Parallel Axis Theorem

Parallel Axis Theorem:

In physics, the parallel axis theorem or Huygens-Steiner theorem can be used to determine the moment of inertia of a rigid body about any axis, given the moment of inertia of the object about the parallel axis through the object’s center of mass and the perpendicular distance between the axes.

The moment of inertia about the new axis z is given by:

I_z = I_cm + md²

Where:

I_cm is the moment of inertia of the object about its center of mass;

m is the object’s mass;

d is the perpendicular distance between the two axes.

This rule can be applied with the stretch rule and perpendicular axis theorem to find moments of inertia for a variety of shapes.

The parallel axes rule also applies to the second moment of area (area moment of inertia) for a plane region D:

I_z = I_x + Ad²,

Where:

I_z is the area moment of inertia of D relative to the parallel axis;

I_x is the are amoment of inertia of D relative to its centroid;

A is the area of the plane region D;

d is the distance from the new axis z to the centroid of the plane region D.

Note: The centroid of D coincides with the center of gravity (CG) of a physical plate with the same shape that has constant density.

In classical mechanics

In classical mechanics, the Parallel axis theorem (also known as Huygens-Steiner theorem) can be generalized to calculate a new inertia tensor J_ij from an inertia tensor about a center of mass I_ij when the pivot point is a displacement a from the center of mass:

J_ij = Ii_j + m (a²δ_ij – a_ia_j)

where

a = a₁x + a₂y + a₃z (where x, y, and z are unit vectors)

is the displacement vector from the center of mass to the new axis, and

δ_ij

is the Kronecker delta.

We can see that, for diagonal elements (when i=j), displacements perpendicular to the axis of rotation results in the above simplified version of the parallel axis theorem.