Physical Pendulum Calculator

Understanding Physical Pendulum Calculator

A physical pendulum is any rigid body that oscillates about a fixed pivot point. Unlike a simple pendulum, its period depends on the moment of inertia.

How to use this calculator

1. Enter the object’s moment of inertia about the pivot, or the values needed to determine it.
2. Type the mass of the object in kilograms.
3. Enter the distance from the pivot to the center of mass in meters.
4. Click calculate to find the period in seconds.

The formula explained

The formula computes the period of a physical pendulum, \(T\), which is how long one complete oscillation takes. It depends on rotational inertia \(I\), mass \(m\), gravity \(g\), and the pivot to center of mass distance \(d\).

• T = period of the pendulum, in seconds
• I = moment of inertia about the pivot, in \(\mathrm{kg\,m^2}\)
• m = mass of the object, in kilograms
• g = acceleration due to gravity, about \(9.8\,\mathrm{m/s^2}\)
• d = distance from the pivot to the center of mass, in meters

Step by step method

1. Find or calculate the moment of inertia \(I\) about the pivot.
2. Multiply the mass \(m\), gravity \(g\), and distance \(d\) to get the denominator.
3. Divide \(I\) by \(mgd\), take the square root, then multiply by \(2\pi\).

Worked example

Problem. A meter stick of mass \(0.50\,\mathrm{kg}\) is pivoted \(0.20\,\mathrm{m}\) from one end. Its moment of inertia about the pivot is \(0.043\,\mathrm{kg\,m^2}\). Find the period of small oscillations.

1. Use \(T = 2\pi\sqrt{\frac{I}{mgd}}\).
2. Substitute the values, \(T = 2\pi\sqrt{\frac{0.043}{(0.50)(9.8)(0.20)}}\). This gives \(T = 2\pi\sqrt{0.04387}\).
3. Compute the square root and multiply by \(2\pi\), so \(T \approx 1.32\,\mathrm{s}\).

Tips and common mistakes

• Make sure \(d\) is the distance from the pivot to the center of mass, not the total length of the object.
• Use consistent units, because \(I\) must be in \(\mathrm{kg\,m^2}\), mass in kilograms, and distance in meters.

Frequently asked questions