Harmonic Motion

By the equation we find that $sin(\omega t)$ and $cos(\omega t)$ are parameters to terminate, Now we have , $\vec s = a \sin \omega t \vec i + b \cos \omega t \vec j$ $asin(\omega t)=x$

By squaring,

$a^2sin^2(\omega t)=x^2$ $sin^2(\omega t)=\frac{x^2}{a^2}$ Similarly, $cos^2(\omega t)=\frac{y^2}{a^2}$ By adding $sin^2(\omega t)$ and $cos^2(\omega t)$ , $sin^2(\omega t)+cos^2(\omega t)=\frac{x^2}{a^2}+\frac{y^2}{b^2}$ $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ Hence the equation of trajectory is an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ .

A harmonic motion's equation can be described as $\vec s = x\vec i + y \vec j$ .

Now comparing this equation with $\vec s = a \sin \omega t \vec i + b \cos \omega t \vec j$

We get $x =a \sin \omega t$ and $y = b \cos \omega t$

Hence $\dfrac{x}{a} = \sin \omega t$ _ _ __ I And $\dfrac{y}{b} = \cos \omega t$ ------------------------II

Squaring and adding I and II $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = \sin^2 \omega t\ + \cos^2 \omega t$ Hence $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2}$ = 1 [ As $sin^2 x + cos^2x = 1$ ]