Pendulum simulator & period calculator
Change the length, gravity and starting angle, press Play and time the swing. For small angles the period of a simple pendulum is T = 2π√(L/g): it depends only on the length and the gravitational acceleration, not on the mass of the bob. A 1 m pendulum on Earth has a period of about 2.01 s.
What if you changed…
Make the pendulum twice as slow. How much longer must it be?
See the calculation
- T = 2π√(L/g) = 2π√(1/9.81) = 2.01 s.
- Frequency = 1/T = 0.498 Hz.
- Small-angle motion: θ(t) = 10 cos(2πt/T) degrees.
What this model assumes
Small-angle approximation, massless rod/string, point bob, no damping. Angles are limited to 15°.
Keep an eye on this
Large swings have a longer period than the small-angle approximation predicts.
Understand the physics ↗Formulas used in this simulation
- Period (small angles)
- T = 2π√(L/g)
- Accurate below about 15°
- Frequency
- f = 1/T = (1/2π)√(g/L)
- Complete swings per second
- Length for a given period
- L = gT² / (4π²)
- Rearranged period formula
- Gravity from a measured period
- g = 4π²L / T²
- A classic lab measurement of g
- Angle over time
- θ(t) = θ₀ cos(2πt/T)
- Released from rest at θ₀
Pendulum explorer: common questions
Wondering about the why? Start here.
How do you calculate the period of a pendulum?
Use T = 2π√(L/g) with the length L in metres and g in m/s². For L = 0.5 m on Earth, T = 2π√(0.5/9.81) ≈ 1.42 s.
How long is a pendulum with a 1 second period?
L = gT² / (4π²) = 9.81 × 1² / 39.48 ≈ 0.248 m, so about 25 cm. A “seconds pendulum”, which takes 1 s for each one-way swing (a 2 s period), is about 0.994 m long.
Why does the formula only work for small angles?
Its derivation uses sin θ ≈ θ in radians. At 15° the true period is only about 0.4% longer than 2π√(L/g), but at 60° it is about 7% longer.
Does a heavier bob swing faster?
Not in this ideal pendulum model: mass cancels out. Length and gravity set the small-angle period.
Why does amplitude barely change the period?
The formula assumes small angles. The animation follows that approximation; for large angles, amplitude does affect period.