement I is true but Statement II is false

Question 69: Palette 69/141

JEE Mains 2021 MCQ

Assume that a tunnel is dug along a chord of the earth, at a perpendicular distance (R/2) from the earth's centre, where 'R' is the radius of the Earth. The wall of the tunnel is frictionless. If a particle is released in this tunnel, it will execute a simple harmonic motion with a time period :

A

$2\pi \sqrt {{R \over g}} $

B

${g \over {2\pi R}}$

C

${{2\pi R} \over g}$

D

${1 \over {2\pi }}\sqrt {{g \over R}} $

Question 70: Palette 70/141

JEE Mains 2021 MCQ

If two similar springs each of spring constant K₁ are joined in series, the new spring constant and time period would be changed by a factor :

A

${1 \over 2},2\sqrt 2 $

B

${1 \over 4},2\sqrt 2 $

C

${1 \over 2},\sqrt 2 $

D

${1 \over 4},\sqrt 2 $

Question 71: Palette 71/141

JEE Mains 2021 MCQ

Y = A sin($\omega$t + $\phi$₀) is the time-displacement equation of a SHM. At t = 0 the displacement of the particle is $Y = {A \over 2}$ and it is moving along negative x-direction. Then the initial phase angle $\phi$₀ will be:

A

${{5\pi } \over 6}$

B

${{\pi } \over 3}$

C

${{2\pi } \over 3}$

D

${{\pi } \over 6}$

Question 72: Palette 72/141

JEE Mains 2021 MCQ

Two identical springs 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. The time period of oscillations of this system is :

A

$2\pi \sqrt {{m \over k}} $

B

$\pi \sqrt {{m \over k}} $

C

$2\pi \sqrt {{m \over {2k}}} $

D

$\pi \sqrt {{m \over {2k}}} $

Question 73: Palette 73/141

JEE Mains 2021 MCQ

The point A moves with a uniform speed along the circumference of a circle of radius 0.36 m and covers 30$^\circ$ in 0.1 s. The perpendicular projection 'P' from '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

9.87 N

B

0.49 N

C

50 N

D

100 N

Question 74: Palette 74/141

JEE Mains 2021 MCQ

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

A

16 m/s²

B

2$\pi$² ms^$-$2

C

$\pi$² ms^$-$2

D

9.8 ms^$-$2

Question 75: Palette 75/141

JEE Mains 2021 MCQ

In the given figure, a body of mass M is held between two massless springs, on a smooth inclined plane. The free ends of the springs are attached to firm supports. If each spring has spring constant k, the frequency of oscillation of given body is :

A

${1 \over {2\pi }}\sqrt {{{2k} \over {Mg\sin \alpha }}} $

B

${1 \over {2\pi }}\sqrt {{k \over {Mg\sin \alpha }}} $

C

${1 \over {2\pi }}\sqrt {{{2k} \over M}} $

D

${1 \over {2\pi }}\sqrt {{k \over {2M}}} $

Question 76: Palette 76/141

JEE Mains 2021 MCQ

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

A

circular

B

straight line

C

parabolic

D

elliptical

Question 77: Palette 77/141

JEE Mains 2021 MCQ

The period of oscillation of a simple pendulum is $T = 2\pi \sqrt {{L \over g}} $. Measured value of 'L' is 1.0 m from meter scale having a minimum division of 1 mm and time of one complete oscillation is 1.95 s measured from stopwatch of 0.01 s resolution. The percentage error in the determination of 'g' will be :

A

1.30%

B

1.33%

C

1.13%

D

1.03%

Question 78: Palette 78/141

JEE Mains 2021 MCQ

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 oscillations will be :

A

$A\sqrt {{M \over {M - m}}} $

B

$A\sqrt {{{M - m} \over M}} $

C

$A\sqrt {{{M + m} \over M}} $

D

$A\sqrt {{M \over {M + m}}} $

Question 79: Palette 79/141

JEE Mains 2020 MCQ

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) = y₀ sin² $\omega $t, where 'y' is measured from the lower end of unstretched spring. Then $\omega $ is:

