# A frequency of variation of kinetic energy of a simple harmonic motion of frequency n is?

## 2n n $\frac{n}{2}$ 3n

Mar 29, 2018

$2 n$

#### Explanation:

The velocity of a simple harmonic oscillator varies as

${v}_{0} \sin \left(2 \pi n t + \phi\right)$

The kinetic energy is

$K = \frac{1}{2} m {v}^{2} = \frac{1}{2} m {v}_{0}^{2} {\sin}^{2} \left(2 \pi n t + \phi\right)$

Now, ${\sin}^{2} \theta = \frac{1}{2} \left(1 - \cos \left(2 \theta\right)\right)$ and so, we have

$K = \frac{1}{2} m {v}_{0}^{2} \left(\frac{1}{2} - \frac{1}{2} \cos \left(4 \pi n t + 2 \phi\right)\right)$
$q \quad = \frac{1}{4} m {v}_{0}^{2} - \frac{1}{4} m {v}_{0}^{2} \cos \left(2 \pi \left(2 n\right) t + 2 \phi\right)$

Here, the time dependence comes from the last term and its frequency is $2 n$.