Perpendicular-Axis Theorem

The perpendicular-axis theorem states that the moment of inertia of a planar object about an axis perpendicular to its plane equals the sum of the moments of inertia about two mutually perpendicular axes lying in the plane of the body. All three axes must be concurrent, meaning they intersect at the same point, typically the object's center.

Circular symmetry ensures that the moments of inertia about two perpendicular planar axes are equal. For a circular disc or hoop, this symmetry means the moment of inertia along the x-axis equals the moment of inertia along the y-axis, simplifying calculations using the perpendicular-axis theorem.

For a circular hoop, the moment of inertia about the z-axis (perpendicular to the plane) equals the mass multiplied by the radius squared. Due to circular symmetry, the moment of inertia along each planar axis equals half the moment of inertia along the z-axis, derived directly from the perpendicular-axis theorem.

For a circular disc, the moment of inertia along the z-axis (perpendicular to the disc) equals the sum of the moments of inertia along the x-axis and y-axis. Since circular symmetry makes these planar moments equal, the z-axis moment is twice the moment along either planar axis.

The three axes must be concurrent—intersecting at a single point—because the theorem relates moments of inertia calculated about the same reference point. Typically, this point is the object's center of mass, ensuring consistent and comparable measurements across all three perpendicular axes.

For a circular hoop of mass M and radius R, the moment of inertia about a planar axis equals half the product of its mass and radius squared, or (1/2)MR². This result comes from applying the perpendicular-axis theorem and using the hoop's circular symmetry.

The perpendicular-axis theorem simplifies calculations by allowing you to find the moment of inertia about a perpendicular axis if you know the moments about two planar axes, or vice versa. For symmetric objects like discs and hoops, circular symmetry further reduces the number of independent calculations needed.