Question

A particle is subjected to two simple harmonic motions, one along the X-axis and the other on a line making an angle of 45° with the X-axis. The two motions are given by x = x₀ sin ωt and s = s₀ sin ωt. Find the amplitude of the resultant motion.

Answer 1

Given:
Equation of motion along X axis, x = x₀sinωt
Equation of motion along Y axis, s = s₀sinωt
Angle between the two motions,\[\theta\] 45^₀

^{Resultant motion (R) will be,}

\[R = \sqrt{\left( x^2 + s^2 + 2\left( x \right)\left( s \right)\cos45^\circ\right)}\] 

\[   = \sqrt{\left\{ x_0^2 sin\omega t + s_0^2 sin\omega t  + 2 x_0 s_0 \sin^2 \omega t\left( \frac{1}{\sqrt{2}} \right) \right\}}\] \[   =  \left[ x_0^2 + s_0^2 + \sqrt{2 x_0 s_0} \right]^{1/2} sin\omega t\]

Hence, the resultant amplitude is

\[\left[ x_0^2 + s_0^2 + \sqrt{2 x_0 s_0} \right]^{1/2}\]