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Physics
Forced oscillations and mechanical resonance
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Change the driving rhythm, watch the spring cart, and build a resonance curve to compare amplitude and phase lag.
Rail scale ±10 cm
Displacement amplitude A =
Displacement lag φ =
Drag the handle to change frequency. f₀: 1 Hz.
- Natural frequency f₀
- 1.00 Hz
- Amplitude peak frequency
- 0.977 Hz
The driving frequency is below the natural frequency; displacement lags force by less than 90°. Compare with the amplitude peak frequency to find the largest response.
┄ Force F/F₀ · dashed
━ Displacement x/A · solid
Both curves are normalized to compare phase, not the sizes of force and displacement.
Displacement lags force by phase φ; its corresponding maximum occurs later.
The system
The rail starts at ±10 cm, expands only for larger amplitudes and stays until Reset. When its scale changes, compare A values rather than distances on screen.
Showing the steady state
m = 1 kg · k = 39.48 N/m. Viscous damping: F damping = −bv, with b = 2ζ√(km).
Model notes
- A single mass on a linear spring moves in one dimension: m = 1 kg, k = (2π)² N/m, viscous damping proportional to velocity, and driving force F = F₀ cos(2πft).
- The scene and readouts show the steady solution after transients have died away. Parameter changes immediately select a new steady solution; amplitude build-up and settling time are not simulated.
- The scan samples 37 analytical solutions from 0.20 to 2.00 Hz at 0.05 Hz intervals; the dots are not experimental measurements. The peak frequency is calculated separately and may lie between samples or outside the scan range.
- Damping ratio ζ ranges from 0.05 to 0.80 to keep amplitudes finite. This is the operating range of this resource, not a physical limit. Dry friction, nonlinear springs, extension limits, and material failure are omitted.
- The time graph separately normalizes force and displacement to compare phase, not magnitudes with different units. At F₀ = 0 the steady response is zero and phase is undefined.
- The rail starts at ±10 cm and expands only for larger amplitudes, keeping the largest scale until Reset. Cart, ruler and ±A band share one labelled metre scale; compare numerical A values across different scales. Cart and spring dimensions are illustrative.