Question

Three simple harmonic motions in the same direction having the same amplitude A and same period are superposed. If each differs in phase from the next by ${45}^0$, then

• A. the resulting amplitude is $(1+\sqrt{3})A$.
• B. the resulting motion is not simple harmonic
• C. The energy associated with the resulting motion ($3+2\sqrt{2}$) times the energy associated with any single motion.
• D. The phase of the resultant motion relative to the first is ${90}^0$

Answer 1

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

[C] $Y_1=A\sin \left(wt-\frac{\pi }{4}\right),Y_2=A\sin \left(wt\right),Y_3=A\left(\sin wt+\frac{\pi }{4}\right)$

on superimposing, Resultant SHM =

$\begin{array}{c}Y=A\left[2\sin wt\cos \frac{\pi }{4}+\sin wt\right]\Rightarrow A\left[\sqrt{2}\sin wt+\sin wt\right]\\ Y=A\left(1+\sqrt{2}\right)\sin wt,ResultantAMP=\left(1+\sqrt{2}\right)A\\ \frac{\in resultant}{\in single}=\frac{{\left\{A\left(1+\sqrt{2}\right)\right\}}^2}{A^2}\Rightarrow \in Resultant=\left(3+2\sqrt{2}\right)single\end{array}$