Formula & Calculator
Period of a Simple Pendulum
Calculates the time for one complete swing of a simple pendulum from its length and local gravitational acceleration, valid for small swing angles.
Interpretation
Period of a simple pendulum: T = 2π√(L/g), where L is length and g is gravity. It depends only on length and gravity, not mass. Example: L=1 m, g=9.81 → T≈2.006 s.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| T | Period of oscillation | s |
| L | Length of the pendulum | m |
| g | Gravitational acceleration | 9.81 m/s2 |
What it means
The simple pendulum is a classic example of simple harmonic motion. The period T is the time for one complete oscillation. For small amplitudes (sinθ ≈ θ), the period is given by T = 2π√(L/g). This formula shows that the period is independent of the mass and amplitude (for small angles), depending only on the length of the pendulum and the acceleration due to gravity. This makes pendulums useful for clocks and as gravimeters to measure g. In physics, it is a standard experiment to determine g. In engineering, it is used in seismometers and inertial sensors. Understanding this formula is essential for designing time‑keeping devices and for analysing oscillatory systems.
Worked example
Simple Pendulum Period – Two Examples
Real‑World| Parameter | Value |
|---|---|
| L | 1 m |
| Parameter | Value |
|---|---|
| L | 0.5 m |
Common mistakes
- Small angle approximation: T = 2π√(L/g) is valid only for small amplitudes (θ < ~15°). For larger angles, the period depends on amplitude.
- Length L: The distance from the pivot to the centre of mass of the bob – not the length of the string if the bob has size.
- Gravity g: Use local g – varies with altitude and location.
- Units: L in m, g in m/s² → T in seconds.
- Damping: This is the ideal simple pendulum; real pendulums have damping (amplitude decays).
Applications
The period of a simple pendulum, T = 2π√(L/g), gives the time for one complete oscillation. It is used in the design of pendulum clocks, seismometers, and gravitational instruments. In engineering, it helps design vibration isolation systems and resonant devices. The formula is also applied in geophysics to measure local gravity variations. In education, it is a classic example of simple harmonic motion. By understanding the pendulum period, engineers can design accurate timekeeping devices, measure gravitational acceleration, and analyse the dynamic response of suspended systems. The simplicity of the formula belies its wide applicability in both scientific research and practical engineering.
- Design of pendulum clocks and timekeeping devices
- Seismometers and gravimeters for geophysical surveys
- Vibration isolation and tuned mass dampers
- Educational demonstrations of simple harmonic motion
- Measurement of local gravitational acceleration
Frequently Asked Questions
For small oscillations (θ < 15°), the period is T = 2π·√(L/g), where L is the length of the pendulum (from pivot to centre of mass) and g is the gravitational acceleration. It is independent of the mass and amplitude (for small angles).
Applying it to large angles where the small‑angle approximation is invalid. For large amplitudes, the period depends on the initial angle and is longer than T = 2π√(L/g).
- The pendulum is a point mass on a massless string.
- The oscillations are small (< 15°).
- No air resistance.
- The pivot is frictionless.
The period is proportional to √L. Doubling the length increases the period by √2 (≈ 1.41).
The period is inversely proportional to √g. On the Moon (g ≈ 1.62 m/s²), the period is longer than on Earth for the same pendulum.
The period is the time for one complete swing (back and forth). It is used in clocks (pendulum clocks) and in measuring g (gravimeter).
The exact period is given by an elliptic integral: T = 4·√(L/g)·K(k), where k = sin(θ₀/2) and K is the complete elliptic integral of the first kind. This converges to the small‑angle formula as θ₀→0.
Measure the period T and the length L accurately; then g = 4π²L/T². This is a classic physics experiment.