Parallel Axis Theorem

Parallel-axis theorem: I about any parallel axis = I_CM + M·d², where I_CM is the moment of inertia about the parallel axis through the center of mass, and d is the distance between the two axes.

Lets you find I about any axis if you know I_CM.

I about COM is always MINIMUM (parallel-axis adds a positive term).

Perpendicular-axis theorem (for PLANAR bodies only): I_z = I_x + I_y, where x, y are in the plane and z is perpendicular.

Used together: parallel + perpendicular axis theorems let you compute I for many standard shapes.

Example: rod about end: I_end = I_CM + M·(L/2)² = ML²/12 + ML²/4 = ML²/3.

Example: disk about edge: I_edge = ½MR² + MR² = (3/2)MR².

Theorem is universal — applies to any rigid body.