$A$ 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 ........

$A$ particle is placed at the lowest point of a smooth wire frame in the shape of a parabola,lying in the vertical $xy-$ plane having equation $x^2 = 5y$ ($x, y$ are in meters). After a slight displacement,the particle is set free. Find the angular frequency of oscillation in $rad/s$ (take $g = 10 \ m/s^2$).

$A$ 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 ........

$A$ rectangular block of mass $m$ and cross-sectional area $A$ floats on a liquid of density $\rho$. It is given a small vertical displacement from equilibrium,and it starts oscillating with frequency $n$. The frequency $n$ is equal to (where $g$ is the acceleration due to gravity):

$A$ 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

In both figures shown below, there is a hole along the diameter of the Earth. In the first figure, a particle is released from point $A$ and it oscillates with a time period $T_1$. In the second figure, the same particle is released from point $B$ and it oscillates with a time period $T_2$. Then, [$O$ is the centre of the Earth]