Obtain the differential equation of linear simple harmonic motion.

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Solution

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,
$m \frac{d^{2} x}{d t^{2}} = - k x$
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.

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