$A$ metal ball released from height $h$ makes a perfectly elastic collision with the ground. The frequency of the periodic vibratory motion is $\quad(g = \text{acceleration due to gravity})$

$A$ uniform rod of length $2.0 \, m$ is suspended through an end and is set into oscillation with small amplitude under gravity. The time period of oscillation is approximately .... $\sec$.

$A$ particle of mass $m$ is located in a one-dimensional potential field where the potential energy is given by: $V(x) = A(1 - \cos px)$, where $A$ and $p$ are constants. The period of small oscillations of the particle is

$A$ uniform stick of mass $M$ and length $L$ is pivoted at its centre. Its ends are tied to two springs each of force constant $K$. In the position shown in the figure,the springs are in their natural length. When the stick is displaced through a small angle $\theta$ and released,the stick:

$A$ uniform rod of length $L$ and mass $M$ is pivoted at the centre. Its two ends are attached to two springs of equal spring constants $k$. The springs are fixed to rigid supports as shown in the figure,and the rod is free to oscillate in the horizontal plane. The rod is gently pushed through a small angle $\theta$ in one direction and released. The frequency of oscillation is

An air chamber of volume $V$ has a neck of cross-sectional area $a$ into which a ball of mass $m$ just fits and can move up and down without any friction (Figure). Show that when the ball is pressed down a little and released,it executes $SHM$. Obtain an expression for the time period of oscillations assuming pressure-volume variations of air to be isothermal.