The potential energy of a particle of mass $10 \ g$ as a function of displacement $x$ is $(50 x^2 + 100) \ J$. The frequency of oscillation is

One end of a $U$-tube containing mercury is connected to a suction pump and the other end to the atmosphere. $A$ small pressure difference is maintained between the two columns. Show that, when the suction pump is removed, the column of mercury in the $U$-tube executes simple harmonic motion.

$A$ body performs $S.H.M.$ under the action of force $F_1$ with period $T_1$ seconds. If the force is changed to $F_2$,it performs $S.H.M.$ with period $T_2$ seconds. If both forces $F_1$ and $F_2$ act simultaneously in the same direction on the body,the period in seconds will be

$A$ ring is suspended from a point $S$ on its rim as shown in the figure. When displaced from equilibrium,it oscillates with a time period of $1 \, s$. The radius of the ring is ..... $m$ (take $g = \pi^2$).

$A$ 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 ........

$A$ 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