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

• A. 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 motionis $(3+2\sqrt{2})$ 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 resultant amplitude is $(1+\sqrt{2})$$A$
C the energy associated with the resulting motionis $(3+2\sqrt{2})$ times the energy associated with any single motion.

(a) By principle of superposition
$y=y_1+y_2+y_3$
$=\mathrm{A\; sin}(\omega t+{45}^0)+\mathrm{A\; sin}\omega t+\mathrm{A\; sin}(\omega t-{45}^0)$
$=\mathrm{A\; sin}(\omega t+{45}^0)+\mathrm{A\; sin}(\omega t-{45}^0)+$ $A$ sin($\omega t$)
$=2\sin \omega t\cos {45}^0+$ $A\sin \omega t$
$=\sqrt{2}\mathrm{A\; sin}\omega t+\mathrm{A\; sin}\omega t=(1+\sqrt{2})\mathrm{A\; sin}\omega t$
$\therefore$ Amplitude of resultant motion $=(1+\sqrt{2})A\dots$ (i)

(b) The option is incorrect as the phase of the resultant motion relative to the first is ${45}^{\circ }$.

(c) Energy in SHM is proportional to (amplitude) ${}^2$
$\therefore \frac{E_R}{E_s}=\frac{(1+\sqrt{2}{)}^2{a}^2}{a^2}\therefore \frac{E_R}{E_s}=\frac{(1+2+2\sqrt{2})}{1}$
$E_R=(3+2\sqrt{2})E_s$

(d) Resultant motion is $y=(1+\sqrt{2})$$A\sin \omega t$. This states that it is $SHM$ . So option $\left(d\right)$ is incorrect.

Hence only options (a) and (c) are correct.