Parallel axis theorem
, where is the moment of inertia about a parallel axis through the centre of mass, is the distance between the two axes, and is the total mass of the body.
Let the moment of intertia through the parallel axis (not through the center of mass) be , and the moment of inertia through the centre of mass be . We have
where is the total mass of the body, and is the distance between the two axes.
Proof
Express all positions relative to the centre of mass. Choose an origin on the parallel axis, and let be the position of particle relative to this origin. Then
where is the position of the centre of mass, and is the position of particle relative to the centre of mass, so
Then
Since , the middle term vanishes, and
noting that and .
The centre-of-mass axis is minimal
The theorem implies that is lower than about any parallel axis. In some cases, it may still be easiest to consider axes that do not pass through the centre of mass.
Disc about an edge
Consider a rotating disc of radius about an axis through its edge. Using for a disc about its centre,
Related
Stated in
- Theorem 5.4 (Parallel Axis Theorem)§5.4 Parallel Axis Theorem