A

$\sqrt {{g \over {{y_0}}}} $

B

${1 \over 2}\sqrt {{g \over {{y_0}}}} $

C

$\sqrt {{{2g} \over {{y_0}}}} $

D

$\sqrt {{g \over {2{y_0}}}} $

Question 80: Palette 80/141

JEE Mains 2020 MCQ

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

1

B

${1 \over 2}$

C

$\sqrt 2 $

D

${1 \over {\sqrt 2 }}$

Question 81: Palette 81/141

JEE Mains 2020 MCQ

The displacement time graph of a particle executing S.H.M is given in figure :
(sketch is schematic and not to scale)
Which of the following statements is/are true for this motion?
(A) The force is zero at t = ${{3T} \over 4}$
(B) The acceleration is maximum at t = T
(C) The speed is maximum at t = ${{T} \over 4}$
(D) The P.E. is equal to K.E. of the oscillation at t = ${{T} \over 2}$

A

(B), (C) and (D)

B

(A), (B) and (C)

C

(A) and (D)

D

(A), (B) and (D)

Question 82: Palette 82/141

JEE Mains 2020 MCQ

A spring mass system (mass m, spring constant k and natural length $l$) rest in equilibrium on a horizontal disc. The free end of the spring is fixed at the centre of the disc. If the disc together with spring mass system, rotates about it's axis with an angular velocity $\omega $, (k $ \gg m{\omega ^2}$) the relative change in the length of the spring is best given by the option :

A

${{m{\omega ^2}} \over {3k}}$

B

${{m{\omega ^2}} \over k}$

C

${{2m{\omega ^2}} \over k}$

D

$\sqrt {{2 \over 3}} \left( {{{m{\omega ^2}} \over k}} \right)$

Question 83: Palette 83/141

JEE Mains 2020 MCQ

A simple pendulum is being used to determine th value of gravitational acceleration g at a certain place. Th length of the pendulum is 25.0 cm and a stop watch with 1s resolution measures the time taken for 40 oscillations to be 50 s. The accuracy in g is :

A

4.40%

B

3.40%

C

2.40%

D

5.40%

Question 84: Palette 84/141

JEE Mains 2019 MCQ

The displacement of a damped harmonic oscillator is given by
x(t ) = e^–0.1t cos (10$\pi $t + f).
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

27 s

B

13 s

C

7 s

D

4 s

Question 85: Palette 85/141

JEE Mains 2019 MCQ

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 1/1000 of the original amplitude is close to :-

A

100 s

B

10 s

C

20 s

D

50 s

Question 86: Palette 86/141

JEE Mains 2019 MCQ

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, ${3 \over 2}$ s

B

5 cm, ${2 \over 3}$ s

C

5 cm, ${3 \over 2}$ s

D

10 cm, ${2 \over 3}$ s

Question 87: Palette 87/141

JEE Mains 2019 MCQ

Two light identical springs of spring constant k are attached horizontally at the two ends of a uniform horizontal rod AB of length $\ell $ 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

${1 \over {2\pi }}\sqrt {{{3k} \over m}} $

B

${1 \over {2\pi }}\sqrt {{{6k} \over m}} $

C

${1 \over {2\pi }}\sqrt {{k \over m}} $

D

${1 \over {2\pi }}\sqrt {{{2k} \over m}} $

Question 88: Palette 88/141

JEE Mains 2019 MCQ

A pendulum is executing simple harmonic motion and its maximum kinetic energy is K₁. 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 K₂. Then :

A

${K_2}$ = ${{{K_1}} \over 2}$

B

K₂ = 2K₁

C

K₂ = K₁

D

K₂ = ${{{K_1}} \over 4}$

Question 89: Palette 89/141

JEE Mains 2019 MCQ

The mass and the diameter of a planet are three times the respective values for the Earth. The period of oscillation of simple pendulum on the Earth is 2 s. The period of oscillation of the same pendulum on the planet would be :

