Separate a pendulum length offset from a clock-scale error
A simple pendulum model has a point-like bob, a massless fixed-length support and small angular displacement. Its approximate period is T=2π√(L/g). Length is measured from pivot to the bob’s center, not merely the visible support segment. The approximation becomes less accurate at large amplitudes; amplitude independence is a model statement, not a universal fact about swinging systems.
Learning outcome: Fit the first and last generated records and retain the middle as a check. Vary length offset separately from time gain, then propose an independent length or time reference that distinguishes their effects.
Starting recap
Declare temperature and time references, system boundaries and model regimes. Use absolute temperatures for ratios; distinguish state, path, oscillator phase and wave propagation.
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At true length 1 m, the small-angle period is 2π/√10≈1.98692 s. Multiplying the physical length by four doubles the ideal period, whereas adding a fixed label offset does not obey that physical scaling.
Open the experimentPredict
A common multiplicative clock error primarily changes the period-squared fit’s:
Experiment
Fit the first and last generated records and retain the middle as a check. Vary length offset separately from time gain, then propose an independent length or time reference that distinguishes their effects.
Use a period-squared slope without hiding length offsets
Worked teaching model · synthetic values · Ideal small-angle pendulum with g=10; true lengths .5,1,1.5 m. Indicated length=true length+offset and indicated period=clock scale×true period. Fit first and last records and reserve the middle. Point bob, massless string; no physical swinging.
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Fit and held-out pendulum record
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: two-point fitted relation (solid).2: held-out middle observation (filled markers). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Fit and held-out pendulum record
two-point fitted relation
| indicated length (m) | indicated period squared (s²) |
|---|---|
| 0.55 | 1.97392 |
| 1.55 | 5.92176 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
held-out middle observation
| indicated length (m) | indicated period squared (s²) |
|---|---|
| 1.05 | 3.94784 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
| Quantity | Value | Unit |
|---|---|---|
| Period-squared slope against indicated length | 3.94784 | s²/m |
| Period-squared intercept | -0.197392 | s² |
| Gravity inferred without clock correction | 10 | m/s² |
| Length offset inferred from intercept and slope | 0.05 | m |
| Indicated middle period | 1.98692 | s |
A length offset appears in the intercept, while a clock gain changes the inferred gravitational acceleration.
For a point bob at small angle, T=2pi sqrt(L/g). Squaring yields Tread²=(4pi² clockGain²/g)(Lread−offset). A midpoint from the same disclosed generator checks consistency but does not independently calibrate the clock. Large amplitudes, extended mass distributions and elastic supports require revised models; amplitude independence is not universal.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
A length offset appears in the intercept, while a clock gain changes the inferred gravitational acceleration.
For a point bob at small angle, T=2pi sqrt(L/g). Squaring yields Tread²=(4pi² clockGain²/g)(Lread−offset). A midpoint from the same disclosed generator checks consistency but does not independently calibrate the clock. Large amplitudes, extended mass distributions and elastic supports require revised models; amplitude independence is not universal.
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Why does a visually straight period-squared plot not establish the correct gravitational acceleration?
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