Simple Pendulum Calculator
Calculate pendulum period, frequency, length, or local gravity using the simple pendulum formula. This student-friendly calculator includes unit conversion, Earth/Moon/Mars gravity presets, small-angle checks, an animated pendulum visual, formula work, and step-by-step explanations.
Background
A simple pendulum is an ideal model of a mass hanging from a light string or rod. For small swing angles, its period depends mainly on the pendulum length L and gravitational acceleration g, not on the mass of the bob.
How to use this calculator
- Choose what you want to solve for: period, frequency, length, or gravity.
- Enter the known values and choose units where needed.
- Select a gravity preset, or choose custom gravity for lab or planet problems.
- Enter the amplitude angle to check whether the small-angle assumption is reasonable.
- Click Calculate to see the answer, diagram, formula, interpretation, and steps.
How this calculator works
- It converts all inputs to SI units before calculating.
- It uses T = 2π√(L/g) for the small-angle ideal pendulum model.
- It rearranges the formula when solving for length or gravity.
- It calculates frequency using f = 1/T and angular frequency using ω = 2πf.
- It estimates finite-amplitude correction using T ≈ T₀(1 + θ₀²/16 + 11θ₀⁴/3072).
- It reinforces a key idea: in the ideal model, bob mass does not affect the period.
- It explains whether the amplitude angle is safely in the small-angle range.
Formula & Equations Used
Small-angle period: T₀ = 2π√(L/g)
Amplitude-corrected estimate: T ≈ T₀(1 + θ₀²/16 + 11θ₀⁴/3072), where θ₀ is in radians.
Frequency: f = 1/T
Angular frequency: ω = 2πf = √(g/L)
Length from period: L = g(T/2π)²
Gravity from period and length: g = 4π²L/T²
Example Problem & Step-by-Step Solution
Example 1 — Find the period of a 1.00 m pendulum on Earth
- Use L = 1.00 m and g = 9.80665 m/s².
- Use the small-angle formula T₀ = 2π√(L/g).
- Substitute: T₀ = 2π√(1.00/9.80665).
- The result is about 2.01 s.
- The frequency is f = 1/T ≈ 0.50 Hz.
Example 2 — Find the length for a 2.00 s pendulum clock
- Use T = 2.00 s and g = 9.80665 m/s².
- Rearrange the period formula to L = g(T/2π)².
- Substitute: L = 9.80665(2.00/2π)².
- The length is about 0.994 m.
- This is why a seconds pendulum is almost 1 meter long on Earth.
Example 3 — Estimate gravity from lab data
- Suppose a lab pendulum has L = 0.80 m and measured period T = 1.80 s.
- Use g = 4π²L/T².
- Substitute: g = 4π²(0.80)/(1.80)².
- The result is about 9.75 m/s².
- Small differences from 9.81 m/s² can come from timing error, angle size, air resistance, or measuring length to the wrong point.
Frequently Asked Questions
Q: Does mass affect the period of a simple pendulum?
No. In the ideal simple pendulum model, the bob mass does not affect the period.
Q: What makes a pendulum swing slower?
A longer pendulum swings more slowly, so its period is larger. Weaker gravity also makes the pendulum swing more slowly.
Q: When is the simple pendulum formula accurate?
It is most accurate for small swing angles, often around 15° or less, when air resistance and string mass can be ignored.
Q: What is the difference between period and frequency?
Period is the time for one complete swing cycle. Frequency is how many cycles happen per second.
Q: Why does the calculator show an amplitude correction?
The classic formula assumes very small angles. At larger angles, the real pendulum period is slightly longer, so the corrected estimate helps students see the size of that effect.
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