A

${{\sqrt 3 } \over 2}$ s

B

${3 \over 2}$ s

C

${2 \over {\sqrt 3 }}$ s

D

$2\sqrt 3 $ s

Question 90: Palette 90/141

JEE Mains 2019 MCQ

A simple pendulum of length 1 m is oscillating with an angular frequency 10 rad/s. The support of the pendulum starts oscillating up and down with a small angular frequency of 1 rad/s and an amplitude of 10^–2 m. The relative change in the angular frequency of the pendulum is best given by :

A

1 rad/s

B

10^$-$3 rad/s

C

10^$-$1 rad/s

D

10^$-$5 rad/s

Question 91: Palette 91/141

JEE Mains 2019 MCQ

A particle undergoing simple harmonic motion has time dependent displacement given by x(t) = Asin${{\pi t} \over {90}}$. The ratio of kinetic to potential energy of this particle at t = 210 s will be:

A

${1 \over 9}$

B

3

C

2

D

1

Question 92: Palette 92/141

JEE Mains 2019 MCQ

A particle executes simple harmonic motion with an amplitude of 5 cm. When the particle is at 4 cm from the mean position, the magnitude of its velocity in SI units is equal to that of its acceleration. Then, its periodic time in seconds is -

A

${{4\pi } \over 3}$

B

${3 \over 8}\pi $

C

${7 \over 3}\pi $

D

${{8\pi } \over 3}$

Question 93: Palette 93/141

JEE Mains 2019 MCQ

A closed organ pipe has a fundamental frequency of 1.5 kHz. The number of overtones that can be distinctly heard by a person with this organ pipe will be (Assume that the highest frequency a person can hear is 20,000 Hz)

A

4

B

7

C

6

D

5

Question 94: Palette 94/141

JEE Mains 2019 MCQ

A cylindrical plastic bottle of negligible mass is filled with 310 ml of water and left floating in a pond with still water. If pressed downward slightly and released, it starts performing simple harmonic motion at angular frequency $\omega $. If the radius of the bottle is 2.5 cm then $\omega $ is close to – (density of water = 10³ kg/m³).

A

2.50 rad s^$-$1

B

3.75 rad s^$-$1

C

5.00 rad s^$-$1

D

7.90 rad s^$-$1

Question 95: Palette 95/141

JEE Mains 2019 MCQ

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 radio m/M is close to :

A

0.77

B

0.57

C

0.37

D

0.17

Question 96: Palette 96/141

JEE Mains 2019 MCQ

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

${A \over 2}$

B

${A \over {2\sqrt 2 }}$

C

${A \over {\sqrt 2 }}$

D

A

Question 97: Palette 97/141

JEE Mains 2018 MCQ

An oscillator of mass M is at rest in its equilibrium position in a potential
V = ${1 \over 2}$ k(x $-$ X)². A particle of mass m comes from right with speed u and collides completely inelastically with M and sticks to it. This process repeats every time the oscillator crosses its equilibrium position. The amplitude of oscillations after 13 collisions is : (M = 10, m = 5, u = 1, k = 1)

A

${1 \over {\sqrt 3 }}$

B

${1 \over 2}$

C

${2 \over 3}$

D

${3 \over {\sqrt 5 }}$

Question 98: Palette 98/141

JEE Mains 2018 MCQ

A particle executes simple harmonic motion and is located at x = a, b and c at times t₀, 2t₀ and 3t₀ respectively. The freqquency of the oscillation is :

A

${1 \over {2\,\pi \,{t_0}}}{\cos ^{ - 1}}\left( {{{a + c} \over {2b}}} \right)$

B

${1 \over {2\,\pi \,{t_0}}}{\cos ^{ - 1}}\left( {{{a + b} \over {2c}}} \right)$

C

${1 \over {2\,\pi \,{t_0}}}{\cos ^{ - 1}}\left( {{{2a + 3c} \over b}} \right)$

D

${1 \over {2\,\pi \,{t_0}}}{\cos ^{ - 1}}\left( {{{a + 2b} \over {3c}}} \right)$

Question 99: Palette 99/141

JEE Mains 2018 MCQ

A silver atom in a solid oscillates in simple harmonic motion in some direction with a frequency of 10¹²/sec. What is the force constant of the bonds connecting one atom with the other? (Mole wt. of silver = 108 and Avogadro number = 6.02 × 10²³ gm mole^–1)

