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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 __________.

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 \( x = \) __________.

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 ________.

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 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 \( \pi = \dfrac{22}{7} \))

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:

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 \( \_\_\_\_ \).

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:

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 \( x = \) ____________.

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.

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JEE Main 2024 4th April Evening Shift

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: