# Why can any general motion of a rigid body be represented as translation + rotation about center of mass?

1. Why can any general motion of a rigid body be represented as translation + rotation about center of mass?

2. I am beginning to read rotational dynamics and my textbook states this fact without proof. I am wondering - Is this fact only true for a center of mass?

3. Then - The phrase "rotation about center of mass" strikes as vague to me. Rotation about which axis?

• You only need to prove that the rotation about CM, first about a certain axis with a certain angle, followed by another axis with another angle, is equivalent to a single rotation about a certain axis with a certain angle. – velut luna Aug 1 '16 at 21:21
• This fact is true not only about CM. The axis is to be determined by the actual configuration of the object. You can find the proof in e.g., Goldstein: Chasle's theorem --- the most general displacement of a rigid body is a translation plus a rotation. – velut luna Aug 1 '16 at 21:24

## 2 Answers

If you want to describe the position of a rigid object in space, it is clearly not sufficient to give just the position of the center of mass - you also need to specify the orientation.

That orientation can be reached with a rotation about the center of mass - but you need to figure out what the direction of the rotation axis has to be, and what the angle is through which you rotate.

You need pick any three points in the object (not on the same line) in order to describe how it was rotated; that is the necessary and sufficient number of parameters. You can then write three equations in three unknowns, and solve for three parameters - exactly the number of parameters needed to describe an axis of rotation (2 parameters) and angle of rotation (third parameter).

You could pick a rotation about a different point than the center of mass - and while it is possible to describe motion with any 3+3 parameters, it is MUCH harder when the axis is not going through the center of mass (because, for example, there will be an apparent centrifugal force in the rotating frame of reference because the center of mass is not on the axis of rotation).

• @KyleOman thanks - my bad. I meant not on the same line. Fixed now. – Floris Aug 17 '16 at 4:20

The fact that the motion of a rigid body can be represented as a translation and a rotation about the center of mass is a consequence of a mathematical theorem that states that every function that goes from R^3 to R^3 such that, for all x, y, d(x,y) = d(f(x),f(y)) (where d(x,y) means distance between x and y) can be expressed uniquely as the composition of a translation and a rotation about a certain axis. You can find the proof of this theorem in Peter Lax: Linear Algebra, in the chapter of kinematicas and dynamics (some prior knowledge of linear algebra is required). After studying the proof, you will realize that in fact, for every point in space, there exists an axis passing through that point such that the motion of the rigid body can be expressed as a translation (that depends on the the point you have chosen) and a rotation around its axis.