Parallel Axis Theorem
Calculate the moment of inertia about any axis using the parallel axis theorem.
Adds Md² to the center-of-mass moment of inertia.
The Formula
The parallel axis theorem (also called Steiner's theorem) allows you to find the moment of inertia about any axis parallel to an axis through the center of mass. You only need to know the center-of-mass moment of inertia (I_cm) and the perpendicular distance between the two axes (d).
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| I | Moment of inertia about the new (shifted) axis | kg·m² |
| I_cm | Moment of inertia about the axis through the center of mass | kg·m² |
| M | Total mass of the object | kg |
| d | Perpendicular distance between the two parallel axes | meters (m) |
Common center-of-mass moments of inertia:
- Solid disk/cylinder: I_cm = ½MR²
- Solid sphere: I_cm = (2/5)MR²
- Hollow sphere (thin shell): I_cm = (2/3)MR²
- Thin rod about center: I_cm = (1/12)ML²
- Rectangular plate about center: I_cm = (1/12)M(a² + b²)
Example 1 — Disk Rotating About Its Edge
A solid disk of mass 5 kg and radius 0.3 m rotates about an axis at its rim. Find the moment of inertia.
I_cm = ½MR² = ½ × 5 × 0.3² = ½ × 5 × 0.09 = 0.225 kg·m²
d = R = 0.3 m (the rim axis is R away from the center axis)
I = I_cm + Md² = 0.225 + 5 × 0.3² = 0.225 + 0.45
I = 0.675 kg·m² — exactly 1.5× the center-of-mass value (= (3/2)MR²)
Example 2 — Rod Rotating About One End
A thin rod of mass 2 kg and length 1.2 m rotates about one end. Find I.
I_cm = (1/12)ML² = (1/12) × 2 × 1.44 = 0.24 kg·m²
d = L/2 = 0.6 m (center of mass is at the midpoint)
I = I_cm + Md² = 0.24 + 2 × 0.36 = 0.24 + 0.72
I = 0.96 kg·m² — this matches the known formula (1/3)ML² = (1/3)×2×1.44 = 0.96 ✓
When to Use It
Use the parallel axis theorem when:
- The axis of rotation does not pass through the object's center of mass
- Computing the rotational inertia of a compound object made of multiple simpler shapes
- Analyzing pendulums swinging about a pivot point at one end
- Engineering rotating machinery — flywheels, crankshafts, wheels
- Sports science — calculating swing weight of bats, rackets, and clubs
The theorem only applies when the two axes are parallel. For non-parallel axes, the full inertia tensor must be used.