The Parallel Axis Theorem

$\boxed{I = I_{cm} + Md^2}$

• Validity: Any rigid body (2D or 3D)
• Condition: The two axes must be parallel
• Use case: Shifting the axis from the CM to any parallel axis
• Key insight: $I_{cm}$ is always the MINIMUM for parallel axes
• Never confuse $d$ with the radius — $d$ is the distance between the two axes

Examples:

• Rod: $I_{end} = ML^2/12 + M(L/2)^2 = ML^2/3$ ✓
• Disc tangent (in-plane): $I = MR^2/4 + MR^2 = 5MR^2/4$ ✓
• Sphere tangent: $I = 2MR^2/5 + MR^2 = 7MR^2/5$ ✓

The Perpendicular Axis Theorem

$\boxed{I_z = I_x + I_y}$

• Validity: ONLY flat (planar/2D) bodies — discs, rings, laminas
• Condition: x, y, z are mutually perpendicular; z is perpendicular to the plane of the body
• Use case: Relating in-plane moments to the out-of-plane moment
• NEVER use for: Spheres, cylinders, cones, or any 3D body

Examples:

• Disc diameter: $I_x = I_y = I_d$; $I_z = MR^2/2$; so $I_d = MR^2/4$ ✓
• Ring diameter: $I_z = MR^2$; so $I_{diameter} = MR^2/2$ ✓

NEET Trap

A very common trap is applying the perpendicular axis theorem to a solid sphere or cylinder. Always ask: "Is this body flat/planar?" If not, do NOT use the perpendicular axis theorem.