Define simple harmonic motion. Show that the motion of projection of a particle performing uniform circular motion, on any diameter, is simple harmonic

The periodic motion, in which a particle moves to and from about a fixed point such that its acceleration is always directly proportional to its displacement but in opposite direction and acceleration always acts towards its mean position is defined as simple harmonic motion.

If “a” is the acceleration of the body at a displacement “y” from the mean position the

$\mathrm{a\alpha }-\mathrm{y}\text{ (or) }\mathrm{a}=-{\mathrm{\omega }}^2\mathrm{y}\text{ here }{\mathrm{\omega }}^2\text{ is constant }$
The projection of uniform circular motion on any diameter is SHM : 

$\text{ i.e }\underset{ }{\angle \mathrm{POX}}=\underset{ }{\angle \mathrm{OPN}}=\mathrm{\theta }=\omega t$

Now the displacement of projection N is “y”

From ${∆}^{\mathrm{le}}$ OPN

$\begin{array}{l}\sin \mathrm{\theta }=\frac{\mathrm{ON}}{\mathrm{OP}}=\frac{\mathrm{y}}{\mathrm{A}}\\ \therefore \mathrm{y}=\mathrm{Asin}\mathrm{\theta }\dots .\left(1\right) \mathrm{y}=\mathrm{Asin}\mathrm{wt}\dots \left(1\right) \end{array}$

The centripetal acceleration of particle P is A${\mathrm{\omega }}^2$  and it acts towards the centre “o” A${\mathrm{\omega }}^2$ is resolved into two components

${\mathrm{A\omega }}^2\cos \mathrm{\theta },{\mathrm{A\omega }}^2\sin \mathrm{\theta }$

$\text{ The projection }\mathrm{N}\text{ of acceleration }{\mathrm{A\omega }}^2\sin \mathrm{\theta } \mathrm{acts} \mathrm{toward} \mathrm{mean} \mathrm{position} \mathrm{then}$

$\mathrm{a}={\mathrm{A\omega }}^2\sin \mathrm{\theta }={\mathrm{A\omega }}^2\sin \mathrm{\omega t} ...\left(2\right)$

From (1) & (2)

$\mathrm{a}=-{\mathrm{\omega }}^2\mathrm{y}-\left(3\right)\text{ or }\mathrm{a\alpha }-\mathrm{y} ...\left(3\right)$

Here –ve sign indicates that acceleration “a” and displacement y of projection N are acts in opposite direction
∴ equation gives the S.H.M hence, the projection of uniform circular motion on any diameter is S.H.M