Question

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

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

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