Question

If the equation $y_1=A\sin \omega t$ and $y_2=\frac{A}{2}\sin \omega t+\frac{A}{2}\cos \omega t$ represent SHM then the ratio of the amplitudes of the two motions is

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

Answer 1

The correct option is B $\sqrt{2}$

Given, $y_1=A\sin \omega t$
and $y_2=\frac{A}{2}\sin \omega t+\frac{A}{2}\cos \omega t$
or $y_2=\frac{A}{2}(\sin \omega t+\cos \omega t)$
$=\frac{A}{3}\sqrt{2}\left[\sin \right(\omega t+{45}^{\circ }]$
$=\frac{A}{\sqrt{2}}\sin (\omega t+{45}^{\circ })$
$\therefore$ The ration of amplitudes of two motions
$\frac{A_1}{A_2}=\frac{A}{A\sqrt{2}}=\sqrt{2}$.