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For a periodic motion represented by the equation \( y = \sin \omega t + \cos \omega t \), the amplitude of the motion is:

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

The velocity of a particular executing SHM varies with displacement \( (x) \) as \( 4v^{2} = 50 - x^{2} \). The time period of oscillations is \( \dfrac{x}{7} \, \text{s} \). The value of \( x \) is ____. \( \left( \text{Take } \pi = \dfrac{22}{7} \right) \)

Two massless springs with spring constants \( 2k \) and \( 9k \), carry \( 50 \ \text{g} \) and \( 100 \ \text{g} \) 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 particle performs simple harmonic motion with a period of \( 2 \, \text{second} \). The time taken by the particle to cover a displacement equal to half of its amplitude from the mean position is \( \frac{1}{a} \, \text{s} \).
The value of \( 'a' \) to the nearest integer is __________ .

The maximum potential energy of a block executing simple harmonic motion is \( 25\,\text{J} \). \( A \) is amplitude of oscillation. At \( \frac{A}{2} \), the kinetic energy of the block is:

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 simple pendulum of string length \( 30 \, \text{cm} \) performs \( 20 \) oscillations in \( 10 \, \text{s} \). The length of the string required for the pendulum to perform \( 40 \) oscillations in the same time duration is _______ \( \text{cm} \).
[Assume that the mass of the pendulum remains same]

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 __________.

Two simple harmonic motions are represented by the equations \( x_1 = 5 \sin \left( 2\pi t + \frac{\pi}{4} \right) \) and \( x_2 = 5\sqrt{2}\left(\sin 2\pi t + \cos 2\pi t\right) \). The amplitude of second motion is ________ times the amplitude in first motion.

The amplitude of a particle executing SHM is \( 3 \, \text{cm} \). The displacement at which its kinetic energy will be \( 25\% \) more than the potential energy is: \( \_\_\_\_ \) \( \text{cm} \).

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:

The kinetic energy of a simple harmonic oscillator is oscillating with angular frequency of \( 176 \; \text{rad/s} \). The frequency of this simple harmonic oscillator is __________ \( \text{Hz} \), \( \left[ \text{take } \pi = \dfrac{22}{7} \right] \).

A mass \( m \) attached to free end of a spring executes SHM with a period of \( 1 \, \text{s} \). If the mass is increased by \( 3 \, \text{kg} \) the period of oscillation increases by one second, the value of mass \( m \) is ________ \( \text{kg} \).

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 \( \dfrac{\pi}{x} \, \text{s} \), where \( x = \) __________.

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:

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

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

Two waves executing simple harmonic motions travelling in the same direction with same amplitude and frequency are superimposed. The resultant amplitude is equal to the \( \sqrt{3} \) times of amplitude of individual motions. The phase difference between the two motions is __________ (degree).

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

Two simple harmonic motion, are represented by the equations
\( y_1 = 10 \sin\left(3\pi t + \dfrac{\pi}{3}\right) \)
\( y_2 = 5 \left( \sin 3\pi t + \sqrt{3} \cos 3\pi t \right) \)
Ratio of amplitude of \( y_1 \) to \( y_2 \) = \( x : 1 \). The value of \( x \) is _______.

The position, velocity and acceleration of a particle executing simple harmonic motion are found to have magnitudes of \( 4 \, \text{m} \), \( 2 \, \text{m s}^{-1} \) and \( 16 \, \text{m s}^{-2} \) at a certain instant. The amplitude of the motion is \( \sqrt{x} \) \( \text{m} \), where \( x \) is ________.

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 \( \omega = \sqrt{\dfrac{K}{m}} \). Suppose that at time \( t = 0 \), the position of mass is \( x(0) \) and velocity \( v(0) \), then its displacement can also be represented as \( x(t) = C \cos( \omega t - \phi) \), where \( C \) and \( \phi \) are:

A simple pendulum doing small oscillations at a place \( R \) height above earth surface has time period of \( T_{1} = 4 \, \text{s} \). \( T_{2} \) would be it’s time period if it is brought to a point which is at a height \( 2R \) from earth surface. Choose the Correct relation \([R = \text{radius of earth}]\):

A particle is executing simple harmonic motion. Its amplitude is \( A \) and time period is \( 5 \, \text{sec} \). The time required by it to move from \( x = A \) to \( x = \frac{A}{\sqrt{2}} \) is _____ \( \text{sec} \).

