First slide

Simple harmonic motion

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

Moderate

A

The resultant amplitude is $\sqrt{2} a$
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

Solution

Let simple harmonic motions be represented by
$y_{1} = a \sin (ω t - \frac{π}{4}); y_{2} = a \sin ω t and y_{3} = a \sin (ω t + \frac{π}{4})$
$Resultant will be y = a [\sin (ω t - \frac{π}{4}) + \sin ω t + \sin (ω t + \frac{π}{4})]$
$\begin{array}{l} = a [2 \sin ω t \cos \frac{π}{4} + \sin ω t] \\ = a [\sqrt{2} \sin ω t + \sin ω t] = a (1 + \sqrt{2}) \sin ω t \end{array}$
Resultant amplitude $= (1 + \sqrt{2}) a$
Energy is S.H.M. $\propto (Amplitude)^{2}$
$\begin{array}{l} ∴ \frac{E_{Resultant}}{E_{Single}} = {(\frac{A}{a})}^{2} = (\sqrt{2} + 1)^{2} = (3 + 2 \sqrt{2}) \\ \Rightarrow E_{Resultant} = (3 + 2 \sqrt{2}) E_{Single} \end{array}$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App