First slide

Simple harmonic motion

Question

A body is oscillating simple harmonically with a period T = 2 s. How much time would it take for its kinetic energy to decrease from $K_{max}$ to 75 % of $K_{max}$ ?

Moderate

A

$\frac{1}{12} s$
B

$\frac{1}{3} s$
C

$\frac{1}{4} s$
D

$\frac{1}{6} s$

Solution

$\begin{array}{l} K = \frac{1}{2} {mv}^{2} = \frac{m}{2} [A^{2} ω^{2} \cos^{2} (ω t)] and K_{max} = \frac{{mA}^{2} ω^{2}}{2} \\ Given that K = 0.75 K_{max} \\ \Rightarrow \cos (ω t) = \frac{\sqrt{3}}{2} \Rightarrow ω t = \frac{π}{6} \Rightarrow (\frac{2 π}{T}) t = \frac{π}{6} . \\ Simplifying this we get t = \frac{T}{12} = \frac{2}{12} = \frac{1}{6} s \end{array}$

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