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Parallel axis theorem
Intermediate
Shift axis by d from the centre of mass.
I = I_cm + md²I about any axis = I_cm + md², where d is the distance between the axes (parallel). The centre-of-mass axis gives the minimum I.
Disc about a point on its rim: ½mR² + mR² = 1.5 mR².
One of the axes must pass through the centre of mass.
In more detail
The parallel axis theorem gives the moment of inertia about any axis parallel to one through the centre of mass: I = I_cm + Md², where d is the distance between the axes.
Real-world uses
- A rod pivoted at one end has I = ML²/3, from the centre-of-mass value plus Md².
- Designers compute moments of inertia about pivot points for swinging parts.
- A physicist finds the moment of inertia of a rod about its end by adding the mass times the square of the offset to the centre value.
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Frequently asked questions
When do I use it?
Whenever the axis of rotation does not pass through the centre of mass.
Does the theorem apply to any shape?
Yes, for any rigid body, provided you use the centre-of-mass moment.
When can I use the parallel axis theorem?
Whenever you know the moment of inertia about the centre of mass and need it about a parallel axis.
Related topics in Rotational Motion
Moment of inertia of a disc · Moment of inertia of a solid sphere · Moment of inertia of a hollow sphere · Perpendicular axis theorem · Radius of gyration · Newton's second law for rotation