Question

Two simple harmonic motions are represented by the equations $y_1=10\sin (3\pi t+\pi /4)$ and $y_2=5(\sin 3\pi t+\sqrt{3}\cos 3\pi t)$. Their amplitudes are in the ratio of _________.

• A. $2:1$.
• B. $1:1$.
• C. $3:1$.
• D. $1:4$.

Answer 1

The correct option is B $1:1$.

$y_2=5(\sin 3\pi t+\sqrt{3}\cos 3\pi t)$

$y_2=5\times 2(\cos \frac{\pi }{3}\sin 3\pi t+\sin \frac{\pi }{3}cos3\pi t)$

$y_2=5\times 2\left[\sin (3\pi t+\frac{\pi }{3})\right]=10\sin (3\pi t+\frac{\pi }{3})$

Now, $y_1=10\sin (3\pi t+\frac{\pi }{4})$

$\therefore$ Ratio of amplitudes $=\frac{a_1}{a_2}=\frac{10}{10}=\frac{1}{1}$

$\therefore$ Ratio of amplitudes $=1:1$