Question

Obtain the differential equation of linear simple harmonic motion.

Answer 1

A mass on a spring can be considered as the simplest kind of Simple Harmonic Oscillator.
With a displacement of x on mass m , the restoring force on the spring is given by Hooke's law, withing the elastic limit, F=-kx where k is the spring constant.
Newton’s Second law in the x-direction in differential form therefore becomes,
$m\frac{d^{2}x}{d{t}^2}=-kx$
or
$\frac{d^{2}x}{d{t}^2}=-\frac{k}{m}x$
The above equation represents the differential form of SHM .
Here the spring force depends on the distance x, the acceleration is proportional to the negative of displacement.