Physical Pendulum Bench: Exploring Parallel-Axis Theorem and Radius of Gyration

What is a Physical Pendulum?

A physical pendulum consists of any object that can oscillate around a pivot point. Unlike a simple pendulum with a point mass suspended by a string, a physical pendulum has an extended shape and its center of mass does not necessarily lie on the axis of rotation.

The period of a physical pendulum depends on the moment of inertia about the pivot point, the distance from the pivot to the center of mass (the radius of gyration), and the gravitational acceleration.

Parallel-Axis Theorem

The parallel-axis theorem states that the moment of inertia of an object about any axis is equal to its moment of inertia about a parallel axis through its center of mass plus the product of its mass and the square of the distance between these axes.

This theorem allows us to calculate the moment of inertia for complex shapes by relating it to simpler, known values.

Radius of Gyration

The radius of gyration (k) is a measure that relates the distribution of mass in an object to its rotational properties. It is defined as the distance from the axis of rotation at which all the mass could be concentrated so that the moment of inertia remains unchanged.

By understanding k, we can simplify complex problems involving moments of inertia.

Minimum Period Pivot

The minimum period pivot is a special point where the period of oscillation for the physical pendulum is at its shortest. This occurs when the distance from the pivot to the center of mass (d) equals the radius of gyration (k).

This concept is crucial in optimizing the design and performance of various mechanical systems, such as clocks or balance wheels.

Frequently asked questions

Try it live

Everything above runs in your browser — open Physical Pendulum Bench: Parallel-Axis Theorem, Radius of Gyration and Minimum-Period Pivot Lab and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.