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

$A$ block of mass $m$,attached to a spring of spring constant $k$,oscillates on a smooth horizontal table. The other end of the spring is fixed to a wall. The block has a speed $v$ when the spring is at its natural length. Before coming to an instantaneous rest,if the block moves a distance $x$ from the mean position,then

The vertical extension in a light spring by a weight of $1\, kg$ suspended from the wire is $9.8\, cm$. What is the period of oscillation?

$A$ mass $M$ is suspended from a light spring. An additional mass $m$ added displaces the spring further by a distance $x$. Now the combined mass will oscillate on the spring with period:

Two masses $m_1 = 1 \, kg$ and $m_2 = 0.5 \, kg$ are suspended together by a massless spring of spring constant $k = 12.5 \, N/m$. When the masses are in equilibrium,$m_1$ is removed without disturbing the system. The new amplitude of oscillation will be .......... $cm$.

Two bodies $A$ and $B$ of equal mass are suspended from two separate massless springs of spring constants $K_1$ and $K_2$ respectively. The two bodies oscillate vertically such that their maximum velocities are equal. The ratio of the amplitude of $B$ to that of $A$ is