Question

Two simple harmonic motions are represented by the equations $x_1=5\sin (2\pi t+\frac{\pi }{4})$ and $x_2=5\sqrt{2}(\sin 2\pi t+\cos 2\pi t)$. The amplitude of second motion is _____ times the amplitude of first motion

• A. 2
• B. 2.0
• C. 2.00

Answer 1

Answer: (A), (B), (C)

Given,

$x_1=5\sin (2\pi t+\frac{\pi }{4}).....\left(1\right)$

$x_2=5\sqrt{2}(\sin 2\pi t+\cos 2\pi t).....\left(2\right)$

Multiplying and dividing equation $\left(2\right)$ by $\sqrt{2}$, we have:

$x_2=5\left(\sqrt{2}\right)\left(\sqrt{2}\right)(\frac{1}{\sqrt{2}}\sin 2\pi t+\frac{1}{\sqrt{2}}\cos 2\pi t)$

$x_2=10(\cos \frac{\pi }{4}\sin 2\pi t+\sin \frac{\pi }{4}\cos 2\pi t)$

$x_2=10\sin (\sin 2\pi t+\frac{\pi }{4})......\left(3\right)$

Using equation $\left(1\right)$ and $\left(3\right)$ we have,

$\frac{A_1}{A_2}=\frac{1}{2}$

$\Rightarrow A_2=2A_1$