Physics

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Physical Pendulum Calculator.

Calculate ideal small-angle physical pendulum period from inertia and center-of-mass distance.

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Period: 1.158401 s

Period

0.000000s
Frequency: 0.863259 Hz

Frequency

0.000000Hz

Results update automatically as you type.

Use Cases

Physics Lab Experiments

Verify the theoretical period of a physical pendulum in a classroom or lab setting, using measured values of inertia, mass, and center-of-mass distance.

Example: A meter stick pivoted at one end with I=0.333 kg·m², m=0.5 kg, d=0.5 m, g=9.81 m/s² gives T≈1.64 s.

Engineering Design of Oscillating Systems

Estimate the natural oscillation period of mechanical components like pendulums in clocks or sensors, aiding in design and tuning.

Example: Designing a metronome with a specific period by adjusting mass distribution and pivot location.

Frequently Asked Questions

What is a physical pendulum?
A physical pendulum is any rigid body that oscillates about a pivot point not through its center of mass. Its period depends on its moment of inertia, mass, and the distance from the pivot to the center of mass.
What does 'small-angle' mean in this calculator?
The calculator assumes small angular amplitudes (typically less than about 15 degrees) where sin(θ) ≈ θ. This simplifies the equation of motion, giving a period independent of amplitude. For larger angles, the period increases slightly.
How is the period calculated?
The period T is calculated using T = 2π√(I / (m g d)), where I is the moment of inertia about the pivot, m is the mass, g is gravitational acceleration, and d is the distance from the pivot to the center of mass.

Tips & Common Mistakes

Tips

  • Ensure the moment of inertia is about the pivot point, not the center of mass. If you have I_cm, use the parallel axis theorem: I = I_cm + m d².
  • Use consistent units: kilograms for mass, meters for distance, and kg·m² for inertia. Gravity should be in m/s² (standard 9.81 m/s²).
  • This calculator is valid only for small angular amplitudes. For large swings, the period becomes longer and this formula underestimates it.
  • Double-check that the center-of-mass distance is measured from the pivot to the center of mass, not the length of the pendulum.

Common Mistakes to Avoid

  • Using the moment of inertia about the center of mass instead of the pivot point without applying the parallel axis theorem.
  • Entering the full length of the pendulum as the center-of-mass distance, which is only correct if the mass is concentrated at the end.
  • Forgetting to convert units (e.g., using grams instead of kilograms) leading to incorrect period calculations.

Last updated: August 13, 2026