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Damped oscillation energy decreases with time

Short answer

Forced oscillation is motion driven by a perodic external force. In damped free oscillation, amplitude decays as a0 e^(−γt), so total energy E = ½kA² decreases exponentially as E0 e^(−2γt) = E0 e^(−bt/m). Also, dE/dt = –b v² ≤ 0 confirms continuous energy loss.

Define forced oscillation. Show that the total energy of the damped oscillation decreases with increasing time.

Forced oscillation

A forced oscillation is the oscillation of a system under the continuous action of an external periodic force. After the initial transient, the system vibrates at the frequency of the applied force (not its natural frequency), with constant amplitude. Example: a mass on a spring driven by F0 cos pt, or an LCR circuit driven by an AC source.

Energy of damped oscillation decreases with time

For free damped oscillation, m x'' + b x' + kx = 0.

Method 1 (from the solution). x = a0 e^(−γt) cos(ω′t + φ), γ = b/2m. For light damping, the amplitude is a0 e^(−γt) and the energy is

E = ½ k a² = ½ k a0² e^(−2γt) = E0 e^(−2γt) = E0 e^(−bt/m)

Since the exponent is negative, E falls exponentially with time.

Method 2 (rate of loss of energy).

  1. E = ½ m v² + ½ k x².
  2. dE/dt = m v v' + k x v = v (m x'' + k x).
  3. From the equation of motion, m x'' + kx = −b x' = −b v:
dE/dt = −b v²
  1. Since b > 0 and v² ≥ 0, dE/dt ≤ 0 always. The energy decreases continuously, and the lost energy is converted into heat by the damping force. The rate of loss is largest when the speed is maximum (at the mean position).
 E |\
   | \__
   |     \___
   |__________\____ t     E = E0 e^(−t/τ)

Relaxation time τ = m/b is the time for the energy to fall to 1/e of E0. In a forced oscillation, the external force supplies energy at the same average rate as is dissipated, so the amplitude stays constant.

✓ Answer checked by Claude Opus 5.5 (high effort)

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