$A$ particle executes linear simple harmonic motion with an amplitude of $2 \ cm$. When the particle is at $1 \ cm$ from the mean position,the magnitude of its velocity is equal to that of its acceleration. Then its time period in seconds is

$A$ 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 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

$A$ 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 ............

$A$ circular disc of mass $10 \; kg$ is suspended by a wire attached to its centre. The wire is twisted by rotating the disc and released. The period of torsional oscillations is found to be $1.5 \; s$. The radius of the disc is $15 \; cm$. Determine the torsional spring constant of the wire in $N \; m \; rad^{-1}$. (Torsional spring constant $\alpha$ is defined by the relation $J = -\alpha \theta$,where $J$ is the restoring couple and $\theta$ is the angle of twist).

$A$ solid cube of side $l$ is made to oscillate about a horizontal axis passing through one of its edges. Its time period will be