Library / Foundations: Physics / FPH-21
Foundations: Physics
Period of a simple pendulum
Swing time of a pendulum from its length.
- Ref
- FPH-21
- Rev
- v1
- Field
- Foundations: Physics
- Result
- T [s]
Assumes a simple pendulum of length L with small angular amplitude. g is the local gravitational field strength.
A longer pendulum swings more slowly; stronger gravity makes it swing faster.
Calculator
s
m
N/kg
- Frequency0.4985 Hz
- Length for a 2 s period0.994 m
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
Period (s) current value 2.006 s
LLengthKey driver
0.4498.97
gGravitational field strength
1.264.97
- A pendulum of length about 1 m has a period of about 2 s.
- Measuring T and L gives a way to measure g.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| L Length | 0.05 to 20 m | 0.4486 | 8.971 | 8.523 | +10% | +4.88 % |
| g Gravitational field strength | 1.6 to 25 N/kg | 1.257 | 4.967 | 3.711 | +10% | −4.65 % |
Sweep of Length
| Length (m) | Period (s) |
|---|---|
| 0.05 | 0.4486 |
| 0.1057 | 0.6523 |
| 0.2236 | 0.9486 |
| 0.4729 | 1.379 |
| 1 | 2.006 |
| 2.115 | 2.917 |
| 4.472 | 4.242 |
| 9.457 | 6.169 |
| 20 | 8.971 |
Sweep one variable
Period (s)Hover, touch, or use the arrow keys on the chart
Variables and how each one contributes
T
Period
What the equation gives you. Everything else feeds it.
L
Length
Length to the centre of the bob. Raises the period with the square root.
g
Gravitational field strength
9.81 on Earth, 1.62 on the Moon. Lowers the period with the square root.
Worked example with the values above
- L/g = 1 / 9.81 = 0.10194.
- Square root: 0.31928.
- T = 2π × 0.31928 = 2.006 s.
Assumptions and limits
- Small angular amplitude.
- Light string, point-like bob.
Common mistakes
- Measuring L to the top of the bob instead of the centre.
- Using large swing angles.
Statistics and uncertainty
Time 20 swings to cut timing error; the small-angle approximation adds under 1% error for amplitudes below about 15°.
Where it is used
- Measuring g and timing.
- Sway and swing of suspended loads.
Maths you need
- Small angles: sin θ ≈ θ
- Square roots: period depends on √L