A particle executes simple harmonic motion with a frequency $^{'} f^{'}$ . The frequency with which its kinetic energy oscillates is?

Question

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Answer 1

Let, $x = A sin ω t$
$v = \frac{d x}{d t} = A ω cos ω t$
Kinetic energy, $K = \frac{1}{2} m v^{2}$
$⟹ K = \frac{1}{2} m ω^{2} A^{2} cos^{2} ω t$
$⟹ K = \frac{1}{2} m ω^{2} A^{2} (\frac{1 + c o s 2 ω t}{2})$
$⟹ K = \frac{1}{4} m ω^{2} A^{2} (1 + cos^{2} ω t)$
$∴ ω_{K} = 2 ω$
$∴$ Frequency of oscillation of K.E $= 2 f [f = \frac{ω}{2 π}]$

Answer 2

Let,

x = A sin ω t

v = \frac{d x}{d t} = A ω cos ω t

Kinetic energy,

K = \frac{1}{2} m v^{2}

⟹ K = \frac{1}{2} m ω^{2} A^{2} cos^{2} ω t

⟹ K = \frac{1}{2} m ω^{2} A^{2} (\frac{1 + c o s 2 ω t}{2})

⟹ K = \frac{1}{4} m ω^{2} A^{2} (1 + cos^{2} ω t)

∴ ω_{K} = 2 ω

∴

Frequency of oscillation of K.E

= 2 f [f = \frac{ω}{2 π}]