First slide

Simple hormonic motion

Question

Two SHMs are represented by the equations $\large {y_1}\, = \,10\sin \left( {3\pi t + \frac{\pi }{4}} \right)\,\,and\,\,{y_2}\, = \,5\left[ {\sin 3\pi t + \sqrt 3 \cos 3\pi t} \right]$ . Their amplitudes are in the ratio

Moderate

A

1:2
B

2:1
C

1:3
D

1 :1

Solution

$\large {y_1}\, = \,10\sin \left[ {3\pi t + \frac{\pi }{4}} \right]$
$\large {y_2}\, = \,5\sin \left[ {(3\pi t) + \sqrt 3 \,\cos 3\pi t} \right]$
$\large \, = \,5 \times 2\left[ {\frac{1}{2}\sin 3\pi t + \frac{{\sqrt 3 }}{2}\cos 3\pi t} \right]$
$\large \, = 10\left[ {\cos 60.\sin 3\pi t + \sin {{60}^0}.\cos 3\pi t} \right]$
$\large {y_2}\, = \,10\sin \left( {3\pi t + {{60}^o}} \right);\,\,{y_1}\, = \,10\sin \left( {3\pi t + \frac{\pi }{3}} \right)$
Ratio of amplitudes = $\large \frac{{10}}{{10}}\, = \,\frac{1}{1}$

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