Question

Show that the period of oscillation of main on helical spring is given by 

T = 2π √(k/m)

Answer 1

The period of oscillations of a mass oh a helical spring depends on the spring constant and mass. According to Newton's second law:

$$F=ma=m\frac{dx^2}{d^2t}.$$

On the other hand, this is equal to

$$F=-kx.$$

So, we have

$$m\frac{dx^2}{d^2t}=-kx.$$

We know that displacement depends on time according to the following equation:

$$x=A\cos\bigg(\frac{2\pi t}{T}\bigg).$$

Substitution in the equation according to Newton's second law gives

$$-m\frac{(2\pi)^2}{T^2}A\cos\bigg(\frac{2\pi t}{T}\bigg)=-kA\cos\bigg(\frac{2\pi t}{T}\bigg),\\\space\\ m\frac{(2\pi)^2}{T^2}=k,\\\space\\ T=2\pi\sqrt{m/k}.$$