# Work done by internal forces of a rigid body

I am reading Goldstein's Classical Mechanics book, and I came across that:

In a rigid body the internal forces do no work

Is this statement based on the assumption that the internal forces are central? Or, is it true even when the internal forces not central?

Goldstein defines a rigid body as a system of particles in which the distance between the particles remain constant, and does not require that the internal forces to be central.

• If the distance between the particles remains constant, how could there be any internal work? – garyp Feb 23 at 14:32
• On pg 56, in Appendix A of "Physically Based Modeling - Rigid Body Simulation" by David Baraff of Pixar: [pdfs.semanticscholar.org/902c/… he gives a derivation which he describes as follows: "The derivation in this appendix is (we feel) much shorter and considerably more elegant than the one found in traditional sources such as Goldstein." I am finding these notes by Baraff very useful. – Simon Feb 23 at 14:45

Recall the difference between the weak and strong Newton's third law, cf. e.g. this Phys.SE post.

1. If the internal forces satisfy the weak Newton's third law (but not the strong Newton's third law, i.e. without the collinarity assumption), then it is not guaranteed that the internal forces do no work, cf. e.g. Fig. 1.

^ F
|
|                  2
x------------------x
1                  |
|
v F


$\uparrow$ Fig. 1: A rigid body consisting of 2 point-particles with a pair of non-collinear internal forces $F$. Note that there in principle could be other internal forces holding the 2 point-particles together.

2. If the internal forces satisfy the strong Newton's third law, then the internal forces do no work, cf. e.g. this Phys.SE post.

• Then, the definition of a rigid body as used in Goldstein's assumes that the strong Newton's third law holds for internal forces. – user74261 Jul 22 '16 at 18:49
• $\uparrow$ Yes, apparently. – Qmechanic Jul 22 '16 at 18:51

In a rigid body, according to Goldstein's definition, the distance between any two constituent particles does not change. Work done is force times distance moved in the direction of the force. There is no relative movement in the direction of any force. Therefore, regardless of the form of the internal forces, no work is done by or against them.