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,
$$\begin{gathered} R=\sqrt{\left(x^2+s^2+2\left(x\right)\left(s\right)\cos 45°\right)} \\ =\sqrt{\left\{x_0^2\mathrm{sin\omega }t+s_0^2\mathrm{sin\omega }t+2x_0{s}_0{\sin }^2\mathrm{\omega }t\left(\frac{1}{\sqrt{2}}\right)\right\}} \\ ={\left[x_0^2+s_0^2+\sqrt{2x_0{s}_0}\right]}^{1/2}\mathrm{sin\omega }t \end{gathered}$$

Hence, the resultant amplitude is ${\left[x_0^2+s_0^2+\sqrt{2x_0{s}_0}\right]}^{1/2}$.