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JEE Mains | PYQ

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

Assume that the earth is a solid sphere of uniform density and a tunnel is dug along its diameter throughout the earth. It is found that when a particle is released in this tunnel, it executes a simple harmonic motion. The mass of the particle is \( 100 \, \text{g} \). The time period of the motion of the particle will be (approximately):
(Take \( g = 10 \, \text{ms}^{-2} \), radius of earth = \( 640 \, \text{km} \))

In the figure given below, a block of mass \( M = 490 \, \text{g} \) placed on a frictionless table is connected with two springs have same spring constant \( \left( K = 2 \, \text{N m}^{-1} \right) \). If the block is horizontally displaced through ‘\( X \)’ \( \text{m} \) then the number of complete oscillations it will make in \( 14\pi \) seconds will be_____.

A particle executes simple harmonic motion between \( x = -A \) and \( x = +A \). If time taken by particle to go from \( x = 0 \) to \( x = \frac{A}{2} \) is \( 2 \, \text{s} \); then time taken by particle in going from \( x = \frac{A}{2} \) to \( A \) is:

A simple pendulum with length \( 100 \, \text{cm} \) and bob of mass \( 250 \, \text{g} \) is executing S.H.M. of amplitude \( 10 \, \text{cm} \). The maximum tension in the string is found to be \( \dfrac{x}{40} \, \text{N} \). The value of \( x \) is __________.

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

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 \( m \) is:

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 block of a mass \( 2 \, \text{kg} \) is attached with two identical springs of spring constant \( 20 \, \text{N/m} \) each. The block is placed on a frictionless surface and the ends of the springs are attached to rigid supports (see in figure). When the mass is displaced from its equilibrium position, it executes a simple harmonic motion. The time period of oscillation is \( \frac{\pi}{\sqrt{x}} \) in SI unit. The value of \( x \) is ________.

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 \( t = \frac{T}{\beta} \). The value of \( \beta \) is ________.

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

In a linear Simple Harmonic Motion \( (\text{SHM}) \)
(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 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 \( t \) is \( x = A \sin(\omega t + \delta) \), then the value \( \delta \) will be:

A rectangular block of mass \( 5 \, \text{kg} \) attached to a horizontal spiral spring executes simple harmonic motion of amplitude \( 1 \, \text{m} \) and time period \( 3.14 \, \text{s} \). The maximum force exerted by spring on block is ______ \( \text{N} \).

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 \( 40 \, \text{cm} \) and kinetic energy at that time is \( 200 \, \text{J} \), the value of force constant is \( 1.0 \times 10^{x} \, \text{N m}^{-1} \). The value of \( x \) is ________.

A particle of mass \( 250 \, \text{g} \) executes a simple harmonic motion under a periodic force \( F = (-25 x) \, \text{N} \). The particle attains a maximum speed of \( 4 \, \text{m} \, /\text{s}^{} \) during its oscillation. The amplitude of the motion is ______ cm.