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.