A simple pendulum swings under gravity along a circular arc. Its period depends on effective length, gravity and swing amplitude. Choose a target period to calculate the ideal length, including the finite-amplitude correction, and compare it with the small-angle result.
Ideal Pendulum Length Explorer
Choose a target full period, local gravitational acceleration and peak angle on either side of vertical. The worksheet calculates the ideal point-mass effective length including the finite-amplitude correction.
Equation Used
- The bob is an ideal point mass and the suspension is massless and inextensible.
- The pivot is frictionless and motion is planar.
- The selected angle is peak amplitude on each side of vertical.
- The target is one complete oscillation period, not one beat.
- Rod and bob mass distribution, damping, drive, escapement, suspension elasticity, thermal effects, construction tolerances and clock accuracy are excluded.
The length calculation includes the finite-amplitude period correction. The drawing keeps a fixed visual length and uses sinusoidal motion to illustrate the swing; it is not the exact nonlinear angle history.
An ideal gravity pendulum
A simple pendulum is an ideal point mass suspended by a massless, inextensible connection. Gravity provides the restoring torque. The bob follows a circular arc while the suspension length remains constant.
For very small angles, replacing sin(theta) with theta gives simple harmonic motion. At finite amplitude the exact nonlinear period is slightly longer, so an ideal pendulum built for the same target period is slightly shorter.
Compare period, gravity and amplitude
Use the explorer to compare an ideal effective length for a selected complete period and local gravitational acceleration. The half-period is also shown because many clock descriptions call one side-to-side swing a beat.
A physical pendulum with distributed mass needs an equivalent length based on its inertia and centre of mass. This worksheet does not size a real clock, suspension or structure.
Exact finite-amplitude period
The exact ideal period is T=2pi sqrt(L/g)/A, where alpha is peak angle and AGM is the arithmetic-geometric mean. Solving for L gives the displayed finite-amplitude effective length. Setting alpha to zero makes A=1 and recovers L0=g[T/(2pi)] squared.
A two-second ideal pendulum
For T=2 s, g=9.80665 m/s squared and alpha=2 degrees, the small-angle reference length is 993.621 mm. Including finite amplitude gives 993.470 mm, a reduction of about 0.151 mm, and the half-period beat is 1.000 s.
Ideal effective length is not clock accuracy
The model excludes pivot friction, air drag, drive impulses, escapement error, suspension elasticity, thermal expansion, bob and rod mass distribution, mounting motion and construction tolerances. It therefore does not predict seconds gained or lost by a real clock.
The displayed length already accounts for the entered amplitude and target period. Its difference from the small-angle length is an ideal geometric adjustment, not a prediction of a real clock’s daily error.
Pendulum calculation questions
Is period a single swing?
No. T is one complete return to the same position and direction. A side-to-side beat is T/2.
Why does amplitude shorten the calculated length?
Finite amplitude lengthens the period at fixed length. To retain the selected period, the ideal effective length must therefore be slightly shorter.
Does bob mass affect the ideal result?
No, not for the ideal point-mass model. Mass distribution matters for a real physical pendulum.
Does this predict clock accuracy?
No.
Pendulum references
- OpenStax University Physics, Pendulums: ideal simple-pendulum definition and small-angle period.
- Michigan State University, The Simple Pendulum laboratory manual, Appendix B: finite-amplitude integral and series correction.
- NIST Digital Library of Mathematical Functions, AGM algorithms: arithmetic-geometric-mean evaluation of elliptic integrals.
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