Q. Two simple harmonic motions are given by $x=A \sin (\omega t+\delta )$and $y=A \sin \bigg(\omega t+\delta +\frac{\pi}{2}\bigg)$act on a particle simultaneously, then the motion of particle will be
Oscillations
Solution:
Given, $ \, \, \, \, \, \, x=A \sin (\omega t +\delta) \, \, \, \, \, \, \, \, \, \, \, \, \, \, $... (i)
and $ \, \, \, \, \, y=A \sin \bigg(\omega t +\delta +\frac{\pi}{2}\bigg)$
$ \, \, \, \, \, \, \, \, =A \cos (\omega t +\delta) \, \, \, \, \, \, \, \, \, \, \, \, \, ... (ii) $
Squaring and adding Eqs. (i) and (ii), we get
$ \, \, \, \, \, \, \, x^2+y^2=A^2[\sin^2(\omega t + \delta )+\cos^2 (\omega t+\delta)]$
$or \, \, \, \, \, \, \, x^2+y^2=A^2$
which is the equation of a circle .
Now,At $(\omega t +\delta )=0, x=0, y=0$
At $ \, \, \, \, (\omega t+\delta )=\frac{\pi}{2}, x =A , y=0$
At $ \, \, \, \, (\omega t+\delta )=\pi,x=0,y=-A$
At $ \, \, \, \, (\omega t+\delta )=\frac{3\pi}{2},x=-A, y=0$
At $ \, \, \, \, (\omega t+\delta )=2\pi,x=A,y=0$
From the above data, the motion of a particle is a circle transversed in clockwise direction
