C.1 · IB Physics
Damped oscillations
Change the damping coefficient of a mass–spring oscillator and watch it move from light to critical to heavy damping, with displacement, the exponential envelope and the energy plotted against time.
m·d²x/dt² + b·dx/dt + kx = 0
bc = 2√(mk), γ = b/(2m), ω₀ = √(k/m)
Light (b < bc): x = A₀e^(−γt)[cos ωdt + (γ/ωd) sin ωdt], ωd = √(ω₀² − γ²)
E = ½kx² + ½mv²
- bc
- b / bc
- γ
- ω₀
- ωd
- t
- x
- v
- E
- E / E₀
Try this
- Set b = 1.00 kg s⁻¹ and count the oscillations needed for the amplitude to halve. Now double m and predict the change using γ = b/(2m) before you play it.
- Press “Set critical b”, then try values just below and just above bc. Which one returns to x = 0 fastest without overshooting?
- With light damping, watch the energy graph fall in steps. Where in each cycle is energy lost fastest, and why does the power lost, bv², explain this?
Common error: saying heavy damping brings a system to rest fastest. Critical damping returns it to equilibrium in the shortest time without overshooting, while heavy damping creeps back more slowly.