A mass $0.4 \mathrm{~kg}$ performs S.H.M. with a frequency $\frac{16}{\pi} \mathrm{Hz}$. At a certain…

A mass $0.4 \mathrm{~kg}$ performs S.H.M. with a frequency $\frac{16}{\pi} \mathrm{Hz}$. At a certain displacement it has kinetic energy $2 \mathrm{~J}$ and potential energy $1.2 \mathrm{~J}$. The amplitude of oscillation is
  1. 0.15 m
  2. 0.125 m
  3. 0.075 m
  4. 0.1 m

Solution

$\begin{aligned} & \mathrm{m}=0.4 \mathrm{~kg}, \mathrm{f}=\frac{16}{\pi} \mathrm{Hz}, \mathrm{K} \cdot \mathrm{E} \cdot=2 \mathrm{~J}, \text { P.E. }=1.2 \mathrm{~J} \\ & \omega=2 \pi \mathrm{f}=2 \pi \times \frac{16}{\pi}=32 \mathrm{rad} / \mathrm{s} \end{aligned}$ Total energy T.E. $=2+1.2=3.2 \mathrm{~J}$ $\begin{aligned} & \mathrm{TE}=\frac{1}{2} \mathrm{~m} \omega^2 \mathrm{~A}^2 \\ & \therefore \mathrm{A}^2=\frac{2(\mathrm{TE})}{\mathrm{m} \omega^2}=\frac{2 \times 3.2}{0.4 \times(32)^2}=\frac{6.4}{0.4 \times(32)^2}=\frac{16}{(32)^2} \\ & \therefore \mathrm{A}=\frac{4}{32}=\frac{1}{8}=0.125 \mathrm{~m} \end{aligned}$

Asked in: MHT CET 2021 (22 Sep Shift 1)

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