Physical Pendulum Calculator: T = 2 pi sqrt(I / (m g d))

How long a swinging rigid body takes for one small swing.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

Choose which quantity you want to find.

Enter the other values in the units shown.

The calculator gives the answer with the formula, the substituted numbers, and the answer in other units.

Formulas used

  • T = 2\pi\sqrt{\frac{I}{mgd}}
  • I = \frac{mgdT^2}{4\pi^2}

Worked example

With the default numbers in the calculator above:

  1. The formula, arranged to find the period: period = (2 x pi / sqrt(g) x moment of inertia about the pivot^0.5) / (mass^0.5 x distance from the pivot to the centre of mass^0.5)
  2. Put the values in (every value in its SI unit): period = (2 x pi / sqrt(g) x 0.5^0.5) / (2^0.5 x 0.4^0.5)
  3. The answer: period: 1.58621 s

Change the numbers in the calculator to see the full working, every step, for your own case.

A second worked example

A different case, with these numbers: What do you want to find? Moment of inertia about the pivot; Period (leave it as it is if this is what you are finding) 1.5864 s; Mass (leave it as it is if this is what you are finding) 2 kg; Distance from the pivot to the centre of mass (leave it as it is if this is what you are finding) 0.4 m.

  1. The formula, arranged to find the moment of inertia about the pivot: moment of inertia about the pivot = ((period x mass^0.5 x distance from the pivot to the centre of mass^0.5) / (2 x pi / sqrt(g)))^2
  2. Put the values in (every value in its SI unit): moment of inertia about the pivot = ((1.5864 x 2^0.5 x 0.4^0.5) / (2 x pi / sqrt(g)))^2
  3. The answer: moment of inertia about the pivot: 0.500122 kg m2

Common questions

How is a physical pendulum different from a simple one?

A simple pendulum is a point mass on a string. A physical pendulum is any rigid body, so its shape matters through the moment of inertia.

What is the period of a swinging rod?

For a rod pivoted at its end, T = 2 pi sqrt(2L / 3g).

Why use the moment of inertia about the pivot?

Because the body rotates about the pivot. Use the parallel axis theorem to move it from the centre of mass.

Related calculators

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Any physics or dynamics text (the physical pendulum).

Need a calculator we do not have?

Tell us what you would like to calculate. We read every request and build the most asked-for ones.

We store your request (and your email if you give it) to decide what to build. We do not store the numbers you type into calculators.

Embed this calculator on your own page

Copy this code into your page. The calculator runs inside the frame and links back to Drafting Central.

Need this drawn, designed or checked?

Drafting Central is a mechanical engineering and drafting business working with customers across Australia and around the world. A calculator gives you the numbers; we can turn them into drawings and designs.

Need help with a real project?

Talk to Drafting Central →

← All engineering calculators