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JIPMER 2006
Exam
JIPMER
Year
2006
Subject
Physics
Chapter
Oscillations

JIPMER 2006 · Physics · Oscillations

A pendulum is undergoing SHM with frequency f . What is the...

A pendulum is undergoing SHM with frequency ff. What is the frequency of its kinetic energy ?

Options

  1. Af/2f / 2
  2. B2f2 f
  3. C3f3 f
  4. D4f4 f

Correct answer

B. 2f2 f

Step-by-step solution

Let, x=Asin⁡ωtv=dxdt=Aωcos⁡ωt\mathrm{x}=\mathrm{A} \sin \omega \mathrm{t} \mathrm{v}=\frac{\mathrm{dx}}{\mathrm{dt}}=\mathrm{A} \omega \cos \omega \mathrm{t} Kinetic energy, K=12mv2\mathrm{K}=\frac{1}{2} \mathrm{mv}^2⇒K=12 mω2 A2cos⁡2ωt⇒K=12 mω2 A2(1+cos⁡2ωt2)⇒K=14 mω2 A2(1+cos⁡2ωt)∴ωK=2ω\begin{aligned} & \Rightarrow \mathrm{K}=\frac{1}{2} \mathrm{~m} \omega^2 \mathrm{~A}^2 \cos ^2 \omega \mathrm{t} \\ & \Rightarrow \mathrm{K}=\frac{1}{2} \mathrm{~m} \omega^2 \mathrm{~A}^2\left(\frac{1+\cos 2 \omega \mathrm{t}}{2}\right) \\ & \Rightarrow \mathrm{K}=\frac{1}{4} \mathrm{~m} \omega^2 \mathrm{~A}^2\left(1+\cos ^2 \omega \mathrm{t}\right) \\ & \therefore \omega_{\mathrm{K}}=2 \omega\end{aligned}∴\therefore Frequency of oscillation of K.E=2f[f=ω2π]\mathrm{K} . \mathrm{E}=2 \mathrm{f} \quad\left[\mathrm{f}=\frac{\omega}{2 \pi}\right]
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