Question

A mass m, free to move along a line is attracted towards a given point on the line with a force proportional to its distance from the given point . If the mass starts from rest at a distance s from the given. Show that the mass moves in a simple harmonic motion.

Answer 1

In simple harmonic motion the acceleration of the system, and therefore the net force, is proportional to the displacement and acts in the opposite direction of the displacement.

$$F=-kx$$ $$F=m\ddot{x}$$

Differential equation

$$m\ddot{x}+kx=0$$

This is a second-order linear differential equation.

The initial conditions: 

$$x(0)=s, \dot{x}(0)=0$$

Let

$$\omega^2=k/m$$

The auxiliary equation is

$$r^2+\omega^2=0$$

Then

$$x(t)=A\cos(\omega*t)+B\sin(\omega*t)$$ $$x(0)=A=s$$

$$x(t)=s\cos(\omega*t)+B\sin(\omega*t)$$

$$\dot{x}(t)=-s\omega\sin(\omega*t)+B\omega\cos(\omega*t)$$

$$\dot{x}(0)=B\omega=0$$ $$B=0$$

$$x(t)=s\cos(\omega*t)$$

$$x(t)=s\cos(\sqrt{k/m}*t)$$

The equation of a simple harmonic motion.