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 ${45}^o$, 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 with any single motion
• D. The resulting motion is not simple harmonic

Answer 1

Answer: (A), (C)

The correct options are
A The resultant amplitude $(1+\sqrt{2})a$
C The energy associated with the resulting motion is $(3+2\sqrt{2})$ times the energy associated with any single motion

Let $y_1=a\sin (\omega t-\frac{\pi }{4})$
$y_2=a\sin \left(\omega t\right)$
$y_3=a\sin (\omega t+\frac{\pi }{4})$
On super imposing, resulting SHM-
$y=a[\sin (\omega t-\frac{\pi }{4})+\sin \omega t+\sin (\omega t+\frac{\pi }{4})]$
$⟹y=a[2\sin \omega t\cos \frac{\pi }{4}+\sin \omega t]$
$⟹y=a(1+\sqrt{2})\sin \omega t$
$\therefore$ Resultant amplitude $=(1+\sqrt{2})a$
Also, $\frac{E_{resultant}}{E_{single}}={\left(\frac{A}{a}\right)}^2$
$⟹\frac{E_{resultant}}{E_{single}}=(\sqrt{2}+1{)}^2$
$⟹\frac{E_{resultant}}{E_{single}}=(3+2\sqrt{2})$
$\therefore E_{resultant}=(3+2\sqrt{2})E_{single}$