You hang a small weight from a string and let it swing. How long does each swing take? The answer is the simple pendulum period, given by T = 2pi sqrt(L/g). Galileo Galilei first noticed this around 1602 while watching a swinging chandelier in Pisa Cathedral. He timed the swings with his own pulse. The period depends only on the length of the string and the local gravity. It does not depend on the mass of the bob or on how far you pull it back, as long as the angle stays small. This property is called isochronism. It is the reason pendulums became the heart of accurate clocks for nearly 300 years.
What This Calculator Does
This tool solves the simple pendulum equation for any one of three unknowns. You can solve for the period T, the length L, or the gravitational acceleration g. Pick what you want to find, enter the other two values, and the calculator does the rest. It also reports the frequency and angular frequency. Gravity presets let you switch between Earth, the Moon, Mars, Jupiter, and the Sun with one click.
Pendulums connect to many other physics topics. The energy in a swinging bob shifts between gravitational and kinetic forms, so you may find our Potential Energy Calculator and Kinetic Energy Calculator useful. The bob accelerates as it swings, which ties in with the Acceleration Calculator.
Inputs Required
- Length (L): The distance from the pivot to the center of the bob, in meters, centimeters, millimeters, feet, or inches
- Period (T): The time for one complete swing, in seconds, milliseconds, or minutes
- Gravity (g): The local gravitational acceleration, entered directly or chosen from a planet preset
Outputs Provided
- Period (T): Time for one full back-and-forth swing in seconds
- Frequency (f): Number of swings per second, equal to 1/T
- Angular Frequency (omega): The rate of oscillation in radians per second, equal to 2pi/T
- Length (L): The required pendulum length in meters
- Gravity (g): The inferred gravitational acceleration in m/s^2
How the Calculation Works
The simple pendulum formula comes from Newton's second law applied to a small mass on a light string. When the bob is displaced by a small angle, the restoring force is proportional to the displacement. That produces simple harmonic motion. The math gives three equivalent forms of the same equation, depending on which variable you solve for.
T = 2 * PI * sqrt(L / g)
L = g * (T / (2 * PI))^2
g = L * (2 * PI / T)^2
f = 1 / T
omega = 2 * PI / T
The small angle approximation is the key simplification. The exact period includes correction terms that grow with the swing angle. For angles under about 15 degrees, the error stays below 0.5 percent. Most classroom and clock problems fall well inside this range, so the simple formula is accurate enough.
Notice that mass never appears in the formula. A heavier bob feels a stronger gravitational pull, but it also has more inertia. These two effects cancel exactly. A 1 gram bob and a 1 kilogram bob on the same length string swing with the same period. This is why the pendulum formula is so clean.
How to Use the Calculator
- Choose what to solve for using the Solve For dropdown: Period, Length, or Gravity.
- Enter the two known values in their input fields.
- Pick the right unit for each length and period value from the unit dropdowns.
- Select a gravity preset, or choose Custom and type your own gravity value.
- Read the primary result in the highlighted box at the top of the results card.
- Use the Copy Result button to copy the value to your clipboard.
- Check the calculation breakdown and the formula reference to verify your work.
Example Calculations
Grandfather clock in London. A grandfather clock in a London townhouse needs a pendulum that ticks once per second, meaning a full period of 2 seconds. Using Earth gravity of 9.80665 m/s^2, the required length is L = 9.80665 * (2 / (2 * 3.14159))^2 = 0.994 meters. That is why old longcase clocks stand about 1.8 meters tall, with the bob near the bottom of the case.
Pendulum on the Moon. An astronaut sets up a 1 meter pendulum inside a lunar habitat where gravity is 1.62 m/s^2. The period is T = 2 * 3.14159 * sqrt(1 / 1.62) = 4.94 seconds. On Earth the same 1 meter pendulum would swing with a period of about 2.01 seconds. The lower lunar gravity stretches each swing to roughly two and a half times as long.
Real-World Scenarios
Foucault pendulum at the Pantheon, Paris. In 1851 Leon Foucault hung a 28 kilogram brass bob from a 67 meter wire inside the Pantheon in Paris. Its period was T = 2 * 3.14159 * sqrt(67 / 9.80665) = 16.4 seconds. As it swung, the plane of oscillation slowly rotated, giving the first simple visual proof that the Earth turns. At Paris latitude the plane completes a full rotation in about 31 hours and 47 minutes.
Seismometer design in San Francisco. Engineers at a seismic lab in San Francisco tune a horizontal pendulum sensor to a 1.0 second period so it matches the dominant frequency of nearby fault ground motion. Solving for length gives L = 9.80665 * (1.0 / (2 * 3.14159))^2 = 0.248 meters. The 24.8 centimeter arm length keeps the sensor responsive to the waves that matter most for early warning systems.
Pendulum gravity survey in a South African mine. Geophysicists lower a pendulum apparatus down a shaft in a gold mine near Johannesburg. At the surface a 0.5 meter pendulum has a period of 1.419 seconds. At a depth of 3 kilometers the period drops to 1.418 seconds. Plugging back into g = L * (2 * PI / T)^2 gives a gravity change of about 0.003 m/s^2, which reveals the density of the rock the shaft passes through.
Common Mistakes to Avoid
- Large angle errors: The simple formula breaks down above 15 degrees. A 45 degree swing has a period about 4 percent longer than the formula predicts. Use the full elliptic integral correction for large swings.
- Confusing physical and simple pendulum: A real object like a baseball bat is a physical pendulum, not a point mass on a string. It needs the moment of inertia in its formula, which this tool does not include.
- Forgetting mass independence: People often assume a heavier bob swings slower. It does not. Adding mass changes nothing about the period, so do not expect different results when you swap bobs.
- Using the wrong gravity value: Gravity varies with latitude and altitude. Standard gravity is 9.80665 m/s^2, but local values can differ by up to 0.5 percent. For precision work, look up the local value rather than using the default.
Limitations of This Calculator
This tool models an ideal simple pendulum: a point mass on a massless, inextensible string swinging in a frictionless vacuum. Real pendulums behave differently. The calculator does not handle large angle correction, which requires elliptic integrals. It does not model physical or compound pendulums, where the mass is distributed along a rigid body. It ignores damping from air resistance and friction at the pivot, so it cannot predict how a pendulum slows and stops. It also does not cover coupled pendulums that transfer energy back and forth. For those cases you need more advanced tools or a full numerical simulation.
Authoritative Research and Resources
For deeper study, these sources are reliable and free to access.
- NIST SI Units - The National Institute of Standards and Technology defines the meter and second used in the pendulum formula. Their reference explains the exact SI definitions and standard gravity value of 9.80665 m/s^2.
- HyperPhysics Pendulum - Georgia State University's HyperPhysics site walks through the derivation of the pendulum period step by step. It also covers the large angle correction and physical pendulums with clear diagrams.
- Khan Academy Physics - Khan Academy offers free video lessons on simple harmonic motion, including worked pendulum problems. It is a good place to build intuition before tackling harder derivations.