M062 · Nonlinear oscillations / period metrology

Finite-Amplitude Pendulum

Drag the hollow release bob on a rigid, massless rod. Compare its nonlinear motion with a harmonic reference, acquire consecutive bottom crossings, and test the measured period against an independent elliptic-integral prediction. Explore large swings, near-barrier slowing, damping and complete rotations.

Recorded time $0\,\mathrm{s}$

Principal angle $0\,\mathrm{rad}$

Unwrapped angle $0\,\mathrm{rad}$

Angular velocity $0\,\mathrm{rad\,s^{-1}}$

Relative kinetic energy $0\,J$

Gravitational energy $0\,J$

Mechanical energy $0\,J$

Viscous dissipation $0\,J$

Recorded energy-balance defect $0\,J$

Step-halving state difference $0$

Actual integration step $0\,\mathrm{ms}$

Initial-energy undamped period $0\,s$

Small-angle period $0\,s$

Latest acquired crossing interval $\text{Collecting}$

Measured vs initial-energy reference $0\,\%$

Initial conservative orbit $\text{Libration}$

Rod force toward pivot $0\,N$

RELEASE / ACQUIRE / VERIFY

Does swing amplitude change the clock?

Bob mass

m

1.00\,\mathrm{kg}

Rod length

L

1.00\,\mathrm{m}

Gravity

g

9.81\,\mathrm{m\,s^{-2}}

Release angle from the bottom

\theta_0

75.00\,{}^\circ

Initial angular velocity

\dot\theta_0

0.00\,\mathrm{rad\,s^{-1}}

Viscous damping ratio

\zeta

0.00\,

Maximum numerical step

h_{\max}

10.00\,\mathrm{ms}

Inspect recorded time

t/t_{\mathrm{end}}

0.00\,\%

Purple outline sets release angle; teal is velocity. The faint motion curve is the small-angle solution for the same initial state. Period dots use only crossings acquired before the cursor; the period curve is a prediction for release from rest. Negative rod force means compression. Every drag has an equivalent keyboard slider.

Point bob on a rigid rod, fixed pivot and uniform gravity. The rod supports both tension and compression; this is not a slack-string model. Viscous pivot torque is optional. Elliptic periods describe the undamped orbit at the initial energy; a damped crossing interval changes as energy is lost. RK4 power quadrature, step halving and a signed rod force expose model and numerical limits.

Start with the essentials

Focus question: How much timing error does the small-angle approximation create?
One-sentence intuition: Separate an undamped orbit prediction from a period actually acquired at two successive crossings.

Core mathematical model

Nonlinear equation

\ddot\theta+2\zeta\omega_0\dot\theta+\omega_0^2\sin\theta=0,\quad \omega_0=\sqrt{g/L}

The viscous torque scales with inertia and angular velocity.

Undamped libration period

T=\frac{4}{\omega_0}K(k),\quad k=\sin\frac{\theta_{\max}}2

The turning angle is inferred from energy; the release may include initial velocity.

Independent elliptic integral

K(k)=\frac{\pi}{2\operatorname{AGM}(1,\sqrt{1-k^2})}

The prediction uses an independent numerical algorithm rather than the trajectory integrator.

Energy and dissipated heat

E=\tfrac12mL^2\dot\theta^2+mgL(1-\cos\theta),\quad E+Q=E_0

Heat is integrated from the viscous power; the residual remains an independent check.

Signed rigid-rod force

F_{\mathrm{rod}}=m(g\cos\theta+L\dot\theta^2)

Positive force pulls toward the pivot; negative force pushes away. A rigid rod supports both.

Common difficulties

Prediction is not acquisition

Typical misconceptionThe period display should already be a measurement at release.

Better mental modelThe reference is available immediately. The instrument needs two same-direction bottom crossings.

Damping changes orbit energy

Typical misconceptionOne elliptic period describes every damped cycle.

Better mental modelThe fixed reference uses initial energy; later crossing intervals change as the swing shrinks.

A rod can push

Typical misconceptionEvery high-angle release must lose its constraint.

Better mental modelThat conclusion belongs to a tension-only string. This apparatus deliberately uses a rigid rod.

Run the experiment

01
Calibrate near the bottom
Acquire the small-angle preset, then change bob mass.
What to observe: The period stays fixed while energy and force scale with mass.
02
Increase the swing
Compare large-swing and near-barrier presets with the gray motion trace.
What to observe: The harmonic curve gets ahead and the measured interval matches the longer elliptic prediction.
03
Change the orbit family
Choose complete rotations, then reverse initial angular velocity.
What to observe: The principal angle folds, while the unwrapped trace and same-direction revolution timing remain continuous.
04
Separate physics from integration
Use damping, then halve the requested time step.
What to observe: Dissipation persists; the independent energy residual and state difference decrease with refinement.