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

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 $=\frac{10}{10}=1$