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JEE Mains | PYQ

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 ________.