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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 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 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 _____ \( \text{sec} \).

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 __________.

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 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 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 _______ \( \text{N/m} \).

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 ______.

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.

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 ________.

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JEE MAIN 2026 24TH Jan Morning Shift

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 _______ \( \text{N/m} \).