Question #103276

Show that the average energy of a weakly damped oscillator is given by:

< E > = E0 exp (-2bt)

< E > = E0 exp (-2bt)

Expert's answer

The energy of the system is

The instantaneous displacement of a weakly damped harmonic oscillator is

(open the squared parentheses).

The potential energy is

Remember that

"U=\\frac{1}{2}kx^2=\\\\\n\\space\\\\\n=\\frac{1}{2}m\\omega_0^2a_0^2 \\text{exp} (-2bt)\\cdot \\text{cos}^2 (\\omega_dt + \\phi)."

During one oscillation, for small amplitudes, the average values of sine and cosine squared are equal

Remember that the average values of sine per one oscillation is 0.

Therefore, after adding the KE with opened parentheses and U, we get

Since

we can finally write

"<E>=E_0\\text{exp}(-2bt)."

Learn more about our help with Assignments: MechanicsRelativity

## Comments

## Leave a comment