A swinging pendulum looks like it should depend on how heavy the bob is, but it doesn't — under the small-angle approximation, the period depends only on the pendulum's length and the local strength of gravity. This calculator applies T = 2π√(L/g) to find the period and frequency, and works the equation backward to size a pendulum for a target swing time.
How the pendulum calculator works
The calculator applies the classical small-angle period formula T = 2π√(L/g), where L is the length from the pivot to the bob's center of mass and g is the local gravitational acceleration. From the period it derives the frequency (f = 1/T) and angular frequency (ω = 2π/T). The Solve Length tab rearranges the same equation to find the length that produces a target period.
This is the small-angle approximation: it assumes the swing amplitude is small enough (roughly under 15°) that sin θ ≈ θ in radians, which turns the pendulum's true (nonlinear) motion into simple harmonic motion with an exact closed-form period. For everyday swings — a grandfather clock, a metronome, a swinging lamp — this approximation is extremely accurate.
Why mass doesn't matter
It's counterintuitive, but a bowling ball and a tennis ball on strings of the same length swing at exactly the same rate. The reason is that both the restoring force (gravity acting on the bob) and the bob's inertia (its resistance to acceleration) scale with mass in exactly the same way — so mass cancels out of the equation of motion entirely. Galileo is credited with first observing this isochronism by timing swinging chandeliers against his own pulse.
What does matter is length and gravity. Because period scales with the square root of length, doubling the length does not double the period — it multiplies it by √2 ≈ 1.41. Quadrupling the length is what doubles the period, which the examples above demonstrate directly.
Gravity changes everything — the Gravity tab
Because g sits inside the square root in the denominator, a lower local gravity stretches out the period. The same 1 m pendulum that swings in about 2 seconds on Earth (g = 9.81 m/s²) takes nearly 5 seconds on the Moon (g = 1.62 m/s²) and speeds up to roughly 0.38 seconds near the Sun's surface (g ≈ 274 m/s²). The Gravity tab holds length fixed and compares the resulting period across several worlds side by side, which is a handy sanity check for physics coursework on gravitational acceleration.
Limits and edge cases
The T = 2π√(L/g) formula is exact only in the small-angle limit. As swing amplitude grows past roughly 15–20°, the true period runs slightly longer than this formula predicts, and the correction grows without bound as the amplitude approaches 180°. This calculator does not model large-amplitude corrections, air resistance, or a physical (extended, non-point-mass) pendulum's moment of inertia — all of which shift the real-world period away from the idealized simple pendulum.
Edge cases the calculator handles cleanly: a length or gravity of zero or less is rejected rather than producing NaN or infinite results, since the square root and division are undefined there. Solving for length similarly rejects a non-positive target period.