What your result means
True period is the time for one complete there-and-back swing, calculated without approximating. Small-angle estimate is the same period from the textbook formula that assumes tiny swings, and approximation error is how far apart the two have drifted.
Maximum bob speed is reached at the lowest point of the arc, where all the height the bob lost has become motion. That exchange is the same energy accounting covered by the kinetic energy calculator.
Why this one is different
Two periods are given, not one: the true value and the small-angle approximation taught in schools, with the error between them stated as a figure. That is the whole point of the tool — the textbook formula quietly drifts as the swing widens, and here you can watch it happen rather than being told it happens.
How it works
Gravity pulls the bob straight down, but the rod only lets it move along an arc, so the useful part of that pull is the component along the arc — proportional to the sine of the angle. That gives a restoring force which always points back towards the bottom, and the bob overshoots each time, producing a swing.
Because the force follows the sine of the angle rather than the angle itself, the motion is not quite a simple oscillation. For small swings the difference is negligible and the classic formula works; for large ones the pendulum spends longer near the top of its arc, where the restoring force is weakest, and the real period stretches out. This simulator integrates the true equation step by step, which is why its animation and its period agree even at 170°.
How to use this simulator
- Set the length — the distance from pivot to the centre of the bob.
- Set the release angle. Start at 10°, then push it to 90° and watch the approximation error climb.
- Switch gravity to the Moon and the same pendulum slows to roughly half its Earth rate.
- Press play to release, or read the figures straight off the panel as you drag.
Formula
true period: T = T₀ · (2/π) · K( sin( θ₀ ÷ 2 ) )
max speed: v = √( 2 g L · ( 1 − cos θ₀ ) )
frequency: f = 1 ÷ T
- L — length from pivot to bob centre, in metres
- θ₀ — release angle from vertical, in degrees
- g — gravitational field strength, in m/s² (9.80665 on Earth)
- K — the complete elliptic integral of the first kind, which is what makes the true period exact rather than approximate
Example calculation
A 1 metre pendulum released at 10° on Earth:
T = 2.006 × (2/π) × K( sin 5° ) = 2.010 s
error = 0.19 %
v = √( 2 × 9.80665 × 1 × ( 1 − cos 10° ) ) = 0.546 m/s
Set the controls to those values and the panel shows the same figures. Raise the release angle to 90° and the true period grows to 2.368 s while the estimate stays at 2.006 s — an 18% error.
How far the approximation drifts
Frequently asked questions
Does the mass of the pendulum change its period?
No. Period depends on length and gravitational field strength, not on mass. A heavy bob and a light bob on the same string swing in step, because gravity accelerates both equally. Mass does affect the tension in the string and the energy stored, but not the timing.
Why does the textbook formula go wrong at large angles?
The familiar formula assumes sin of the angle equals the angle, which only holds for small swings. At a 10 degree release it is out by 0.19 percent, at 90 degrees by 18 percent, and at 170 degrees by 144 percent. This simulator reports both the true period and the approximation so you can see the gap open up.
What length gives a one-second swing?
A pendulum about 0.994 metres long has a period near 2 seconds on Earth, meaning one second per swing in each direction. That is the basis of the seconds pendulum used in clock design, and it is why longcase clocks are roughly a metre tall inside.
Related calculators
Assumptions & limitations
- No friction or air resistance. The swing never dies away here. A real pendulum loses amplitude and eventually stops.
- The rod is rigid and weightless. All the mass sits at a single point, so this is an idealised simple pendulum rather than a swinging bar.
- The bob is a point. A real bob has size and spin, which shifts the effective length slightly.
- The pivot is fixed. No support flexing, and no driving force, so resonance is out of scope.
- Motion stays in one plane. A real pendulum can trace an ellipse; this one only swings back and forth.
Further reading
Our guide to the pendulum and simple harmonic motion works through the period formula, the small-angle approximation and exactly how large the error grows at wide swings. For the energy exchange that sets the bob's top speed at the bottom of its arc, see force, work, energy and power.
Sources & references
The constants and both period formulas used on this page come from the following. These are specific to pendulum motion and gravitational field strength.
- NIST — standard acceleration of gravity, the exact 9.80665 m/s² used as the Earth value
- OpenStax College Physics — the simple pendulum, and the small-angle derivation
- Nelson & Olsson (1986), American Journal of Physics — the large-angle period correction, and why the elliptic integral is needed
- NASA NSSDC — Moon fact sheet, surface gravity 1.62 m/s²