A

5.5 N/m

B

6.4 N/m

C

7.1 N/m

D

2.2 N/m

Question 100: Palette 100/141

JEE Mains 2018 MCQ

Two simple harmonic motions, as shown below, are at right angles. They are combined to form Lissajous figures.
x(t) = A sin (at + $\delta $)
y(t) = B sin (bt)

Identify the correct match below.

A

Parameters   A $ \ne $ B, a = b; $\delta $ = 0;
Curve    Parabola

B

Parameters    A = B, a = b; $\delta $ = ${\raise0.5ex\hbox{$\scriptstyle \pi $} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}$
Curve    Line

C

Parameters    A $ \ne $ B, a = b; $\delta $ = ${\raise0.5ex\hbox{$\scriptstyle \pi $} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}$
Curve    Ellipse

D

Parameters    A = B, a = 2b; $\delta $ = ${\raise0.5ex\hbox{$\scriptstyle \pi $} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}$
Curve    Circle

Question 101: Palette 101/141

JEE Mains 2017 MCQ

A block of mass 0.1 kg is connected to an elastic spring of spring constant 640 Nm⁻¹ and oscillates in a damping medium of damping constant 10⁻² kg s⁻¹ . The system dissipates its energy gradually. The time taken for its mechanical energy of vibration to drop to half of its initial value, is closest to :

A

2 s

B

3.5 s

C

5 s

D

7 s

Question 102: Palette 102/141

JEE Mains 2017 MCQ

The ratio of maximum acceleration to maximum velocity in a simple harmonic motion is 10 s⁻¹ . At, t = 0 the displacement is 5 m. What is the maximum acceleration ? The initial phase is ${\pi \over 4}$.

A

500 m/s²

B

500 $\sqrt 2 m/$ s²

C

750 m/s²

D

750 $\sqrt 2 $m / s²

Question 103: Palette 103/141

JEE Mains 2017 MCQ

A 1 kg block attached to a spring vibrates with a frequency of 1 Hz on a frictionless horizontal table. Two springs identical to the original spring are attached in parallel to an 8 kg block placed on the same table. So, the frequency of vibration of the 8 kg block is :

A

${1 \over 4}Hz$

B

${1 \over {2\sqrt 2 }}Hz$

C

${1 \over 2}Hz$

D

$2$ $Hz$

Question 104: Palette 104/141

JEE Mains 2017 MCQ

A particle is executing simple harmonic motion with a time period T. At time t = 0, it is at its position of equilibrium. The kinetic energy – time graph of the particle will look like:

A

B

C

D

Question 105: Palette 105/141

JEE Mains 2016 MCQ

In an engine the piston undergoes vertical simple harmonic motion with amplitude 7 cm. A washer rests on top of the piston and moves with it. The motor speed is slowly increased. The frequency of the piston at which the washer no longer stays in contact with the piston, is close to :

A

0.1 Hz

B

1.2 Hz

C

0.7 Hz

D

1.9 Hz

Question 106: Palette 106/141

JEE Mains 2016 MCQ

Two particles are performing simple harmonic motion in a straight line about the same equilibrium point. The amplitude and time period for both particles are same and equal to A and I, respectively. At time t = 0 one particle has displacement A while the other one has displacement ${{ - A} \over 2}$ and they are moving towards each other. If they cross each other at time t, then t is :

A

${T \over 6}$

B

${5T \over 6}$

C

${T \over 3}$

D

${T \over 4}$

Question 107: Palette 107/141

JEE Mains 2016 MCQ

A particle performs simple harmonic motion with amplitude $A.$ Its speed is trebled at the instant that it is at a distance ${{2A} \over 3}$ from equilibrium position. The new amplitude of the motion is:

