First slide

Simple harmonic motion

Question

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

Moderate

A

$y = 10 \sin (16 t - \frac{π}{3}) cm$
B

$y = 10 \sin (16 t + \frac{π}{3}) cm$
C

$y = 10 \sin (16 t - \frac{π}{4}) cm$
D

$y = 10 \cos (16 t + \frac{π}{4}) cm$

Solution

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

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