The time period of simple harmonic motion of mass $M$ in the given figure is $\pi \sqrt{\frac{\alpha M}{5K}}$, where the value of $\alpha$ is _______.

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.]