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Pendulum at large amplitude
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Period increases with amplitude.
T ≈ T₀(1+θ₀²/16)For large θ₀ the period increases: T ≈ T₀(1 + θ₀²/16 + ...). At 60° it's 7% longer, and the motion isn't sinusoidal.
At 90° T = 1.18 T₀.
Period isn't amplitude-independent for large swings.
In more detail
At large amplitudes, the pendulum period increases, and the exact period involves an elliptic integral. The simple formula underestimates the period, which increases with amplitude.
Real-world uses
- Precision clocks are designed to swing at small amplitudes to avoid errors.
- Physicists study large-amplitude pendulums as examples of nonlinear dynamics.
- A grandfather clock with a larger swing runs slightly slow, which is a practical effect of the amplitude dependence of the period.
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Frequently asked questions
How much longer is the period at 45 degrees?
About 4% longer than the small-angle period.
Does the mass change the period?
No, not at any amplitude.
Why does a larger swing take longer?
At large angles the restoring force is less than linear in the angle, so the motion is slower on average.
Related topics in Oscillations & SHM
Springs in series/parallel oscillations · Simple pendulum · Small-angle approximation · Physical pendulum · Torsional pendulum · Second's pendulum