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 = \( 640 \, \text{km} \))

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:

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

The displacement of a particle, executing simple harmonic motion with time period \( T \), is expressed as \( x(t) = A \sin \omega t \), where \( A \) is the amplitude. The maximum value of potential energy of this oscillator is found at \( t = \frac{T}{2\beta} \). The value of \( \beta \) is __________.

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

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

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 \( v_1 \) and \( v_2 \) respectively. The time period of its oscillation is given as:

Time period of a simple pendulum in a stationary lift is ' \( T \)'. If the lift acceleration with \( \frac{g}{6} \) vertically upwards then the time period will be: (where \( g \) = acceleration due to gravity)

A particle of mass \( 0.50 \, \text{kg} \) executes simple harmonic motion under force \( F = -50 \, (\text{N m}^{-1}) \, x \).The time period of oscillation is \( \dfrac{x}{35} \, \text{s} \). The value of \( x \) is
(Given \( \pi = \dfrac{22}{7} \))

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 \( \dfrac{x}{7} \, \text{s} \), the value of \( x \) is ______.

A particle executes simple harmonic motion with an amplitude of \( 4 \, \text{cm} \). At the mean position, velocity of the particle is \( 10 \, \text{cm/s} \). The distance of the particle from the mean position when its speed becomes \( 5 \, \text{cm/s} \) is \( \sqrt{\alpha} \, \text{cm} \), where \( \alpha = \) ______________.

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 ____________.

If \( R \) is the radius of the earth and the acceleration due to gravity on the surface of earth is \( g = \pi^{2}\, \mathrm{m/s^{2}} \), then the length of the second's pendulum at a height \( h = 2R \) from the surface of earth will be:

Amplitude of a mass-spring system, which is executing simple harmonic motion decreases with time. If mass \( = 500 \, \text{g} \), Decay constant \( = 20 \, \text{g/s} \) 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 particle executes simple harmonic motion represented by displacement function as \( x(t) = A \sin(\omega t + \phi) \). If the position and velocity of the particle at \( t = 0 \, \text{s} \) are \( 2 \, \text{cm} \) and \( 2\omega \, \text{cm s}^{-1} \) respectively, then its amplitude is \( x\sqrt{2} \, \text{cm} \) where the value of \( x \) is __________.

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:

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:

Consider two identical springs each of spring constant \( k \) and negligible mass compared to the mass \( M \) as shown. Figure \( 1 \) shows one of them and Figure \( 2 \) shows their series combination. The ratio of time period of oscillation of the two SHM is \( \frac{T_b}{T_a} = \sqrt{x} \), where value of \( x \) is _______. (Round off to the Nearest Integer)

A mass of \( 5 \, \text{kg} \) 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 \( 4 \, \text{m} \) 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 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 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 \( \_\_\_\_ \).

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)

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 option given below:

A spring of force constant \( 15 \, \text{N/m} \) is cut into two pieces. If the ratio of their length is \( 1:3 \), then the force constant of smaller piece is _______ \( \text{N/m} \).

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 \( a = -\dfrac{kx}{M + m} \)
\( (x = \) displacement of the blocks 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:

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 \( \frac{v_2}{v_1} \) is ______.

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 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:

The equation of a particle executing simple harmonic motion is given by \( x = \sin \!\pi \left( t + \frac{1}{3} \right) ) \, \text{m} \). At \( t = 1 \, \text{s} \), the speed of particle will be (Given : \( \pi = 3.14 \) ).

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:

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 \( x = \) ____________.

\( T_0 \) is the time period of a simple pendulum at a place. If the length of the pendulum is reduced to \( \dfrac{1}{16} \) times of its initial value, the modified time period is:

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 \( 3 \, \text{s} \), then the period of oscillation in figure (b) will be \( \sqrt{x} \, \text{s} \). The value of \( x \) is __________.

