Question

Three simple harmonic motions in the same direction having the same amplitude $a$ and period are superposed. IF each differs in phase from the next by ${45}^o$ then

• A. the resultant amplitude $(1+\sqrt{2})a$
• B. the phase of the resultant motion relative to the first is ${90}^o$
• C. the energy associated with the resulting motion is $(3+2\sqrt{2})$ times the energy associated any single motion
• D. the resulting motion is not simple harmonic

Answer 1

The correct option is C the energy associated with the resulting motion is $(3+2\sqrt{2})$ times the energy associated any single motion

$\begin{array}{c}Letsimpleharmonicmotionsberepresentedby\\ y_1=a\sin \left(\omega t-\frac{\pi }{4}\right);y_2=s\sin \omega tand\\ y_3=a\sin \left(\omega t+\frac{\pi }{4}\right)\\ Onerimpo\sin gresul\tan tSHMwillbe\\ y=a\left[\sin \left(\omega t-\frac{\pi }{4}\right)+\sin \omega t+\sin \left(\omega t+\frac{\pi }{4}\right)\right]\\ =a\left[2\sin \omega t\cos \frac{\pi }{4}+\sin \omega t\right]\\ =a\left[\sqrt{2}\sin \omega t+\sin \omega t\right]\\ =a\left(1+\sqrt{2}\right)\sin \omega t\\ resul\tan tamplitude=\left(1+\sqrt{2}\right)a\\ Now,\\ \frac{E_{Resul\tan t}}{E_{Single}}={\left(\frac{A}{a}\right)}^2={\left(\sqrt{2}+1\right)}^2=\left(3+2\sqrt{2}\right)\\ E_{Resul\tan t}=\left(3+2\sqrt{2}\right)E_{Single}\end{array}$

Hence, the option $C$ is the correct answer.