Use the chain rule to audit an oscillator clock

Differentiate a phase-aware displacement

The true assigned motion is x=.4 cos(2t+phi) m. Its velocity is −.8 sin(2t+phi) m/s and acceleration −1.6 cos(2t+phi) m/s². Position and acceleration have opposite signs; velocity is a quarter-cycle out of phase. A different phi shifts the time of a given state without changing amplitude or true frequency.

Introduce a declared clock intervention

The reported time is u=g t, where g is a dimensionless positive gain. The same positions become x(u)=.4 cos(2u/g+phi). By the chain rule dx/du=(dx/dt)/g and d²x/du²=(d²x/dt²)/g². The indicated angular frequency is 2/g in radians per reported second and indicated period is g pi reported seconds.

Identify what the record can establish

A perfectly smooth sinusoid identifies frequency in its recorded time coordinate. Without an independent timing reference or independently known k/m, it cannot decide whether an altered period came from clock gain or a different physical oscillator. More repeated points from the same clock do not independently calibrate it. The workbench deliberately supplies the gain as an intervention; it does not pretend to infer it from the curve.

Compare representations at one physical instant

The ledger compares all quantities at true t=.5 s; the reported time for that instant is .5g. Comparing two curves at the same numerical reported time would instead compare different physical instants when gains differ. State which comparison you mean before claiming an amplitude, phase or derivative error.

Distinguish free oscillation from resonance

The natural frequency is the frequency of the undriven system after a displacement. A sinusoidal external force near that natural frequency can repeatedly transfer energy with a favorable phase and increase the oscillation amplitude: resonance. The undriven workbench has no forcing term and must not be used as a driven-response calculator. Real damping and nonlinear response limit amplitudes; a response peak alone does not validate the time axis.

In a separate original supplied comparison, the free-oscillation period is pi seconds. Equal small forcing amplitudes at angular frequencies 1,2,4 rad/s give recorded displacement amplitudes .04,.18,.03 m after the same declared observation protocol. Predict which drive is closest to the natural angular frequency before checking 2 rad/s. The pattern is consistent with resonant enhancement, not proof that all other parameters or instrument gains were held fixed. Propose a repeat with independently checked force amplitude and timing. The three supplied observations are not generated by the undriven model and do not establish an exact resonance curve.

For a changed ideal mass-spring system whose stiffness is quadrupled at fixed mass, natural angular frequency doubles. Predict how the drive should change to remain near resonance; merely increasing the forcing amplitude would not restore a frequency match.

Worked original case

With phi=0 and g=1.2, indicated omega=5/3 rad/reported s and indicated period=1.2pi. First derivatives are scaled by5/6 and second derivatives by25/36. Neither scaling changes the assigned true amplitude .4 m.

Investigation

Compare phase indices0 and1 at gain1, then hold phase fixed while changing gain. Derive first and second derivatives in both time coordinates. Retain a same-physical-instant comparison and propose an independent clock or force/mass check that could distinguish the alternatives.

Public numerical transfer practice

Try a new practice case

If reported time is twice true time, by what factor is dx/du reduced relative to dx/dt at the same physical instant?

Numerical result in the requested units Check this practice result

This public, unsaved check is not a secure assessment. Explain the derivative, integral or boundary in the retained course evidence workspace.

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.