Question

Two simple harmonic motions a re represented by $y_1=5[\sin 2\pi t+\sqrt{3}\cos 2\pi t]$ and $y_2=5\sin [2\pi t+\frac{\pi }{4}]$.The ratio of their amplitude is

• A. 1 : 3
• B. 3 : 1
• C. 1 : 1
• D. 2 : 1

Answer 1

Answer: (D)

$y_1=5[\sin 2\pi t+\sqrt{3}\cos 2\pi t]$
$=10\left[\frac{1}{2}\sin 2\pi t+\frac{\sqrt{3}}{2}\cos 2\pi t\right]$
$=10\left[\cos \frac{\pi }{3}\sin 2\pi t+\sin \frac{\pi }{3}\cos 2\pi t\right]$
$=10\left[\sin \left(2\pi t+\frac{\pi }{3}\right)\right]$
$\Rightarrow A_1=10$
Similarly, $y_2=5\sin \left(2\pi t+\frac{\pi }{4}\right)$
$\Rightarrow A_2=5$
Hence, $\frac{A_1}{A_2}=\frac{10}{5}=\frac{2}{1}$