Question

Two SHM are represented by the equations $x_{1}=20 \sin \left[5 \pi t+\frac{\pi}{4}\right]$ and $x_{2}=10(\sin 5 \pi t+\sqrt{3} \cos 5 \pi t)$ The ratio of the amplitudes of the two motions is

• A. 0.5
• B. 1
• C. 0.25
• D. $\frac{\sqrt{3}}{2}$

Answer 1

Answer: (B)

1

Answer 2

For SHM (1), $x _{1}=20 \sin \left(5 \pi t +\frac{\pi}{4}\right)$ Its amplitude, $A_{1}=20$ For SHM (2), $x _{2} =10[\sin 5 \pi t +\sqrt{3} \cos 5 \pi t ]$ $=10 \times 2\left(\frac{\sin 5 \pi t }{2}+\frac{\sqrt{3}}{2} \cos 5 \pi t \right)$ $=20\left(\sin 5 \pi t \cdot \cos \frac{\pi}{3}+\sin \frac{\pi}{3} \cos 5 \pi t \right)$ $=20 \sin \left(5 \pi t +\frac{\pi}{3}\right)$ It's amplitude, $A_{2}=20$ $\therefore \frac{A_{1}}{A_{2}}=\frac{20}{20}=1$