Simple Pendulum Calculator
Calculate the period, frequency, and energy of a simple pendulum given its length and swing angle. Used in physics labs, clock design, and seismology to model oscillatory motion.
About this calculator
A simple pendulum consists of a mass suspended from a pivot by a massless string of length L. For small angles the period is approximated as T ≈ 2π√(L/g). For larger angles a correction term improves accuracy: T = 2π√(L/g) × (1 + sin²(θ/2)/4), where θ is the maximum swing angle and g is gravitational acceleration. Frequency is simply f = 1/T (Hz). The kinetic energy at the bottom of the swing is KE = ½·m·v²_max, and the potential energy at maximum displacement is PE = m·g·L·(1 − cosθ). By conservation of energy, KE_max = PE_max. Note that the period is independent of mass — only length and gravity matter for timing.
How to use
Take a pendulum with length L = 1 m, maximum angle θ = 15°, mass m = 0.5 kg, and g = 9.81 m/s². Small-angle period: T₀ = 2π√(1/9.81) ≈ 2.007 s. Correction factor: 1 + sin²(7.5°)/4 = 1 + (0.1305)²/4 ≈ 1.00425. Corrected period: T ≈ 2.007 × 1.00425 ≈ 2.016 s. Frequency: f = 1/2.016 ≈ 0.496 Hz. Potential energy at max displacement: PE = 0.5 × 9.81 × 1 × (1 − cos15°) ≈ 0.167 J. Enter these values to instantly obtain all outputs.
Frequently asked questions
How does pendulum length affect the period of oscillation?
Period is proportional to the square root of the pendulum's length: T ∝ √L. Doubling the length increases the period by a factor of √2 ≈ 1.414, meaning the pendulum swings more slowly. Halving the length makes it swing √2 times faster. This is why grandfather clocks have long pendulums for slow, steady ticks, and metronomes use short, adjustable rods for faster tempos. Precise length control is therefore the key tool for tuning a pendulum clock.
Why is a pendulum's period independent of its mass?
Both the restoring force (gravity) and the inertia (mass) scale with m in exactly the same way, so mass cancels out of the equation of motion. Mathematically, the angular acceleration is −(g/L)·sinθ, which contains no mass term. This means a 1 kg bob and a 10 kg bob on identical strings swing in perfect unison — a fact famously observed by Galileo. Mass does, however, affect the energy stored in the pendulum: a heavier bob carries more kinetic and potential energy at the same angle and length.
When does the small-angle approximation break down for a pendulum?
The simple formula T = 2π√(L/g) assumes sinθ ≈ θ (in radians), which introduces less than 1% error for angles below about 14°. Beyond 20° the error grows noticeably, and by 45° the true period is roughly 4% longer than the small-angle estimate. This calculator applies a first-order correction term (1 + sin²(θ/2)/4) that extends accuracy to moderate angles of around 30–40°. For very large swings approaching 90°, higher-order series or numerical integration is needed for precise results.