SCIENCE · INVESTIGATE AND EXPLAIN

Separate physical parameters from a shared measurement gain

A period fit can be internally consistent while leaving important parameters unidentified. For an ideal spring carrying unknown base mass m plus known added mass M, a clock with elapsed-time gain g reports Tread² = (4π²g²/k)(m + M). The intercept-to-slope ratio identifies m, but the slope alone identifies only k/g².

Learning outcome: Freeze used and withheld data, derive identifiable combinations, add an independent static calibration, and propagate the stated bounded errors only under their declared mass condition.

Starting recap

Reconcile independent equations and evidence, distinguish identifiable combinations from parameters, and retain branch, calibration and observation limitations.

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Try

If the true stiffness changes to 40 N/m while clock gain remains 1.05, the static slope changes and the dynamic period-squared slope changes inversely. The intercept-to-slope mass ratio remains 2 kg. If only clock gain changes to 1.1, the static slope stays fixed while the dynamic slope scales by the gain squared.

Open the experiment

Predict

Loaded-period data alone determine which combination without independent stiffness or timing evidence?

Your prediction and reason

Experiment

Freeze used and withheld data, derive identifiable combinations, add an independent static calibration, and propagate the stated bounded errors only under their declared mass condition.

Combine independent static and loaded-period records

Worked teaching model · synthetic values · Original exact generator: unknown base mass 2 kg, ideal linear spring k, period-readout gain g, force-reading additive offset 0.5 N. Independently calibrated extension and force scale. Static readings at extensions 0.1, 0.2 m and periods at added masses 0, 1 kg identify parameters conditionally; half-load and midpoint-extension records are withheld. Error bounds are stipulated deterministic alternatives, not probabilities or measured data.

True linear spring stiffness: 30 N/m Elapsed-time gain: 1.05 ratio

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Loaded-period squared fit with withheld half-load

Unlike graph quantities rescale independently; read units and values, not screen angles. 1: calibration endpoints (solid).2: withheld half-load observation (open-circle observation). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Loaded-period squared fit with withheld half-load

calibration endpoints

added mass (kg)	indicated period squared (s²)
0	2.901664
1	4.352496

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.

withheld half-load observation

added mass (kg)	indicated period squared (s²)
0.5	3.62708

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.

Calculated quantities under the declared measurement model
Quantity	Value	Unit
Static force reading at extension 0.1	3.5	N
Static force reading at extension 0.2	6.5	N
Indicated unloaded period	1.703427	s
Indicated period with 1 kg added	2.086264	s
Period-squared slope per added mass	1.450832	s²/kg
Base mass inferred from intercept divided by slope	2	kg
Independent-static inferred stiffness	30	N/m
Inferred additive force offset	0.5	N
Clock gain inferred after independent stiffness	1.05	ratio
Withheld half-load indicated period	1.904489	s
Predicted half-load indicated period	1.904489	s
Withheld midpoint static force	5	N
Predicted midpoint static force	5	N
Stiffness lower bound from separate force errors	29	N/m
Stiffness upper bound from separate force errors	31	N/m
Clock lower bound conditional on independently known mass 2	1.026291	ratio
Clock upper bound conditional on independently known mass 2	1.073622	ratio

Loaded periods alone identify base mass and k/g², not stiffness and clock gain separately.

Tread²=4pi²g²(m+Mload)/k. Intercept divided by slope yields m; an independent static slope yields k and its intercept gives the force offset. Then infer g. Withheld records test the linear families but not every shared systematic alternative. Separate force bounds ±0.05 N at each extension yield kfit±1 N/m. The displayed clock interval additionally stipulates an independently known base mass exactly 2 kg and period bound ±0.01 s; without that extra mass evidence, mass uncertainty must also propagate. These are deterministic compatible bounds, not confidence intervals.

Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.

Loaded periods alone identify base mass and k/g², not stiffness and clock gain separately.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

Separate physical parameters from a shared measurement gain

The final supplied packet uses base mass 2 kg, stiffness k = 30 N/m and clock gain 1.05. Periods at added masses 0 and 1 kg determine the slope and intercept. Separately calibrated extension and force-scale units give static readings Fread = kx + b at x = 0.1 and 0.2 m, with an unknown additive force offset. Readings 3.5 and 6.5 N identify k = 30 N/m and b = 0.5 N.

Only after that independent static calibration can the dynamic slope identify g = sqrt(slope k/(4π²)) = 1.05. The half-load period and midpoint-extension force are withheld from calibration and checked afterward. Agreement tests these particular linear families; it is not proof that every systematic fault or nonlinear extension is absent.

The bounded alternative gives each static reading a separate ±0.05 N error. Their difference can vary by ±0.1 N, so stiffness lies between 29 and 31 N/m. A displayed clock interval additionally assumes a newly independent exact mass of 2 kg and period bound ±0.01 s. Since g = Tread sqrt(k/m)/(2π), lower and upper bounds use corresponding low and high values. Without that independent mass condition, the fitted mass uncertainty must also propagate.

Keep supplied generator truth, raw observations, fitted quantities and independent evidence distinct in the final dossier. Repeating a period with the same clock is not an independent clock calibration, and differentiating a fitted force law is not a new force measurement. The bounds are deterministic compatibility statements, not statistical confidence levels or proof of practical measurement skill.

Changed-case check

Predict before reading the comparison. If the true stiffness changes to 40 N/m while clock gain remains 1.05, the static slope changes and the dynamic period-squared slope changes inversely. The intercept-to-slope mass ratio remains 2 kg. If only clock gain changes to 1.1, the static slope stays fixed while the dynamic slope scales by the gain squared.

Retained investigation

Freeze used and withheld data, derive identifiable combinations, add an independent static calibration, and propagate the stated bounded errors only under their declared mass condition. How would a force-scale gain rather than an additive offset change what the static experiment identifies?

Sources and evidence limits

Primary conceptual reference checked 30 October 2026: OpenStax University Physics Volume 1. These original derived packets synthesize the course; no source exercise is reproduced. They are supplied mathematical evidence, not observed apparatus or independent practical certification. Preserve the first report, critique, revision and actual reviewer-frozen unfamiliar assignment in the course evidence workflow. A missing human review or unfamiliar response remains missing evidence.

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How would a force-scale gain rather than an additive offset change what the static experiment identifies?

Your explanation and limitations

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