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Question

Physics Question on simple harmonic motion

Two simple harmonic motions are represented by y1=5[sin2πt+3cos2πt]andy2=5sin(2πt+π4)y_{1} = 5\left[sin 2\pi t + \sqrt{3} cos 2\pi t\right] and y_{2} = 5 sin \left(2\pi t +\frac{\pi}{4}\right). The ratio of their amplitudes is

A

1:11 : 1

B

2:1 2 : 1

C

1:3 1 : 3

D

3:1\sqrt{3} : 1

Answer

2:1 2 : 1

Explanation

Solution

Here, y1=5[sin2πt+3cos2πt]y_{1}=5\left[sin\,2\pi t+\sqrt{3}cos\,2\pi t\right] =10[12sin2πt+32cos2πt]=10 \left[\frac{1}{2}sin\,2\pi t+\frac{\sqrt{3}}{2}cos\,2\pi t\right] =10[cosπ3sin2πt+sinπ3cos2πt]=10\left[cos \frac{\pi}{3}sin\,2\pi t+sin\frac{\pi}{3}cos\,2\pi t\right] =10[sin(2πt+π3)]=10\left[sin\left(2\pi t+\frac{\pi}{3}\right)\right] A1=10\therefore\, A_{1}=10 and y2=5sin(2πt+π4)y_{2}=5 sin\left(2\pi t+\frac{\pi}{4}\right) A2=5\therefore A_{2}=5 Hence A1A2=105=21\frac{A_{1}}{A_{2}}=\frac{10}{5}=\frac{2}{1}