In the reported figure, two bodies A and B of masses \( 200 \, \text{g} \) and \( 800 \, \text{g} \) 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 frictionless. The angular frequency will be __________ rad/s when \( k = 20 \, \text{N/m} \).

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

A particle oscillates along the \( x \)-axis according to the law, \(\displaystyle x(t) = x_{0}\sin^{2}\!\left(\frac{t}{2}\right) \) where \( x_{0} = 1 \, \text{m} \). The kinetic energy \( (K) \) of the particle as a function of \( x \) is correctly represented by the graph.

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.

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 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 ________.

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 \( \dfrac{1}{4} \) times that of the bob and the length of the thread is increased by \( \dfrac{1}{3}^{rd} \) of the original length, then the time period of the simple harmonic oscillations will be:

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 ________.

A particle of mass \( 1 \, \text{mg} \) and charge \( q \) is lying at the mid-point of two stationary particles kept at a distance '\( 2 \, \text{m} \)' when each is carrying same charge '\( q \)'. If the free charged particle is displaced from its equilibrium position through distance '\( x \)' (\( x \ll 1 \, \text{m} \)). The particle executes SHM. Its angular frequency of oscillation will be __________ \( \times 10^{5} \, \text{rad/s} \) if \( q^{2} = 10 \, c^{2} \).

A particle is doing simple harmonic motion of amplitude \( 0.06 \, \text{m} \) and time period \( 3.14 \, \text{s} \). The maximum velocity of the particle is ______ cm/s.

The metallic bob of simple pendulum has the relative density \( 5 \). The time period of this pendulum is \( 10\,\text{s} \). If the metallic bob is immersed in water, then the new time period becomes \( 5\sqrt{x}\,\text{s} \). The value of \( x \) will be _____.

Two electrons each are fixed at a distance '\( 2d \)'. A third charge proton placed at the midpoint is displaced slightly by a distance \( x \) (\( x \ll d \)) perpendicular to the line joining the two fixed charges. Proton will execute simple harmonic motion having angular frequency: (\( m \) = mass of charged particle)

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

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

If time \( (t) \), velocity \( (v) \), and angular momentum \( (l) \) are taken as the fundamental units. Then the dimension of mass \( (m) \) in terms of \( t \), \( v \), and \( l \) is:

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

For a simple harmonic motion in a mass spring shown, the surface is frictionless. When the mass of the block is \( 1 \, \text{kg} \), the angular frequency is \( \omega_{1} \). When the mass block is \( 2 \, \text{kg} \) the angular frequency is \( \omega_{2} \). The ratio \( \frac{\omega_{1}}{\omega_{2}} \) is :

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:

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 body is performing simple harmonic with an amplitude of \( 10 \, \text{cm} \). The velocity of the body was tripled by air Jet when it is at \( 5 \, \text{cm} \) from its mean position. The new amplitude of vibration is \( \sqrt{x} \) cm. The value of \( x \) is __________.

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

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 ______ \( \text{N} \).

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:

The displacement of simple harmonic oscillator after \( 3 \, \text{seconds} \) starting from its mean position is equal to half of its amplitude. The time period of harmonic motion is:

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:

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 ________.

The potential energy of a particle of mass \( 4 \, \text{kg} \) in motion along the \( x \)-axis is given by \( U = 4(1 - \cos 4x) \, \text{J} \). The time period of the particle for small oscillation \( (\sin \theta \simeq \theta) \) is \( \left( \dfrac{\pi}{K} \right) \, \text{s} \). The value of \( K \) 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.

A particle of mass \( 1 \, \text{kg} \) is hanging from a spring of force constant \( 100 \, \text{N m}^{-1} \). 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 \( \tfrac{T}{x} \). The value of \( x \) is ________.

The equation of motion of a particle is given by \( x = a \sin\!\left(50 t + \dfrac{\pi}{3}\right)\, \text{cm} \). 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 ________.

In figure (A), mass \( 2m \) is fixed on mass \( m \) which is attached to two springs of spring constant \( k \). In figure (B), mass \( m \) is attached to two springs 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 \( T_1 \) & \( T_2 \) corresponding to (A) and (B) respectively follow the relation.

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JEE Main 2023 8th April Evening Shift

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: