Damped SHM Displacement (x(t) = A·e^(-bt/2m)·cos(ω't + φ), where ω' = √(ω₀² )
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x(t) = A·e^(-bt/2m)·cos(ω't + φ), where ω' = √(ω₀² - (b/2m)²)
What each symbol means
| Symbol | What it stands for |
|---|---|
| A | Initial amplitude (m) |
| b | Damping constant (kg/s) |
| m | Mass (kg) |
| ω' | Damped angular frequency (rad/s) |
| x(t) | Displacement at time t (m) |
| ω₀ | Natural angular frequency = √(k/m) (rad/s) |
| e^(-bt/2m) | Exponential decay factor for amplitude |
When to use this
For damping force F_d = -bv (proportional to velocity). Amplitude decays exponentially as Ae^(-bt/2m). Damped frequency ω' < ω₀ (oscillations slow down). Energy decays as E(t) = ½kA²e^(-bt/m). If b/2m > ω₀: overdamped (no oscillation). If b/2m = ω₀: critical damping. NCERT mentions in Points to Ponder.