Hong Life CalculatorsUnderstanding the pendulum period calculator — concept and principle
A simple pendulum is a small weight on a light string or rod that swings under gravity. Its period, the time for one full swing out and back, depends almost only on its length and the local gravitational acceleration. This calculator gives the classic small-angle period, a corrected period for a finite swing amplitude, and the frequency.
How it is calculated
- Small-angle period: T₀ = 2π √(L ÷ g), with length L in metres (measured to the centre of the bob) and g in m/s². The default g is standard gravity, 9.80665 m/s².
- Amplitude correction: T ≈ T₀ × (1 + θ₀²/16 + 11θ₀⁴/3072), where θ₀ is the amplitude converted to radians.
- Frequency: f = 1 ÷ T₀, in hertz (Hz), based on the small-angle period.
The principle
When the bob is displaced by angle θ, gravity pulls it back along the arc with a force proportional to sin θ. For small angles sin θ ≈ θ, so the restoring force is proportional to displacement and the motion is simple harmonic, with angular frequency √(g ÷ L). Mass cancels out because both the restoring force and the inertia scale with mass, which is why a heavy and a light bob on the same string swing together.
For larger swings sin θ is smaller than θ, the restoring force is relatively weaker, and the period grows slightly. The exact period involves an elliptic integral; the two-term series here comes from expanding it.
Worked example
- Inputs: L = 1 m, amplitude = 10°, g = 9.80665 m/s².
- T₀ = 2π × √(1 ÷ 9.80665) = 2π × 0.319333 ≈ 2.006409293 s.
- θ₀ = 10 × π ÷ 180 = 0.174533 rad, so the factor is 1 + 0.0019039 + 0.0000033 = 1.0019072.
- T ≈ 2.006409293 × 1.0019072 ≈ 2.010235879 s.
- f = 1 ÷ 2.006409293 ≈ 0.4984027953 Hz.
| g (m/s²) | Period, small angle (s) |
|---|---|
| 2.45 | 4.012820631 |
| 4.9 | 2.83749268 |
| 7.35 | 2.316801497 |
| 9.81 | 2.006409293 |
| 12.3 | 1.794587943 |
| 14.7 | 1.638224935 |
| 19.6 | 1.418745617 |
How to read the result
At 10° the amplitude adds only about 0.19%, so the small-angle formula is accurate enough for most purposes. Period grows with the square root of length: four times the length gives twice the period. A "seconds pendulum", whose swing one way takes one second (T = 2 s), is about 0.994 m long at standard gravity. At 90° the series gives a factor of about 1.176, slightly below the exact 1.180, so treat very large amplitudes with care.
Common mistakes
- Measuring L to the top of the bob instead of to its centre of mass.
- Entering the amplitude as the full swing width (side to side) rather than the angle from vertical.
- Treating a rigid bar or a large bob as a simple pendulum; it is a physical pendulum with a different formula.
Frequently asked questions
How can I measure g with a pendulum?
Time many swings, divide to get T, then use g = 4π²L ÷ T². Timing 20 or more swings reduces reaction-time error.
Would a pendulum swing differently on the Moon?
Yes. With g about one-sixth of Earth's, the period is about √6 ≈ 2.45 times longer for the same length.
Sources: Halliday, Resnick and Walker, Fundamentals of Physics, chapter on oscillations · Christiaan Huygens, Horologium Oscillatorium (1673)