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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 __________ .

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.

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

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 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:

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.

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:

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 __________.

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?

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:

\( T_0 \) 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:

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 \( k = 20 \, \text{N/m} \).

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 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:

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 \( q^{2} = 10 \, c^{2} \).

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)

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:

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 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 ________.

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JEE Main 2021 17th March Morning Shift

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