Question

A particle executes simple harmonic motion with a frequency $ f $ . The frequency of its kinetic energy is?
A) $ f $
B) $ f/2 $
C) $ 2f $
D) zero

Answer 1

Hint : In this solution, we will use the formula for displacement and velocity of a simple harmonic oscillator. Then we will use the formula for kinetic energy to determine the period of oscillation of kinetic energy.

Formula used: In this solution, we will use the following formula
Displacement of a harmonics oscillator $ x = A\sin ft $ where $ f $ is the frequency of oscillation and $ t $ is the time
Velocity of a harmonic oscillator $ v = dx/dt $
Kinetic energy of an oscillator $ K = \dfrac{1}{2}m{v^2} $ where $ m $ is the mass of the oscillator.

Complete step by step answer
In this solution, we will find the equation of kinetic energy of a harmonic oscillator using its equation of motion and then determine its frequency of oscillation. Since the displacement of a harmonic oscillator is given as
 $\Rightarrow x = A\sin ft $
We can calculate its velocity equation as
 $\Rightarrow v = dx/dt $
 $\Rightarrow v = Af\cos ft $
Then the kinetic energy of the particle will be
 $\Rightarrow K = \dfrac{1}{2}m{v^2} $
 $\Rightarrow K = \dfrac{1}{2}m{(Af\cos ft)^2} $
Since $ {\cos ^2}(ft) = \dfrac{{1 + \cos (2ft)}}{2} $ , we can write the above equation as
 $\Rightarrow K = \dfrac{1}{2}m{A^2}{f^2}\left( {\dfrac{{1 + \cos (2ft)}}{2}} \right) $
The oscillating term with respect to time in this equation is $ \cos (2ft) $ as rest all terms remain constant, which has an equivalent frequency of $ {f_k} = 2f $ .
Hence the correct choice is option (C).

Note
The velocity of a harmonic oscillator has the same frequency as its displacement. To determine the frequency of the kinetic energy of the oscillator, we cannot find its frequency from the $ {\cos ^2}(ft) $ term as $ f $ since the time period of any oscillation is decided from trigonometric terms having a singular power and are not squared or cubed, etc.