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Physical Pendulum Period Calculator

Enter inertia, mass and pivot distance to get the swing period.

Result

1.6383 s
0.6104 Hz

How it is calculated

The tool uses the small-angle formula T = 2π·√(I / (m·g·d)) and then f = 1 / T. Here I is the moment of inertia about the pivot axis in kg·m², m the mass in kg, d the distance from the pivot to the centre of mass in m, and g the gravitational acceleration in m/s². If you leave gravity empty, it uses standard gravity, 9.80665 m/s². The period is shown in seconds and the frequency in hertz, both to 4 decimals. Inertia, mass and distance must all be positive, otherwise no result is produced. The formula is only valid for small swings (a few degrees).

Worked example

A uniform 2 kg rod, 1 m long, hangs from one end. About that end I = mL²/3 = 0.6667 kg·m², and the centre of mass is d = 0.5 m below the pivot. The product m·g·d is 2 × 9.80665 × 0.5 = 9.80665. Then I / (m·g·d) = 0.0680, its square root is 0.2607, and multiplying by 2π gives the period 1.6383 s. The frequency is 1 / 1.6383 = 0.6104 Hz.

Try this example →

Common mistakes

  • Using the moment of inertia about the centre of mass instead of about the pivot. They differ by m·d² (parallel-axis theorem), and the formula needs the pivot value.
  • Entering the full length of the rod as the distance. d is only the pivot-to-centre-of-mass distance, 0.5 m for a uniform 1 m rod hung from its end.
  • Mixing units, such as grams or centimetres with a kg·m² inertia. Convert everything to kg and m first.
  • Using a large starting angle. The result assumes small oscillations; for wide swings the real period is longer than the one shown.
  • Entering 0 as the distance for a body pivoted at its centre of mass. It would never swing back, and the tool returns no result.

About this calculator

Unlike an idealized simple pendulum (a point mass on a massless string), a physical (compound) pendulum is any rigid body that swings about a pivot, so its period depends on how its mass is distributed rather than just its length. For small oscillations, the period is T = 2π·√(I / (m·g·d)), where I is the moment of inertia about the pivot, m is the total mass, g is gravitational acceleration, and d is the distance from the pivot to the center of mass.

This calculator takes the moment of inertia, mass, and pivot distance and returns the swing period directly from that formula. Physics students verifying lab measurements, and engineers designing pendulum clocks, seismometers, or vibration-damping mechanisms use it because real swinging objects — a metal rod, a clock's pendulum with its rod and bob, a swinging door — are physical pendulums, not the idealized point-mass version taught first in introductory mechanics.

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Comments (1)

  • Editorial team

    For ordinary lab work the choice of g hardly matters. The 2 kg rod from the example has a period of 1.6383 s with standard gravity and 1.6405 s with 9.78, a change of about one part in a thousand. The inertia value matters far more, so get it about the pivot, not the centre of mass. Measure ten swings and divide by ten to check the result. What shape are you swinging, and how did you obtain its inertia?

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