Time Period Does Not Depend On?

Consider the situation where the block is hanging on the spring and is in equilibrium. That is, its weight is balanced by the spring force. Suppose the extension of the spring in this case is $x_0$ , then

$mg = kx_0$

Now if we displace the block by $A$ and release it, then the block will oscillate about this equilibrium position with amplitude $A$ . When the block is at distance $x$ away from this equilibrium position, the spring will exert an additional force of magnitude $kx$ towards the equilibrium point.

The force applied by the spring is always proportional to the displacement and is in the opposite direction. The net force on the block is $F = - k(x_0 + kx) + mg$ . However, we know that $mg = kx_0$ , so they cancel out and we get:

$\begin{aligned} F & = -kx \\ a & = - \frac{k}{m} x \\ \frac{d^2x}{dt^2} & = - \frac{k}{m} x\end{aligned}$

The solution for this differential equation is of the form $x = A \sin \left(\sqrt{\frac{k}{m}}t + \phi\right)$ . Its time period is $2 \pi \sqrt{\dfrac{m}{k}}$ , and it is independent of $g$ .