A

$A\sqrt 3 $

B

${{7A} \over 3}$

C

${A \over 3}\sqrt {41} $

D

$3A$

Question 108: Palette 108/141

JEE Mains 2015 MCQ

The period of oscillation of a simple pendulum is $T = 2\pi \sqrt {{L \over g}} $. Measured value of L is 20.0 cm known to 1 mm accuracy and time for 100 oscillations of the pendulum is found to be 90 s using wrist watch of 1 s resolution. The accuracy in the determination of g is:

A

1 %

B

5 %

C

2 %

D

3 %

Question 109: Palette 109/141

JEE Mains 2015 MCQ

A pendulum made of a uniform wire of cross sectional area $A$ has time period $T.$ When an additional mass $M$ is added to its bob, the time period changes to ${T_{M.}}$ If the Young's modulus of the material of the wire is $Y$ then ${1 \over Y}$ is equal to :
($g=$ $gravitational$ $acceleration$)

A

$\left[ {1 - {{\left( {{{{T_M}} \over T}} \right)}^2}} \right]{A \over {Mg}}$

B

$\left[ {1 - {{\left( {{T \over {{T_M}}}} \right)}^2}} \right]{A \over {Mg}}$

C

$\left[ {{{\left( {{{{T_M}} \over T}} \right)}^2} - 1} \right]{A \over {Mg}}$

D

$\left[ {{{\left( {{{{T_M}} \over T}} \right)}^2} - 1} \right]{{Mg} \over A}$

Question 110: Palette 110/141

JEE Mains 2015 MCQ

For a simple pendulum, a graph is plotted between its kinetic energy $(KE)$ and potential energy $(PE)$ against its displacement $d.$ Which one of the following represents these correctly?
$(graphs$ $are$ $schematic$ $and$ $not$ $drawn$ $to$ $scale)$

A

B

C

D

Question 111: Palette 111/141

JEE Mains 2014 MCQ

A particle moves with simple harmonic motion in a straight line. In first $\tau s,$ after starting from rest it travels a distance $a,$ and in next $\tau s$ it travels $2a,$ in same direction, then:

A

amplitude of motion is $3a$

B

time period of oscillations is $8\tau $

C

amplitude of motion is $4a$

D

time period of oscillations is $6\tau $

Question 112: Palette 112/141

JEE Mains 2013 MCQ

The amplitude of a damped oscillator decreases to $0.9$ times its original magnitude in $5s$. In another $10s$ it will decrease to $\alpha $ times its original magnitude, where $\alpha $ equals

A

$0.7$

B

$0.81$

C

$0.729$

D

$0.6$

Question 113: Palette 113/141

JEE Mains 2013 MCQ

An ideal gas enclosed in a vertical cylindrical container supports a freely moving piston of mass $M.$ The piston and the cylinder have equal cross sectional area $A$. When the piston is in equilibrium, the volume of the gas is ${V_0}$ and its pressure is ${P_0}.$ The piston is slightly displaced from the equilibrium position and released,. Assuming that the system is completely isolated from its surrounding, the piston executes a simple harmonic motion with frquency

A

${1 \over {2\pi }}\,{{A\gamma {P_0}} \over {{V_0}M}}$

B

${1 \over {2\pi }}\,{{{V_0}M{P_0}} \over {{A^2}\gamma }}$

C

${1 \over {2\pi }}\,\sqrt {{{A\gamma {P_0}} \over {{V_0}M}}} $

D

${1 \over {2\pi }}\,\sqrt {{{M{V_0}} \over {A\gamma {P_0}}}} $

Question 114: Palette 114/141

JEE Mains 2012 MCQ

If a simple pendulum has significant amplitude (up to a factor of $1/e$ of original ) only in the period between $t = 0s\,\,to\,\,t = \tau \,s,$ then $\tau \,$ may be called the average life of the pendulum When the spherical bob of the pendulum suffers a retardation (due to viscous drag) proportional to its velocity with $b$ as the constant of proportionality, the average life time of the pendulum is (assuming damping is small) in seconds :

