Question

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

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

Answer 1

The correct option is C $y=10sin(16t-\frac{\pi }{4})cm$

Let $T$ be the time period , then $\frac{T}{2}=\frac{5\pi }{64}-\frac{\pi }{64}=\frac{4\pi }{64}$
$T=\frac{\pi }{8}s$
$\omega =\frac{2\pi }{T}=\frac{2\pi }{\frac{\pi }{8}}=16rad/s$

Also, $A=10cm$ (from the graph)

The equation of the sinusoidal wave can be written as $y=10sin(16t+\varphi )$ cm where $\varphi$ is the initial phase.

From the graph, corresponding to the crest $=10cm$, when
$t=\frac{3\pi }{64}$.
$10cm=10sin\left[16\right(\frac{3\pi }{64})+\varphi ]cm$
$sin[\frac{3\pi }{4}+\varphi ]=1$ or $\frac{3\pi }{4}+\varphi =\frac{\pi }{2}⟹\varphi =-\frac{\pi }{4}$
$y=10sin(16t-\frac{\pi }{4})$