A simple harmonic oscillator has amplitude A, angular velocity Ω , and mass m. Then, average energy in one time period will be :

Question

A simple harmonic oscillator has amplitude A, angular velocity Ω , and mass m. Then, average energy in one time period will be : (A)  (1/4)mω 2A2 (B)  (1/4)m2ω 2A2 (C) ω2A2 (D) zero

Tardigrade Admin · Accepted Answer

Average energy =(∫ limits0T U d t/∫ limits0T d t)=(1/T) ∫ limits0T U d t =(1/2 T) ∫ limits0T m ω2 A2 cos 2(ω t+φ) d t =(1/4) m ω2 A2

Q. A simple harmonic oscillator has amplitude $A$ , angular velocity $Ω$ , and mass $m$ . Then, average energy in one time period will be :

$\frac{1}{4} m ω^{2} A^{2}$

$\frac{1}{4} m^{2} ω^{2} A^{2}$

$ω^{2} A^{2}$

zero

Average energy $= \frac{0 \int T U d t}{0 \int T d t} = \frac{1}{T} 0 \int T U d t$
$= \frac{1}{2 T} 0 \int T m ω^{2} A^{2} cos^{2} (ω t + ϕ) d t$
$= \frac{1}{4} m ω^{2} A^{2}$

Q. A simple harmonic oscillator has amplitude A, angular velocity Ω , and mass m. Then, average energy in one time period will be :

41​mω2A2

41​m2ω2A2

ω2A2