$A$ 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

$A$ particle of mass $4 \,kg$ moves simple harmonically such that its potential energy $U$ varies with position $x$ as shown in the figure. The period of oscillations is ............

$A$ particle of mass $5 \times 10^{-5} \ kg$ is placed at the lowest point of a smooth parabola $x^2 = 40y$ ($x$ and $y$ in $m$). If it is displaced slightly such that it is constrained to move along the parabola,the angular frequency of oscillation (in $rad/s$) will be approximately:

Two light identical springs of spring constant $k$ are attached horizontally at the two ends of a uniform horizontal rod $AB$ of length $l$ and mass $m$. The rod is pivoted at its centre $O$ and can rotate freely in a horizontal plane. The other ends of the two springs are fixed to rigid supports as shown in the figure. The rod is gently pushed through a small angle $\theta$ and released. The frequency of the resulting 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.

$A$ cylindrical piston of mass $M$ slides smoothly inside a long cylinder closed at one end,enclosing a certain mass of gas. The cylinder is kept with its axis horizontal. If the piston is disturbed from its equilibrium position,it oscillates simple harmonically. The period of oscillation will be