Appendix B

A Rigid Body Rotating About aPoint

Imagine a rigid body which is rotating about a point fixed in space. For example, you couldconsider a bicycle wheel rotating about its axis which is held still. More generally, we letthe point about which the body rotates move also. In this case, the point is usually thecenter of mass of the body. However, in this section, this point will be regarded as fixed.Let B(t) denote the set of points in three dimensional space which the body occupies attime t. We will refer to the points in three dimensional space occupied by the body at timet = 0 as the material points of the body.

Recall Theorem 24.3.2 about the existence of the angular velocity vector. The idea isthat you have a material point x0 in the body and some right handed orthonormal system ofbasis vectors {e1 (t) ,e2 (t) ,e3 (t)} which moves with the body such if x(t,x) is the vectorfrom x0 to the point where x is at time t, then x(t,x) = ae1 (t)+ be2 (t)+ ce3 (t) wherea,b,c are constants. Note that here it is assumed that x0 does not change. Thus it is notmoving through space. Then this theorem is summarized in the following lemma.

Lemma B.0.1 For a body which undergoes rigid body motion about a fixed point in threedimensional space, if x(t,x) denotes the position vector of the point x at time t, from somefixed point in the body, then there exists a time dependent vector ω (t) such that the velocityof this point at time t, xt (t,x) is given by

xt (t,x) = ω (t)×x(t,x) .

In particular, letting x= ei, we see that e′i (t) = ω (t)×ei (t) .

Definition B.0.2 The vector, ω (t) whose existence is given by the above lemma is calledthe angular velocity vector.

We are now ready to write the total angular momentum of the rigid body. In doing so,we assume the density equals ρ (x) . Thus at time t the total angular momentum, Ω, wouldbe given by the three dimensional integral,

Ω =∫

B(0)x(t,x)×ρ (x)xt (t,x)dx

=∫

B(0)ρ (x)x(t,x)× (ω (t)×x(t,x))dx. (2.1)

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