Question

The diagram below shows a sinusoidal curve. The equation of the curve will be

• A. $y=10\sin \left(16t-\frac{\pi }{3}\right)cm$
• B. $y=10\sin \left(16t+\frac{\pi }{3}\right)cm$
• C. $y=10\sin \left(16t-\frac{\pi }{4}\right)cm$
• D. $y=10\cos \left(16t+\frac{\pi }{4}\right)cm$

Answer 1

Answer: (C)

Let $T$ be the time period; then $\frac{T}{2}=\frac{5\pi }{64}-\frac{\pi }{64}=\frac{4\pi }{64}$
$T=\frac{\pi }{8}s$
$\therefore \omega =\frac{2\pi }{T}=\frac{2\pi }{(\pi /8)}=16rad/s$
Also, $A=10cm$ (from the graph)
The equation of the sinusoidal wave can be written as $y=10\sin (16t+\varphi )cm$, where $\varphi$ is the initial phase.
From the graph, corresponding to the crest $=10cm$; when $t=3\pi /64$.
$10cm=10\sin \left[16\left(\frac{3\pi }{64}\right)+\varphi \right]cm$
$\sin \left[\frac{3\pi }{4}+\varphi \right]=1$
or $\frac{3\pi }{4}+\varphi =\frac{\pi }{2}$
$\Rightarrow \varphi =-\frac{\pi }{4}$
$\therefore y=10\sin \left(16t-\frac{\pi }{4}\right)$