Simple Pendulum Calculator

A simple pendulum is a point mass hanging from a fixed pivot by a string with no mass and no stretch, swinging back and forth under gravity. This calculator gives you its period, frequency, and angular frequency once you enter the string length and the local gravitational acceleration. You can also work backwards: enter a known period and it solves for length or for the gravity that produced it.

Pendulum-like motion shows up all over the place. A kid on a swing set, a wrecking ball on a cable, a chandelier set off by a draft. As long as the swing stays under about 15 degrees off vertical, all of them behave like the simple pendulum the formulas below describe.

What is a simple pendulum?

A simple pendulum is an idealization: a point mass (the bob) at the end of a rigid, massless string fixed at the top. Pull the bob sideways and let go, and gravity drags it back toward the bottom. Inertia carries it past, and it swings back and forth in a steady, repeating pattern that physicists call simple harmonic motion.

The restoring force depends on the angle $\theta$ between the string and vertical:

F_{\theta} = -mg\sin\theta

For small angles, $\sin\theta$ is close to $\theta$ in radians, so the force is roughly proportional to the displacement. That proportionality is what makes the motion simple harmonic in the first place, and it is also why pendulums keep such steady time.

How to use the calculator

Start with the length $L$ , measured from the pivot to the center of the bob. Gravity defaults to Earth's standard $9.80665\ \text{m/s}^2$ . Change it to $1.62$ for the Moon, $3.71$ for Mars, or whatever value you need. Angular frequency ( $\omega$ ), frequency ( $f$ ), and period ( $T$ ) come back instantly. Reverse the calculation by typing a known period into $T$ to back-solve for length or gravity.

Understanding the formula

Three related formulas describe a simple pendulum:

\omega = \sqrt{\dfrac{g}{L}} \qquad f = \dfrac{1}{2\pi}\sqrt{\dfrac{g}{L}} \qquad T = 2\pi\sqrt{\dfrac{L}{g}}

$L$ is the string length in meters, $g$ is gravitational acceleration in $\text{m/s}^2$ , $\omega$ is angular frequency in radians per second, $f$ is frequency in hertz, and $T$ is the period, the time for one complete back-and-forth swing, in seconds.

Try a 1-meter pendulum on Earth. Drop $L = 1\ \text{m}$ and $g = 9.80665\ \text{m/s}^2$ into the period formula:

T = 2\pi\sqrt{\dfrac{1}{9.80665}} = 2\pi \times 0.3193 = 2.006\ \text{s}

Just over two seconds for a complete swing. That is the reason a 1-meter pendulum is the classic "seconds pendulum": each half-swing takes almost exactly one second, which is how mechanical clocks held time for centuries.

Look at what is missing from the formula: mass. A bowling ball and a marble on identical 1-meter strings swing with the same period. Galileo first noticed this watching a chandelier in Pisa Cathedral, and the observation laid the groundwork for the much later idea of gravitational equivalence.

Applications

Pendulums earn their keep outside the physics classroom too. The most familiar use is timekeeping: from Christiaan Huygens' invention of the pendulum clock in the 17th century until the 1930s, pendulums were the most accurate clocks in the world.

Pendulums also measure gravity. Geophysicists use precision pendulums to map small gravitational variations across Earth's surface, which can reveal dense ore deposits or oil reservoirs underground. Foucault's pendulum, hung from a tall ceiling so it can swing freely for hours, shows Earth's rotation by appearing to slowly turn its swing plane. And every metronome, swing set, and church bell runs on the same equation.

Tips for accurate results

The formulas assume an ideal pendulum: point mass, massless string, no friction, small swing angle. Real pendulums deviate from that ideal. Above 15 degrees, the small-angle approximation starts to break down. The actual period is about 1% longer at 20 degrees and about 4% longer at 40 degrees. Air resistance and pivot friction shrink the amplitude over time (damping), but the period stays nearly constant. That constancy is exactly what makes pendulums useful for clocks.

Frequently asked questions

Does mass affect the period of a simple pendulum?

It does not. Mass cancels out of the equation of motion, so a heavy bob and a light bob on identical strings swing with the same period. That cancellation is a consequence of the equivalence principle: gravitational mass and inertial mass are the same number.

What happens at large swing angles?

The small-angle approximation ( $\sin\theta \approx \theta$ ) loses accuracy as the angle grows. The true period gets longer with amplitude: about 4% longer than the formula predicts at $45^\circ$ , and roughly 18% longer at $90^\circ$ .

Can I measure gravity with a pendulum?

Yes. Rearranging the period formula gives $g = 4\pi^2 L / T^2$ . Measure L with a ruler and T by timing many swings (averaging makes T more accurate), and you have local g. Enter L and T above to let the calculator do the same back-solve.

How does the pendulum's period change on the Moon?

Lunar gravity is about $1.62\ \text{m/s}^2$ , roughly one sixth of Earth's. Since $T$ scales as $1/\sqrt{g}$ , a 1-meter pendulum on the Moon has a period near 4.93 seconds, almost 2.5 times slower than the same pendulum on Earth. Swap the gravity value above to try other planets.

What is the difference between angular frequency and frequency?

Frequency ( $f$ ) counts complete cycles per second in hertz. Angular frequency ( $\omega$ ) measures the same motion in radians per second, where one full cycle equals $2\pi$ radians. So $\omega = 2\pi f$ . Both describe the same oscillation in different units.