Pendulum and time

For an ideal pendulum with small swings, the period is approximately T ≈ 2π√(L/g), independent of the bob’s mass. With gravity held fixed, quadrupling the length approximately doubles the period; stronger gravity shortens it. At large amplitudes, this approximation underestimates the period. The experiment displays the small-angle estimate while animating the nonlinear motion, so you can compare them.

Questions

How many times longer is the period when the length increases from 0.5 m to 2 m?

At a fixed length, does stronger gravity make the pendulum swing faster or slower?

Is the small-angle period still accurate when the initial angle increases from 5° to 70°?

Predict

If the length quadruples while gravity and the small amplitude stay the same, does the period also quadruple? Make a prediction before comparing.

Try

Set the initial angle to 5°, press Play, and watch the time needed to return to the highest point on the same side. Pause, change the length from 0.5 m to 2 m, and compare again. Changing a parameter restarts the model from release at rest.

Observe

A longer pendulum swings more slowly; stronger gravity makes it swing faster. The bob is fastest at the bottom and momentarily at rest at either end. With no drag, this model does not stop on its own.

Explain

What affects an ideal pendulum’s period?

At small amplitudes, period grows with √L and decreases with √g. Quadrupling L at the same g therefore doubles the approximate period. Mass cancels from the motion equation: gravity’s restoring torque and the bob’s rotational inertia both scale with mass. Changing mass alone does not change the ideal model’s period.

Why do large swings take longer than the displayed period?

The nonlinear equation θ″ = −(g/L) sin θ describes the motion. Only for small angles does sin θ ≈ θ, giving the approximate period T ≈ 2π√(L/g). For larger displacements, |sin θ| is smaller than |θ|, so the restoring acceleration is weaker than the linear approximation predicts and the period is longer. The displayed period is the small-angle approximation; the motion itself uses RK4 integration of the nonlinear equation.

Energy per unit mass is E/m = ½(Lω)² + gL(1 − cos θ), in J/kg. The fixed simulation step is 1/120 s, split into two RK4 steps of 1/240 s. The numerical solution has error and is not exact.

Check

Changing the length from 0.5 m to 2 m should approximately double the small-angle period. Then set the initial angle to 70° and check whether the actual return time is longer. At an initial angle of zero the bob stays still; the displayed period describes a small perturbation of that equilibrium.

Explore further

Real pendulums lose energy to air resistance and pivot friction. How would damping change the energy? When do the point-mass and inextensible-string assumptions stop being useful?