Damped Oscillations
Exponential decay envelope.
Key Notes
Damped oscillation: amplitude decreases over time due to friction or air drag.
Equation of motion: m·ẍ + b·ẋ + kx = 0, where b is the damping constant.
Three regimes: UNDERDAMPED (oscillates with decaying amplitude), CRITICALLY DAMPED (returns to equilibrium fastest, no oscillation), OVERDAMPED (returns slowly, no oscillation).
Underdamped solution: x(t) = A₀·e^(−bt/2m)·cos(ω'·t + φ), where ω' = √(ω₀² − γ²), γ = b/2m, ω₀ = √(k/m).
Amplitude decays exponentially: A(t) = A₀·e^(−bt/2m). Half-amplitude time: t₁/₂ = (2m·ln 2)/b.
Quality factor Q = ω₀·m/b. High Q = light damping, oscillates many times. Low Q = heavy damping.
Examples: pendulum in air, RLC circuit with R, shock absorbers (critically damped by design).
Energy decays: E(t) ∝ e^(−bt/m) (twice as fast as amplitude).
Formulas
Equation of motion
Damping term b·ẋ represents friction proportional to velocity.
Underdamped solution
γ = b/2m, ω' = √(ω₀² − γ²).
Quality factor
Number of oscillations before amplitude drops by factor 1/e^π ≈ 1/23.
Energy decay
Decays twice as fast as amplitude (E ∝ A²).
Important Points
Underdamped: oscillates with decaying amplitude. Most common.
Critically damped: fastest return to equilibrium without overshoot. Used in vehicle suspensions.
Overdamped: slow exponential return. Used in door closers.
Damped frequency ω' is slightly LESS than ω₀ (natural frequency).
Q-factor: high Q = sharp resonance, lasts long; low Q = broad resonance, fades fast.
Energy decays as e^(−t/τ) with τ = m/b — the energy time constant.
Damped Oscillations notes from sciphylab (also known as SciPhy, SciPhy Lab, SciPhy Labs, Physics Lab). Class 11 physics revision for JEE Mains, JEE Advanced, NEET UG, AP Physics 1/2/C, SAT, and CUET-UG.