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Sharpness of resonance and Q-factor

Damped and forced oscillations · Physics

Simple pendulum

Interactive 3D

Release the bob and watch true small-angle SHM. The period depends only on the string length.

Controls

Drag the sliders or type a value — the simulation updates live.

m
0.3m2m
°
5°45°

Study notes

A resonance curve peaks at f₀ = 200 Hz with half-power bandwidth Δf = 4 Hz. Find Q. How many oscillations does it take for the energy to decay significantly? Step 1: Q = f₀/Δf = 200/4 = 50. Step 2: Energy meaning: Q = 2π × (stored energy)/(lost per cycle) → fraction lost per cycle = 2π/Q = 2π/50 ≈ 0.126 (about 12.6% per cycle). Step 3: After n cycles, energy fraction = (1 − 0.126)ⁿ; it falls to ~1/e after roughly Q/(2π) ≈ 8 cycles. Answer: Q = 50; the oscillation loses ~13% of its energy per cycle, dying out over roughly 8 cycles. A tuning fork (Q ~10⁴) would ring ~1600 cycles in the same relative decay.

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