A

${{0.693} \over b}$

B

$b$

C

${1 \over b}$

D

${2 \over b}$

Question 115: Palette 115/141

JEE Mains 2011 MCQ

Two particles are executing simple harmonic motion of the same amplitude $A$ and frequency $\omega $ along the $x$-axis. Their mean position is separated by distance ${X_0}\left( {{X_0} > A} \right)$. If the maximum separation between them is $\left( {{X_0} + A} \right),$ the phase difference between their motion is:

A

${\pi \over 3}$

B

${\pi \over 4}$

C

${\pi \over 6}$

D

${\pi \over 2}$

Question 116: Palette 116/141

JEE Mains 2011 MCQ

A mass $M,$ attached to a horizontal spring, executes $S.H.M.$ with amplitude ${A_1}.$ When the mass $M$ passes through its mean position then a smaller mass $m$ is placed over it and both of them move together with amplitude ${A_2}.$ The ratio of $\left( {{{{A_1}} \over {{A_2}}}} \right)$ is :

A

${{M + m} \over M}$

B

${\left( {{M \over {M + m}}} \right)^{{1 \over 2}}}$

C

${\left( {{{M + m} \over M}} \right)^{{1 \over 2}}}$

D

${M \over {M + m}}$

Question 117: Palette 117/141

JEE Mains 2009 MCQ

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

$aT/x$

B

$aT + 2\pi v$

C

$aT/v$

D

${a^2}{T^2} + 4{\pi ^2}{v^2}$

Question 118: Palette 118/141

JEE Mains 2007 MCQ

A point mass oscillates along the $x$-axis according to the law $x = {x_0}\,\cos \left( {\omega t - \pi /4} \right).$ If the acceleration of the particle is written as $a = A\,\cos \left( {\omega t + \delta } \right),$ then

A

$A = {x_0}{\omega ^2},\,\,\delta = 3\pi /4$

B

$A = {x_0},\,\,\delta = - \pi /4$

C

$A = {x_0}{\omega ^2},\,\,\delta = \pi /4$

D

$A = {x_0}{\omega ^2},\,\,\delta = - \pi /4$

Question 119: Palette 119/141

JEE Mains 2007 MCQ

Two springs, of force constant ${k_1}$ and ${k_2}$ are connected to a mass $m$ as shown. The frequency of oscillation of the mass is $f.$ If both ${k_1}$ and ${k_2}$ are made four times their original values, the frequency of oscillation becomes

A

$2f$

B

$f/2$

C

$f/4$

D

$4f$

Question 120: Palette 120/141

JEE Mains 2007 MCQ

The displacement of an object attached to a spring and executing simple harmonic motion is given by $x = 2 \times {10^{ - 2}}$ $cos$ $\pi t$ metre. The time at which the maximum speed first occurs is

A

$0.25$ $s$

B

$0.5$ $s$

C

$0.75$ $s$

D

$0.125$ $s$

Question 121: Palette 121/141

JEE Mains 2007 MCQ

A particle of mass $m$ executes simple harmonic motion with amplitude a and frequency $v.$ The average kinetic energy during its motion from the position of equilibrium to the end is

A

$2{\pi ^2}\,m{a^2}{v^2}$

B

${\pi ^2}\,m{a^2}{v^2}$

C

${1 \over 4}\,m{a^2}{v^2}$

D

$4{\pi ^2}m{a^2}{v^2}$

Question 122: Palette 122/141

JEE Mains 2006 MCQ

A coin is placed on a horizontal platform which undergoes vertical simple harmonic motoin of angular frequency $\omega .$ The amplitude of oscillation is gradually increased. The coin will leave contact with the platform for the first time

