Parallel Axis Theorem
I = I_cm + Md² — shift the axis and see I grow quadratically with d.
Key Notes
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.
Formulas
Parallel-axis theorem
d = perpendicular distance between the two axes.
Perpendicular-axis (planar)
Only for FLAT 2D bodies; z perpendicular to plane.
Rod about end
Useful in compound pendulum.
Disk about edge
Standard result.
Important Points
I about ANY axis ≥ I about parallel axis through COM. Minimum I is always at the COM.
Parallel-axis adds M·d² — grows quadratically with axis separation.
Perpendicular-axis theorem is for 2D (planar) bodies only — wouldn't apply to a 3D solid like a sphere.
Combining both theorems lets you find I for many compound configurations.
Always check the I_CM value: for a thin rod about its center I = ML²/12, but about its end (using parallel-axis) it becomes ML²/3.
For a thin lamina, perpendicular-axis lets you derive I about axis perpendicular to the plane from I about two perpendicular axes in the plane.
Parallel Axis Theorem notes from sciphylab (also known as SciPhy, SciPhy Lab, SciPhy Labs, Physics Lab). Class 11 physics revision for JEE Mains, JEE Advanced, NEET UG, AP Physics 1/2/C, SAT, and CUET-UG.