The simple pendulum
A simple pendulum is an idealized point mass on a massless, inextensible string. For small swings its period is T = 2π √(L/g), where L is the length and g the gravitational acceleration. Remarkably, the period does not depend on the mass or (for small angles) on the amplitude — the property of isochronism that Galileo noticed and Huygens used to build the pendulum clock.
Small-angle approximation and beyond
The formula T = 2π √(L/g) comes from approximating sin θ ≈ θ, which is accurate for small swings (within ~1% below about 20°). At larger amplitudes the true period is longer, growing with the swing angle; the tool applies the leading correction so you can see the amplitude dependence.
How to use the calculator
Set the length, gravity (9.81 m/s² on Earth, less on the Moon or Mars) and release angle. The tool returns the period T and the frequency f = 1/T, and lets you invert the relation — for example, the length needed for a one-second (seconds) pendulum.
Note: the model assumes a rigid, massless string, a point bob and no friction or air resistance. A real (physical) pendulum with distributed mass uses its moment of inertia instead of a simple length.
Frequently asked questions
What is the period of a simple pendulum?
For small swings, the period is T = 2*pi*sqrt(L/g), where L is the length and g is the gravitational acceleration. On Earth (g = 9.81 m/s^2), a 1-metre pendulum has a period of about 2.0 seconds.
Does mass affect the period of a pendulum?
No. The period of a simple pendulum depends only on its length and the local gravity, not on the mass of the bob. This is because gravity accelerates all masses equally.
Does amplitude affect the pendulum's period?
For small angles the period is essentially independent of amplitude (isochronism). At large amplitudes the period grows noticeably longer, so the small-angle formula slightly underestimates it.
What length gives a one-second pendulum?
A pendulum whose period is 2 seconds (a 'seconds pendulum', one second per swing) is about 0.994 m long on Earth. Solving T = 2*pi*sqrt(L/g) for L gives L = g*(T/(2*pi))^2.
References
- D. Halliday, R. Resnick & J. Walker, Fundamentals of Physics (Wiley) — oscillations and the simple pendulum.
- J. B. Marion & S. T. Thornton, Classical Dynamics of Particles and Systems (Cengage) — the pendulum and large-amplitude corrections.
- Isochronism of the pendulum: Galileo Galilei (c. 1602) and Christiaan Huygens, Horologium Oscillatorium (1673).