Question

The string, the spring and the pulley shown in figure are light. Find the time period of the mass $m$.

Answer 1

In equilibrium position $k{x}_o=mg$ ....(1)

So $x_o=\frac{mg}{k}$

Lets further extension in the spring due to applied force be x, then the resorting force is $F=k(x+x_o)-mg$

$⟹F=-kx$ (using 1)

or $a=\frac{-k}{m}x$ ...(1)

Also $a=-{\omega }^{2}x$ ..(2)

From 1 and 2, we get

${\omega }^2=\frac{k}{m}$

$T=\frac{2\pi }{\omega }$

$T=2\pi \sqrt{\frac{m}{k}}$