What is the Parallel Axis Theorem?

C.B. Fox

The parallel axis theorem is used in physics to determine an object’s moment of inertia as it rotates about any axis. The theorem states that there is a relationship between the inertia of an object rotating around its center of gravity and an axis parallel to this center. This theorem applies to any solid object in rotation, including irregular shapes.

Scientist with beakers
Scientist with beakers

An object’s resistance to change in rotational speed or direction in terms of its inertia is measured by the parallel axis theorem. Inertia is the resistance a physical object shows to a change in its state of motion. When an object is moving in a linear direction, this resistance is represented by an object’s mass. In rotational dynamics, when describing angular momentum, angular velocity, torque and angular acceleration, this resistance is called moment of inertia.

With regard to regular objects, such as spheres, rods and cylinders, moment of inertia can be solved using simple formulas, specific to the shape of these objects. For irregular shapes, the moment of inertia can be solved using calculus, which allows for the use of continuous variables. In an irregular shape, the rotation of the object around an axis involves a continuous distribution of mass. In an object that is not symmetrical, the mass will not be evenly distributed as it rotates, which means that solving for its moment of inertia will require using multiple variables. The moment of inertia is one variable in the parallel axis theorem equation.

The lowest amount of force needed to change an object’s speed or direction about its center of mass is its moment of inertia. Center of mass, also known as center of gravity, is the point in an object wherein the mass is balanced evenly on all sides. For instance, a see-saw will have a center of mass in the center of the board, which can be demonstrated by balancing the board on a pivot point placed in the center. If an adult and a small child are placed at opposite ends of the see saw, the center of mass will shift toward the adult, until the total mass is even on both sides.

In the parallel axis theorem, the moment of inertia for any axis parallel to the axis in the center of mass can be given with a single formula. The inertia of the parallel axis is equal to the inertia of the center of mass plus the point mass of the object multiplied by the square of the distance between the center of mass and the parallel axis. This formula holds true for any rigid body rotating about an axis.

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