Pendulum Calculator — Period, Frequency, and Length for Simple Pendulums

The Simple Pendulum Formula

For small angles (amplitude ≤ 15° from vertical), the period T of a simple pendulum is:

T = 2π √(L/g)

Where L is the length from the pivot to the centre of mass of the bob (in metres), and g is the local gravitational acceleration (m/s²). This formula is remarkable for what it does not include: the mass of the bob and the amplitude of swing both drop out, meaning the period depends only on the length and local gravity.

The period can be rearranged to solve for each variable: L = g(T/2π)², or g = L(2π/T)². These rearrangements are all handled automatically by the calculator.

Variables in the Pendulum Calculator

How to Use the Pendulum Calculator

1. Open the pendulum calculator.
2. Select what you want to calculate: Period (T), Length (L), or Gravitational acceleration (g).
3. Enter the known values. For period, you need L and g. For L, you need T and g. For g, you need T and L.
4. Click Calculate. The result is displayed with the formula used and the substituted values.
5. To use a non-Earth gravity, select a planet from the dropdown or enter a custom g value in m/s².

Pendulum Period on Different Planets

The Small-Angle Approximation

The formula T = 2π√(L/g) is not exact — it is derived using the small-angle approximation: for angles θ measured in radians, sin(θ) ≈ θ when θ is small. The approximation is accurate to within 1% for amplitudes up to about 7°, and within 2% for amplitudes up to about 10°. At 20° amplitude, the error is about 2%; at 45°, the error exceeds 10%.

For larger amplitudes, the exact period involves an elliptic integral that cannot be expressed in a simple closed form. Practical lab experiments at school and university level stay well within the small-angle regime (typically swinging through 5–10°) so that the simple formula applies.

Using the Pendulum to Measure g

One of the most important experimental applications of the pendulum is measuring the local gravitational acceleration g. By measuring the period T of a pendulum of known length L, you can calculate g = 4π²L/T².

This is one of the classic school physics experiments because it requires only simple equipment (a string, a mass, and a stopwatch) and gives an accurate value of g if done carefully. The main sources of error are:

• Measuring L precisely: L must be from the pivot point to the centre of mass of the bob, not to the top of the bob. Measure to the bob's midpoint.
• Timing the period: Time 10 or 20 complete oscillations and divide, rather than timing a single swing. This averages out human reaction time error.
• Keeping the amplitude small: Large-angle swings violate the small-angle approximation and make T longer than predicted. Keep the amplitude below 10°.
• Air resistance and friction at pivot: These damp the oscillation and introduce systematic error. A smooth pivot (pin or blade) and a dense bob (to minimise drag-to-mass ratio) reduce both effects.

Historical Significance of the Pendulum Clock

Galileo observed in 1581 that a pendulum's period is independent of its amplitude — a property called isochronism (for small angles). Christiaan Huygens built the first practical pendulum clock in 1656, dramatically improving timekeeping accuracy from minutes per day to seconds per day. The pendulum remained the world's most accurate timekeeping mechanism until the quartz oscillator was developed in the 1920s.

The practical challenge of using pendulum clocks at sea — where the ship's motion disrupted the oscillation — drove the invention of the marine chronometer (Huygens and later John Harrison), which was essential for determining longitude on long ocean voyages.

The Seconds Pendulum

A "seconds pendulum" is a pendulum with a period of exactly 2 seconds — it takes 1 second to swing from one extreme to the other (half a period). On Earth at standard gravity (g = 9.81 m/s²), the length of a seconds pendulum is:

L = g(T/2π)² = 9.81 × (2/(2π))² = 9.81 × (1/π)² ≈ 0.994 m ≈ 99.4 cm

The metre was originally proposed to be defined as the length of a seconds pendulum, which is why the metre and the seconds pendulum are so close to being linked by a round number. The definition was later changed to use the Earth's circumference instead, but the pendulum connection is why a 1-metre pendulum has a period so close to exactly 2 seconds.

Compound (Physical) Pendulums

The simple pendulum assumes all mass is concentrated at the bob (a point mass). Real pendulums — including clock pendulums, which are often a rod with a disc — distribute mass along their length. These are compound or physical pendulums.

The formula for a compound pendulum is T = 2π√(I/mgd), where I is the moment of inertia about the pivot, m is the total mass, and d is the distance from the pivot to the centre of mass. For a uniform rod pivoting at one end, I = mL²/3, which gives T = 2π√(2L/3g). The calculator handles the simple pendulum case; for compound pendulums, you need to calculate I first.

Pendulum Waves and Chaos

Multiple pendulums with different lengths swing at different natural frequencies, creating beautiful wave patterns when released together — this is the basis of the "pendulum wave" demonstration common in physics classes. The different periods cause the pendulums to drift in and out of phase, creating the illusion of a wave travelling along the row.

A double pendulum (a pendulum attached to the end of another pendulum) exhibits chaotic motion for large amplitudes — it is one of the simplest systems in physics where the motion is sensitive to initial conditions, making it unpredictable over long time scales despite following deterministic equations of motion.

Pendulums in Modern Context

Seismic pendulums

Long-period pendulums (with very long effective lengths) are used in seismometers to detect slow ground movements. The inertia of the pendulum bob keeps it stationary while the Earth moves beneath it — the relative displacement between bob and support reveals ground motion.

Inverted pendulum control

The inverted pendulum (balanced upright rather than hanging) is a canonical problem in control theory. Segway vehicles, rocket attitude control, and two-legged robot balance all involve inverted pendulum dynamics. Understanding the natural (hanging) pendulum is the mathematical foundation for understanding why the inverted case is so challenging.

Foucault pendulum

A Foucault pendulum (long, free-swinging pendulum) appears to rotate its plane of oscillation over time due to the rotation of the Earth beneath it. The rate of rotation depends on latitude: fastest at the poles (one full rotation per 24 hours) and slowest near the equator. Foucault pendulums are common museum exhibits and provide a direct visual demonstration of Earth's rotation.

Common Questions

Does the mass of the bob affect the period?

No. Mass cancels out of the derivation of the simple pendulum formula. A 1 kg bob and a 10 kg bob on the same length string will have identical periods in the small-angle regime (assuming the bob is dense enough that air resistance is negligible for both). This is a consequence of the equivalence of inertial and gravitational mass — the same property that causes all objects to fall at the same rate in a vacuum.

Why does a shorter pendulum swing faster?

From T = 2π√(L/g): as L decreases, T decreases — the pendulum completes each swing in less time. Physically, a shorter pendulum has less angular displacement to cover per swing and the restoring force (gravity) acts over a shorter lever arm. The relationship is with the square root of length, so to halve the period you need to reduce the length by a factor of four.

What happens to a pendulum in free fall?

In free fall (or in orbit, which is continuous free fall), the effective gravitational acceleration is zero. The restoring force that drives the pendulum's oscillation disappears. The pendulum would not swing at all — any displacement from vertical would be permanent. This is why pendulum clocks stop working in orbital environments.

How accurate is the simple pendulum formula?

At amplitudes below 10°, the formula is accurate to within ~0.5%. At 20°, error is ~2%. At 30°, error is ~4%. For a clock pendulum designed to keep accurate time, these errors matter — clock pendulums keep very small amplitudes (a few degrees) specifically to stay in the accurate small-angle regime.