A

at the mean position of the platform

B

for an amplitude of ${g \over {{\omega ^2}}}$

C

For an amplitude of ${{{g^2}} \over {{\omega ^2}}}$

D

at the height position of the platform

Question 123: Palette 123/141

JEE Mains 2006 MCQ

The maximum velocity of a particle, executing simple harmonic motion with an amplitude $7$ $mm,$ is $4.4$ $m/s.$ The period of oscillation is

A

$0.01$ $s$

B

$10$ $s$

C

$0.1$ $s$

D

$100$ $s$

Question 124: Palette 124/141

JEE Mains 2006 MCQ

Starting from the origin a body oscillates simple harmonically with a period of $2$ $s.$ After what time will its kinetic energy be $75\% $ of the total energy?

A

${1 \over 6}s$

B

${1 \over 4}s$

C

${1 \over 3}s$

D

${1 \over 12}s$

Question 125: Palette 125/141

JEE Mains 2005 MCQ

The function ${\sin ^2}\left( {\omega t} \right)$ represents

A

a periodic, but not $SHM$ with a period ${\pi \over \omega }$

B

a periodic, but not $SHM$ with a period ${{2\pi } \over \omega }$

C

a $SHM$ with a period ${\pi \over \omega }$

D

a $SHM$ with a period ${{2\pi } \over \omega }$

Question 126: Palette 126/141

JEE Mains 2005 MCQ

Two simple harmonic motions are represented by the equations ${y_1} = 0.1\,\sin \left( {100\pi t + {\pi \over 3}} \right)$ and ${y_2} = 0.1\,\cos \,\pi t.$ The phase difference of the velocity of particle $1$ with respect to the velocity of particle $2$ is

A

${\pi \over 3}$

B

${{ - \pi } \over 6}$

C

${\pi \over 6}$

D

${{ - \pi } \over 3}$

Question 127: Palette 127/141

JEE Mains 2005 MCQ

If a simple harmonic motion is represented by ${{{d^2}x} \over {d{t^2}}} + \alpha x = 0.$ its time period is

A

${{2\pi } \over {\sqrt \alpha }}$

B

${{2\pi } \over \alpha }$

C

$2\pi \sqrt \alpha $

D

$2\pi \alpha $

Question 128: Palette 128/141

JEE Mains 2005 MCQ

The bob of a simple pendulum is a spherical hollow ball filled with water. A plugged hole near the bottom of the oscillating bob gets suddenly unplugged. During observation, till water is coming out, the time period of oscillation would

A

first decrease and then increase to the original value

B

first increase and then decrease to the original value

C

increase towards a saturation value

D

remain unchanged

Question 129: Palette 129/141

JEE Mains 2004 MCQ

A particle of mass $m$ is attached to a spring (of spring constant $k$) and has a natural angular frequency ${\omega _0}.$ An external force $F(t)$ proportional to $\cos \,\omega t\left( {\omega \ne {\omega _0}} \right)$ is applied to the oscillator. The time displacement of the oscillator will be proportional to

A

${1 \over {m\left( {\omega _0^2 + {\omega ^2}} \right)}}$

B

${1 \over {m\left( {\omega _0^2 - {\omega ^2}} \right)}}$

C

${m \over {\omega _0^2 - {\omega ^2}}}$

D

${m \over {\omega _0^2 + {\omega ^2}}}$

Question 130: Palette 130/141

JEE Mains 2004 MCQ

In forced oscillation of a particle the amplitude is maximum for a frequency ${\omega _1}$ of the force while the energy is maximum for a frequency ${\omega _2}$ of the force; then

A

${\omega _1} < {\omega _2}$ when damping is small and ${\omega _1} > {\omega _2}$ when damping is large

B

${\omega _1} > {\omega _2}$

C

${\omega _1} = {\omega _2}$

D

${\omega _1} < {\omega _2}$

Question 131: Palette 131/141

JEE Mains 2004 MCQ

The bob of a simple pendulum executes simple harmonic motion in water with a period $t,$ while the period of oscillation of the bob is ${t_0}$ in air. Neglecting frictional force of water and given that the density of the bob is $\left( {4/3} \right) \times 1000\,\,kg/{m^3}.$ What relationship between $t$ and ${t_0}$ is true

