Identify torsional stiffness with a known inertia increment

Derive the torsional oscillator

A linear torsional restoring torque is −kappa theta, where kappa has units N m/rad. Rotational dynamics gives I theta double-dot=−kappa theta, so omega=sqrt(kappa/I) and T=2pi sqrt(I/kappa). The potential energy is kappa theta²/2 and total energy is I omega_angle²/2+kappa theta²/2. Here omega_angle denotes instantaneous angular velocity, distinct from the fixed oscillation angular frequency.

Identify the one-period ambiguity

A single measured T determines I/kappa=T²/(4pi²), not I and kappa separately. Doubling both inertia and stiffness leaves the same period. Assuming one parameter while reporting the other as independently measured would overstate what the observation establishes.

Add a known independent inertia increment

With a known added coaxial inertia DeltaI and unchanged stiffness, T2²−T1²=4pi² DeltaI/kappa. Therefore kappa=4pi² DeltaI/(T2²−T1²) and I=kappa T1²/(4pi²). The added inertia is independently stipulated here; unknown attachment inertia or changed stiffness would undermine the identification.

Challenge the subtraction with a clock alternative

The generator has true I=2 kg m² and kappa=4 N m/rad. The first period is unbiased; the selected gain changes only the second. A small period-squared difference can amplify that fault. If instead both periods shared the same gain g, inferred I would remain correct while inferred kappa would be divided by g². These are two different error models; neither a good algebraic fit nor repeated readings proves a calibrated clock.

Worked original case

With added inertia1 kg m² and gain1, T1²=2pi² and T2²=3pi². Their difference pi² gives kappa=4 and I=2 exactly. If both periods were multiplied by1.1, the same method would still infer I=2 but kappa=4/1.21≈3.305785.

Investigation

Compare two added inertias at unbiased settings, then bias only the second period. Derive both inferred parameters and explain the sensitivity of a difference. Contrast this with a written common-gain case. Propose an independent timing or stiffness check rather than treating the known generator values as measured evidence.

Public numerical transfer practice

Try a new practice case

For a torsional oscillator with I=2 kg m² and kappa=8 N m/rad, what is oscillation angular frequency in rad/s?

Numerical result in the requested units Check this practice result

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Sources and evidence boundary

Original derivations and assigned numerical cases checked 30 October 2026 against the official AP Physics C Mechanics framework and OpenStax harmonic motion and pendulums. These are supplied or generated records, not physical measurements or performed laboratory time. Do not construct rotating equipment, release loads or conduct launch experiments. Public feedback and retained self-reports do not certify independent mastery.