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, $\frac{A_{1}}{A_{2}}=\frac{10}{5}=\frac{2}{1}$