Under the influence of force $F_1$,a body oscillates with a period $T_1$,and due to another force $F_2$,the body oscillates with a period $T_2$. If both forces act simultaneously,what is the resultant period? (Consider the displacement is the same in all three cases.)

$A$ ring is hung on a nail. It can oscillate,without slipping or sliding $(i)$ in its plane with a time period $T_{1}$ and,$(ii)$ back and forth in a direction perpendicular to its plane,with a period $T_{2}$. The ratio $\frac{T_{1}}{T_{2}}$ will be

The potential energy of a particle of mass $1\,kg$ in motion along the $x-$axis is given by $U = 4(1 - \cos 2x)\,J,$ where $x$ is in $meters$. The period of small oscillations (in $sec$) is

If a body is released into a tunnel dug across the diameter of the Earth,it executes simple harmonic motion with a time period of:

$A$ 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).

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