What Is a Simple Pendulum?
A simple pendulum is a point mass (bob) suspended from a weightless, inextensible string or rod attached to a fixed pivot. When displaced from its equilibrium position and released, it oscillates in a regular, repetitive motion called simple harmonic motion — provided the angular displacement remains small (less than about 15°).
Period of a Pendulum — Formula
For small angular displacements, the oscillation period does not depend on the mass of the bob or the amplitude of the swing. The period depends only on the length of the pendulum and the local gravitational acceleration:
- T — period of one complete oscillation (back and forth), in seconds (s)
- L — length of the pendulum (from pivot to center of bob), in meters (m) or feet (ft)
- g — local gravitational acceleration (Earth: 9.80665 m/s² = 32.174 ft/s²)
- π — mathematical constant ≈ 3.14159
Frequency and Angular Frequency
The frequency f is the number of complete oscillations per second:
The angular frequency ω (omega) relates to how fast the pendulum sweeps in radians per second:
How to Analyze a Pendulum in Swing
- Measure the pendulum length — from the pivot point to the center of the bob. For example, L = 2 m.
- Choose the gravitational acceleration — use 9.80665 m/s² for standard Earth, or select a planetary preset (Moon, Mars, Jupiter, Sun) or enter a custom value.
- Calculate the period: T = 2π × √(2 / 9.80665) ≈ 2.837 s.
- Find the frequency: f = 1 / 2.837 ≈ 0.352 Hz.
- Find the angular frequency: ω = 2π × 0.352 ≈ 2.213 rad/s.
Finding the Length from the Period
If you know the period T and want to find the required pendulum length:
Example: A grandfather clock beats once per second (T = 2 s). On Earth, the pendulum length must be: L = 9.80665 × (2 / 2π)² ≈ 0.9937 m ≈ 99.4 cm.
Metric vs. Imperial Units
This calculator supports both unit systems:
- Metric: length in meters (m), gravity in m/s²
- Imperial (US customary): length in feet (ft), gravity in ft/s²
Conversion results are always displayed in m, cm, ft, and inches for easy cross-referencing.
Gravity on Different Celestial Bodies
| Body | g (m/s²) | g (ft/s²) |
|---|---|---|
| Earth | 9.80665 | 32.174 |
| Moon | 1.62 | 5.315 |
| Mars | 3.721 | 12.207 |
| Jupiter | 24.79 | 81.332 |
| Sun | 274.0 | 898.95 |
Frequently Asked Questions
Does the mass of the bob affect the period?
No. For a simple pendulum with small oscillations, the period is independent of mass. Only the length and gravity matter.
Does the amplitude affect the period?
For small angles (< 15°), the approximation T = 2π√(L/g) is accurate to within 0.5%. For larger angles, the period increases slightly and depends on amplitude. Our calculator uses the small-angle formula.
What is the length of a pendulum with a 2-second period on Earth?
Using L = g × (T / 2π)²: L = 9.80665 × (2 / 2π)² ≈ 0.9937 m ≈ 99.37 cm ≈ 3.26 ft.
How is a simple pendulum different from a physical pendulum?
A simple pendulum assumes all mass is concentrated at a single point (the bob) and the string is massless. A physical pendulum is a rigid body that rotates about a pivot — its period depends on the moment of inertia: T = 2π√(I / mgD), where D is the distance from pivot to center of mass.