Question

Two simple harmonic motion, are represented by the equations $y_{1}=10 \sin \left(3 \pi t+\frac{\pi}{3}\right)$
$y_{2}=5(\sin \,3 \pi t+\sqrt{3} \cos 3 \pi t)$
Ratio of amplitude of $y_{1}$ to $y_{2}=x: 1$. The value of $x$ is_____.

Answer 1

$y _{1}=10 \sin \left(3 \pi t +\frac{\pi}{3}\right) $
$\Rightarrow $ Amplitude $=10 $
$y _{2}=5(\sin 3 \pi t +\sqrt{3} \cos 3 \pi t )$
$y _{2}=10\left(\frac{1}{2} \sin 3 \pi t +\frac{\sqrt{3}}{2} \cos 3 \pi t \right)$
$ y _{2}=10\left(\cos \frac{\pi}{3} \sin 3 \pi t +\sin \frac{\pi}{3} \cos 3 \pi t \right)$
$ y _{2}=10 \sin \left(3 \pi t +\frac{\pi}{3}\right) $
$\Rightarrow $ Amplitude $=10$
So ratio of amplitudes $=\frac{10}{10}=1$