A driven 0.50 kg oscillator gives (frequency in Hz, steady amplitude in cm) (0.8, 1.2), (0.9, 2.1), (1, 4.8), (1.1, 2.4), (1.2, 1.4). (a) Estimate resonance and angular frequency. (b) Estimate spring constant. (c) Explain phase well below, near and above resonance. (d) Predict increased damping. (e) Explain why finite peak does not prove zero damping and discuss benefit and hazard.

Question

Accepted Answer

Resonance about 1.0 Hz; ω = 6.28 rad/s; k ≈19.7 N/m; greater damping lowers and broadens the peak. Largest sampled amplitude is 4.8 cm at 1.0 Hz, so resonance is near 1.0 Hz; finer frequency steps are needed for precision. Angular frequency ω = 2πf = 6.28 rad/s. For light damping, k ≈mω² = 0.50 × 6.28² = 19.7 N/m. Displacement is approximately in phase with drive well below resonance, lags about π/2 near resonance and approaches antiphase well above it. Greater damping lowers and broadens the peak and can shift it slightly lower, reducing frequency selectivity. A finite steady peak actually requires dissipation balanced by input. Resonance supports tuning and sensing but can produce destructive structural motion, managed through damping or frequency separation.

Driven-oscillator resonance

What a complete response should do

Driven response

How did this one feel?

Driven response — supplied values
Driving frequency / Hz	Amplitude / cm
0.8	1.2
0.9	2.1
1	4.8
1.1	2.4
1.2	1.4