Question

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

• A. $1 : 1$
• B. $ 2 : 1$
• C. $ 1 : 3$
• D. $\sqrt{3} : 1$

Answer 1

Answer: (B)

Here, $y_{1}=5\left[sin\,2\pi t+\sqrt{3}cos\,2\pi t\right]$
$=10 \left[\frac{1}{2}sin\,2\pi t+\frac{\sqrt{3}}{2}cos\,2\pi t\right]$
$=10\left[cos \frac{\pi}{3}sin\,2\pi t+sin\frac{\pi}{3}cos\,2\pi t\right]$
$=10\left[sin\left(2\pi t+\frac{\pi}{3}\right)\right]$
$\therefore \, A_{1}=10$
and $y_{2}=5 sin\left(2\pi t+\frac{\pi}{4}\right)$
$\therefore A_{2}=5$
Hence $\frac{A_{1}}{A_{2}}=\frac{10}{5}=\frac{2}{1}$