- 2073 Shrawan · 5 marks
Damped oscillation, mechanical vs e.m.
Short answer
Mechanical oscillation involves displacement x, mass m, spring k; electromagnetic involves charge q, inductance L, 1/C. Damped mechanical equation: m d²x/dt² + b dx/dt + kx = 0. With γ=b/2m, ω₀²=k/m, solution gives underdamped frequency ω' = √(ω₀²−γ²). Damping types: underdamped (γ<ω₀, oscillatory decay), critically damped (γ=ω₀, no oscillation quickest return), overdamped (γ>ω₀, non-oscillatory slow return).
Write the differences between mechanical oscillation and e.m. oscillation. Set up the differential equation of damped harmonic mechanical oscillation. Obtain the relation for frequency of such oscillation. Hence explain the conditions for different types of damped oscillation.
Mechanical and e.m. oscillations
| Mechanical oscillation | Electromagnetic oscillation |
|---|---|
| Oscillating quantity: displacement x | Oscillating quantity: charge q |
| Inertia: mass m | Inertia: inductance L |
| Restoring factor: spring constant k | Restoring factor: 1/C |
| Dissipation: friction constant b | Dissipation: resistance R |
| Energies: KE and PE | Energies: magnetic (½Li²) and electric (q²/2C) |
| ω0 = √(k/m) | ω0 = 1/√(LC) |
Differential equation of damped mechanical oscillation
Restoring force −kx, damping force −bv:
m d²x/dt² + b dx/dt + kx = 0
d²x/dt² + 2γ dx/dt + ω0² x = 0 , γ = b/2m , ω0² = k/m
Try x = e^(αt): α² + 2γα + ω0² = 0, so α = −γ ± √(γ² − ω0²).
Frequency of oscillation
For γ < ω0, x = a e^(−γt) cos(ω′t + φ) with
ω′ = √(ω0² − γ²) = √(k/m − b²/4m²)
Types of damping
| Condition | Type | Motion |
|---|---|---|
| γ < ω0 (b² < 4mk) | Underdamped | Oscillation with decaying amplitude, frequency ω′ |
| γ = ω0 (b² = 4mk) | Critically damped | No oscillation; returns to equilibrium in least time without overshoot: x = (A + Bt)e^(−γt) |
| γ > ω0 (b² > 4mk) | Overdamped | No oscillation; returns slowly (aperiodic): x = e^(−γt)(Ae^(βt) + Be^(−βt)), β = √(γ² − ω0²) |
x |\ /\ under
| \ / \_ _ _
| \/ ___-- critical
| ___---- over
|_____________ t
✓ Answer checked by Claude Opus 5.5 (high effort)
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