Question

Three simple harmonic motions in the same direction having the same ampitude $a$ 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
• D. the resulting motion is not simple harmonic

Answer 1

Answer: (A), (C)

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

By principle of superposition

$y=y_1+y_2+y_3$

$=asin(\omega t+{45}^{\circ })+asin\omega t+asin(\omega t-{45}^{\circ })$

$=asin(\omega t+{45}^{\circ })+asin(\omega t-{45}^{\circ })+asin\omega t$

$=2asin\omega tcos{45}^{\circ }+asin\omega t$

$=\sqrt{2a}sin\omega t+asin\omega t=(1+\sqrt{2})asin\omega t$

$\therefore$ Amplitude of resultant motion $=(1+\sqrt{2})a$

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

c) Energy is SHM is proportional to square of the amplitude:
$E\propto a^2$

$\therefore \frac{E_R}{E}=\frac{(1+\sqrt{2}{)}^2{a}^2}{a^2}\therefore \frac{E_R}{E}=\frac{(1+2+2\sqrt{2})}{1}$

or $E_R=(3+2\sqrt{2})E$

d) Resultant motion is $y=(1+\sqrt{2})asin\omega t$
It is a SHM.

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