Three simple harmonic motions in the same direction having the sa… - Vidyadip

MCQ

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 $45^{\circ}$, then

A
The resultant amplitude is $(1+\sqrt{2}) a$
B
The phase of the resultant motion relative to the first is $90^{\circ}$
C
The energy associated with the resulting motion is $(3+2 \sqrt{2})$ times the energy associated with any single motion
✓
A and C both

✓

Answer

Correct option: D.

A and C both

(D) Let simple harmonic motions be represented by$y_1=a \sin \left(\omega t-\frac{\pi}{4}\right) ; y_2=a \sin \omega t$ and$y_3=a \sin \left(\omega t+\frac{\pi}{4}\right)$. On superimposing, resultant SHM willbe $y=a\left[\sin \left(\omega t-\frac{\pi}{4}\right)+\sin \omega t+\sin \left(\omega t+\frac{\pi}{4}\right)\right]$
$ =a\left[2 \sin \omega t \cos \frac{\pi}{4}+\sin \omega t\right] $
$ =a[\sqrt{2} \sin \omega t+\sin \omega t]=a(1+\sqrt{2}) \sin \omega t$
Resultant amplitude $=(1+\sqrt{2}) a$
Energy is S.H.M. $\propto$ (Amplitude)
$\therefore \frac{E_{\text {Resultant }}}{E_{\text {Single }}}=\left(\frac{A}{a}\right)^2=(\sqrt{2}+1)^2=(3+2 \sqrt{2})$
$ \Rightarrow E_{\text {Resultant }}=(3+2 \sqrt{2}) E_{\text {Single }}$

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