Damped Free Displacement

Work through one named subgoal at a time. Each step is checked deterministically against the canonical solver — no AI required to verify correctness. Get an AI explanation when you're stuck.

Step 1

Find the natural angular frequency ω₀ as if there were no damping (b = 0).

Hint

ω₀ = √(k/m). This is the frequency the oscillator would have without any friction.

Step 2

The equation of motion for damped SHM is x″ + γx′ + ω₀²x = 0, where γ = b/m. Calculate γ.

Step 3

Confirm underdamping (ω₀² > γ²/4) and find the damped angular frequency ω_D = √(ω₀² − γ²/4).

Step 4

The amplitude decays as e^(−t/τ) where τ is the time constant. Express τ in terms of γ.

Solution walkthrough

Damping modifies simple harmonic motion by adding a velocity-proportional drag force F_d = −bẋ. The equation x″ + γx′ + ω₀²x = 0 (with γ = b/m = 2 s⁻¹, ω₀ = 5 rad/s) has three possible regimes. Checking ω₀² = 25 against γ²/4 = 1 confirms underdamping: the system oscillates. The damped frequency is ω_D = √(25 − 1) = √24 ≈ 4.90 rad/s — slightly slower than ω₀ because damping robs the oscillator of some effective stiffness. The solution for zero initial velocity is x(t) = x₀ e^(−γt/2)[cos(ω_D t) + (γ/2ω_D) sin(ω_D t)]. At t = 1.5 s, the exponential factor is e^(−1.5) ≈ 0.223, and the trigonometric part evaluates to give x ≈ 0.0136 m — the oscillator has lost most of its initial amplitude in just over one decay time. The time constant τ = 2/γ = 1 s sets the scale for that decay. This same underdamped profile governs a swinging door closer, a car shock absorber, and a plucked guitar string.

Step-by-step solution

Find the natural angular frequency ω₀ as if there were no damping (b = 0).

The equation of motion for damped SHM is x″ + γx′ + ω₀²x = 0, where γ = b/m. Calculate γ.

Confirm underdamping (ω₀² > γ²/4) and find the damped angular frequency ω_D = √(ω₀² − γ²/4).

The amplitude decays as e^(−t/τ) where τ is the time constant. Express τ in terms of γ.

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