Question

A particle is subjected to two simple harmonic motions, one along the X-axis and the other on a line making an angle

${45}^{\circ }$ with the X-axis. The two motions are given by $x=x_{0}sin\omega t$ and $s=s_{0}sin\omega t$

Find the amplitude of the resultant motion.

Answer 1

The particle is subjected to two simple harmonic motions represented by

$x=x_{0}sinwt$

$x=s_{0}sinwt$

and angle between two motions

$=\theta ={45}^{\circ }$

$\therefore$ Resultant motions will be given by

$R=\sqrt{(x^2+s^2+2xs.cos{45}^{\circ })}$

$=\sqrt{\{x_0^{2}si{n}^{2}wt+s_0^{2}si{n}^{2}wt+2x_0{s}_{0}si{n}^2\omega t\left(\frac{1}{\sqrt{2}}\right)\}}$

$=[x_0^2+s_0^2+\sqrt{2x_0{s}_0}{]}^{1/2}sin\omega t$

$\therefore$ Resultant amplitude

$[x_0^2+s_0^2+\sqrt{2x_0{s}_0}{]}^{1/2}$