A thin rod of length L and uniform cross-section is pivoted at its lowest point P inside a stationary homogeneous and non-viscous liquid. The rod is free to rotate in a vertical plane about a horizontal axis passing through P. The density d₁ of the material of the rod is smaller than the density d₂ of the liquid. The rod is displaced by small angle θ from its equilibrium position and then released. Determine its angular frequency in terms of the given parameters.

The position co-ordinates of a particle moving in a $3D$ coordinate system is given by

$x=a\cos \mathrm{\omega }\mathrm{t}$

$y=a\sin \omega t$

and $z=a\omega t$

The speed of the particle is:

An ideal gas enclosed in a vertical cylindrical container supports a freely moving piston of mass $\mathit{\text{M}}$. The piston and the cylinder have equal cross-sectional area $\mathit{\text{A}}$. When the piston is in equilibrium, the volume of the gas is $V_0$ and its pressure is $M_0$. The piston is slightly displaced from the equilibrium position and released. Assuming that the system is completely isolated from its surrounding, the piston executes a simple harmonic motion with frequency
[Assume the system is in space.]

A simple pendulum has time period $T_1$. The point of suspension is now moved upward according to the relation $y=k{t}^2,\left(k=1 \mathrm{m} {\mathrm{s}}^{-2}\right)$  where $y$ is the vertical displacement. The time
period now becomes $T_2$.

The ratio of $\frac{T_1^2}{T_2^2}\left(g=10 \mathrm{m} {\mathrm{s}}^{-2}\right)$ is