Cube3D 立方体数理Physics · Middle school / High school
Does a pendulum swing more slowly when you pull it farther?
A pendulum’s period is the time for a full return to the same state. If you release it from the right, reaching the left-hand turning point takes only half a period. Passing the vertical line happens twice per cycle, once in each direction. Use the model clock to time a full return to the starting side.
Shorten the pendulum while keeping the release angle fixed
Release the bob at 1.00 m and 10°. The model completes its first swing in about 2.010 s. Now choose 0.25 m and 10°, release it again, and compare the first complete swing. It takes about 1.005 s, half the original time.
For small angles, T₀ = 2π√(L/g), where L is measured from the pivot to the center of the bob. Shortening the length to a quarter multiplies the period by √(1/4) = 1/2. This ratio also holds in the ideal nonlinear model when the release angle is unchanged: the angle changes the correction factor, but the √L dependence stays the same.
A large swing takes longer than the small-angle prediction
Set the length back to 1.00 m and choose 60°. The period rises to about 2.153 s, while the small-angle reference stays at 2.006 s, so the model period is about 7.3% longer than that reference. Turn on the overlay and watch the dashed prediction reach its starting angle earlier than the solid curve. Both curves start at the same angle on the same simulated clock.
The angular acceleration is −(g/L) sin θ. The small-angle formula replaces sin θ with θ in radians. At large displacement, sin θ is smaller than θ in magnitude, so the true restoring acceleration is weaker than that linear prediction. The nonlinear period is 4√(L/g) K(sin(θ₀/2)), where K is a complete elliptic integral and θ₀ is the release angle. The experiment computes that period and integrates the angular motion.
Time a cycle without mixing clocks
Enable quarter-speed playback. The display moves more slowly in real time, but the model still reports the same duration per swing in simulated seconds. Compare the model readings, rather than timing the slowed animation with a wall clock.
Choose the 0° preset to test the limit. With zero initial displacement and zero speed the pendulum stays at rest. The small-angle expression still gives a limiting reference as the amplitude tends to zero, but there is no actual swing to time at exactly zero.
Check your understanding
At a fixed 60° release angle, what happens to the period if the length goes from 0.50 m to 2.00 m? Use the length slider to compare.
Show the explanation
It doubles, from about 1.522 s to 3.045 s. The length is multiplied by four, so its square root doubles, while the angular correction is unchanged.
What the model assumes
This is an ideal point mass on a massless rigid link, with no friction or air drag, released from rest at 0–80° with length 0.25–2.00 m and g fixed at 9.81 m/s². The diagram is illustrative. The displayed duration and cycle markers come from the mathematical model, not measurements of a physical pendulum. The lab requires Family Plus or Teacher access; this note is public.