§ 13.4NCERT Class 11 · Physics · Chapter 13
Damped Oscillations, Forced Oscillations & Resonance
Physical Classification of Oscillations:
- Free Oscillations: When a system oscillates under its own internal restoring force with no external or dissipative forces. It oscillates with its natural frequency ω0 = √(k / m) and constant amplitude indefinitely.
- Damped Oscillations: Oscillations in which mechanical energy is continuously dissipated as heat due to resistive frictional forces (e.g., air resistance, viscous drag), resulting in an exponential decrease of amplitude over time.
- Forced (Driven) Oscillations: Oscillations maintained by an external periodic force to counteract damping losses.
1. Damped Simple Harmonic Oscillations
When an oscillator moves through a viscous medium, it experiences a damping force proportional and opposite to its velocity: Fd = −b v = −b (dx/dt) (where b is the damping constant in kg/s or N·s/m).
Total Force: Fnet = −kx − b(dx/dt) ⇒ m (d2x / dt2) + b (dx / dt) + kx = 0
Displacement & Decaying Amplitude
The solution for underdamped oscillations (b2 < 4mk):
x(t) = A(t) cos(ω't + φ) = [A0 e−bt / (2m)] cos(ω't + φ)
- Amplitude decays exponentially: A(t) = A0 e−bt / (2m).
- Angular frequency of damped SHM:
ω' = √[ (k / m) − (b2 / 4m2) ] = √[ ω02 − γ2 ]where γ = b / (2m) is the damping coefficient. Notice that ω' < ω0 (frequency slightly decreases).
Decay of Mechanical Energy
The mechanical energy of a damped oscillator at time t is:
E(t) = ½ k A(t)2 = [½ k A02] e−bt / m = E0 e−bt / m
- Energy decays twice as fast as amplitude!
- Relaxation Time (τ): Time in which energy drops to 1/e (≈ 36.8%) of its initial value:
τ = m / b
2. Forced (Driven) Oscillations
When an oscillator is subjected to an external periodic driving force F(t) = F0 cos(ωd t), where ωd is the driving angular frequency:
m (d2x / dt2) + b (dx / dt) + kx = F0 cos(ωd t)
After initial transient vibrations die out, the system oscillates in a steady state with the frequency of the external driver (ωd), with constant amplitude:
A = F0 / √[ m2(ω02 − ωd2)2 + b2 ωd2 ]
3. The Phenomenon of Resonance
Condition for Resonance: When the driving frequency is close to the natural frequency of the oscillating system:
ωd ≈ ω0 = √(k / m)
The term (ω02 − ωd2) vanishes, and the amplitude reaches its maximum possible value:
Ares = F0 / (b ωd) ≈ F0 / (b ω0)
4. Sharpness of Resonance & Quality Factor (Q)
| Parameter | Small Damping (b → 0) | Large Damping (b >> 0) |
|---|---|---|
| Resonant Amplitude Ares | Very high (theoretically ∞ for b = 0) | Small / Low peak |
| Resonance Peak Shape | Sharp and narrow | Flat and broad |
| Quality Factor Q = ω0 / Δω | Very High: Q = (m ω0) / b | Low |
Real-World Manifestations of Resonance:
- Suspension Bridges: Soldiers are ordered to break step while crossing a bridge to prevent marching frequency from matching the natural frequency of the bridge (Broughton Suspension Bridge collapse, 1831).
- Radio Tuning: Adjusting the capacitor dial in an LCR circuit changes the natural electrical frequency of the receiver until it matches the broadcast station frequency (ωd = 1/√(LC)), maximizing signal current.
- Earthquakes: Buildings whose natural frequency coincides with the seismic wave frequency experience catastrophic resonant collapse!
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