A

$t = 2{t_0}$

B

$t = {t_0}/2$

C

$t = {t_0}$

D

$t = 4{t_0}$

Question 132: Palette 132/141

JEE Mains 2004 MCQ

A particle at the end of a spring executes $S.H.M$ with a period ${t_1}$. While the corresponding period for another spring is ${t_2}$. If the period of oscillation with the two springs in series is $T$ then

A

${T^{ - 1}} = t_1^{ - 1} + t_2^{ - 1}$

B

${T^2} = t_1^2 + t_2^2$

C

$T = {t_1} + {t_2}$

D

${T^{ - 2}} = t_1^{ - 2} + t_2^{ - 2}$

Question 133: Palette 133/141

JEE Mains 2004 MCQ

The total energy of particle, executing simple harmonic motion is

A

independent of $x$

B

$ \propto \,{x^2}$

C

$ \propto \,x$

D

$ \propto \,{x^{1/2}}$

Question 134: Palette 134/141

JEE Mains 2003 MCQ

A mass $M$ is suspended from a spring of negligible mass. The spring is pulled a little and then released so that the mass executes $SHM$ of time period $T.$ If the mass is increased by $m.$ the time period becomes ${{5T} \over 3}$. Then the ratio of ${{m} \over M}$ is

A

${3 \over 5}$

B

${25 \over 9}$

C

${16 \over 9}$

D

${5 \over 3}$

Question 135: Palette 135/141

JEE Mains 2003 MCQ

A body executes simple harmonic motion. The potential energy $(P.E),$ the kinetic energy $(K.E)$ and total energy $(T.E)$ are measured as a function of displacement $x.$ Which of the following statements is true ?

A

$K.E$ is maximum when $x=0$

B

$T.E$ is zero when $x=0$

C

$K.E$ is maximum when $x$ is maximum

D

$P.E$ is maximum when $x=0$

Question 136: Palette 136/141

JEE Mains 2003 MCQ

The displacement of particle varies according to the relation
$x=4$$\left( {\cos \,\pi t + \sin \,\pi t} \right).$ The amplitude of the particle is

A

$-4$

B

$4$

C

$4\sqrt 2 $

D

$8$

Question 137: Palette 137/141

JEE Mains 2003 MCQ

Two particles $A$ and $B$ of equal masses are suspended from two massless springs of spring of spring constant ${k_1}$ and ${k_2}$, respectively. If the maximum velocities, during oscillation, are equal, the ratio of amplitude of $A$ and $B$ is

A

$\sqrt {{{{k_1}} \over {{k_2}}}} $

B

${{{{k_2}} \over {{k_1}}}}$

C

$\sqrt {{{{k_2}} \over {{k_1}}}} $

D

${{{{k_1}} \over {{k_2}}}}$

Question 138: Palette 138/141

JEE Mains 2003 MCQ

The length of a simple pendulum executing simple harmonic motion is increased by $21\% $. The percentage increase in the time period of the pendulum of increased length is

A

$11\% $

B

$21\% $

C

$42\% $

D

$10\% $

Question 139: Palette 139/141

JEE Mains 2002 MCQ

In a simple harmonic oscillator, at the mean position

A

kinetic energy is minimum, potential energy is maximum

B

both kinetic and potential energies are maximum

C

kinetic energy is maximum, potential energy is minimum

D

both kinetic and potential energies are minimum.

Question 140: Palette 140/141

JEE Mains 2002 MCQ

A child swinging on a swing in sitting position, stands up, then the time period of the swing will

A

increase

B

decrease

C

remains same

D

increases of the child is long and decreases if the child is short

Question 141: Palette 141/141

JEE Mains 2002 MCQ

If a spring has time period $T,$ and is cut into $n$ equal parts, then the time period of each part will be

A

$T\sqrt n $

B

$T/\sqrt n $

C

$nT$

D

$T$