$A$ small spherical ball of radius '$r$' is rolling on a curved surface which is frictionless and has a radius of curvature '$R$'. Its motion is simple harmonic. Then its time period of oscillation is proportional to ($g=$ acceleration due to gravity).

$A$ particle of mass $m$ is executing oscillations about the origin on the $X-$axis. Its potential energy is $U(x) = k|x|^3$,where $k$ is a positive constant. If the amplitude of oscillation is $a$,then its time period $T$ 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.

$A$ rectangular block of mass $m$ and cross-sectional area $A$ floats in a liquid of density $\rho$. If we give it a small vertical displacement from equilibrium,it undergoes $SHM$ with a time period $T$. Then:

$A$ body executes simple harmonic motion under the action of a force $F_1$ with a time period $T_1 = \frac{4}{5} \ s$. If the force is changed to $F_2$, it executes $SHM$ with a time period $T_2 = \frac{3}{5} \ s$. If both the forces $F_1$ and $F_2$ act simultaneously in the same direction on the body, what will be its new time period in seconds?

$A$ rod of mass $m$ and length $l$ is suspended from the ceiling with two strings of length $l$ as shown. When the rod is given a small push in the plane of the page and released,the time period is $T_1$. When the rod is given a push perpendicular to the plane,the time period of oscillation is $T_2$. The ratio $\frac{T_1^2}{T_2^2}$ is