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 $45{}^{\circ },$ then (1) the resultant amplitude is $(1+\sqrt{2})a$ (2) the phase of the resultant motion relative to the first is $90{}^{\circ }$ (3) the energy associated with the resulting motion is $(3+2\sqrt{2})$ times the energy associated with any single motion (4) the resulting motion is not simple harmonic

• A. 1, 2 and 3 are correct.
• B. 1 and 2 are correct.
• C. 2 and 4 are correct.
• D. 1 and 3 are correct.

Answer 1

Answer: (D)

Let simple harmonic motion 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 the resultant SHM will be
$y=a\left[\sin \left(\omega t-\frac{\pi }{4}\right)+\sin \omega t+\sin \left(\omega t+\frac{\pi }{4}\right)\right]$
$=a\left[2\sin \omega t\cos \frac{\pi }{4}+\sin \omega t\right]$
$=a[\sqrt{2}\sin \omega t+\sin g\omega t]$
$=a(1+\sqrt{2})\sin \omega t$ Resultant amplitude
$=(1+\sqrt{2})a$ Energy is $SHM\propto (amplitude)$
$\therefore$ $\frac{E_{\mathrm{Re}sul\tan t}}{E_{\sin gle}}={\left[\frac{A}{a}\right]}^2={(\sqrt{2}+1)}^2=(3+2\sqrt{2})$
$\Rightarrow$ $E_{\mathrm{Re}sula\tan t}=(3+2\sqrt{2})E_{\sin gle}$