A body is executing simple harmonic motion with frequency , the frequency of its potential energy is [2021]

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Solution

Q.
A body is executing simple harmonic motion with frequency $‘ n ’$ , the frequency of its potential energy is [2021]

1 $4 n$

2 $n$

3 $2 n$

4 $3 n$

Ans.
(3)

The equation for the S.H.M. is

$y = A \sin ω t = A \sin (2 π n t)$ ...(i)

where, $A$ is amplitude, $ω$ is angular frequency and $y$ is displacement at time $t$ .
The formula of potential energy is

$U = \frac{1}{2} k A^{2} \sin^{2} ω t$

$U = \frac{1}{2} k A^{2} (\frac{1 - \cos 2 ω t}{2}) = \frac{1}{2} k A^{2} [\frac{1 - \cos 2 π (2 n) t}{2}]$ ...(ii)

$(∵ \sin^{2} θ = \frac{1 - \cos 2 θ}{2})$

From eqn. (i) and (ii), we get

Frequency of potential energy = $2 n$

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