$A$ uniform thin ring of radius $R$ and mass $m$ is suspended in a vertical plane from a point on its circumference. Its time period of oscillation is ........

$A$ uniform disc of mass $M$ and radius $R$ is suspended in a vertical plane from a point on its periphery. Its time period of oscillation is ........

$A$ body of mass $m$ is situated in a potential field $U(x) = U_0 (1 - \cos \alpha x)$,where $U_0$ and $\alpha$ are constants. Find the time period of small oscillations.

$A$ 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:

$A$ test tube of mass $6 \ g$ and uniform area of cross-section $10 \ cm^2$ is floating in water vertically when $10 \ g$ of mercury is in the bottom. The tube is depressed by a small amount and then released. The time period of oscillation is (Acceleration due to gravity $= 10 \ m/s^2$) (in $s$)

The relation between the force ($F$ in newton) acting on a particle executing simple harmonic motion and the displacement of the particle ($y$ in metre) is $500 F + \pi^2 y = 0$. If the mass of the particle is $2 \text{ g}$, the time period of oscillation of the particle is (in $\text{ s}$)