Lecture 35: The Inertia Tensor We found last time that the kinetic energy of a rotating obect was: 1 Trot = ωω i Ii where i, ( I m δ x x x i i, k, i, k So the nine numbers represented by the I i tell us all we need to know about the rigid body to determine T for a given ω
But in earlier courses, you probably got the impression that the moment of inertia for an obect was described by three numbers: Ix, Iy, Iz Note, though, that no one tried to write this as a vector Now it seems we need nine numbers to express the same quantity For convenience, can write them as a matrix: I I I I I I I 11 1 13 = 1 3 I I I 31 3 33 The Inertia Tensor So, who s been lying to you? No one, really, though you weren t told the whole story before. More on that later
One step towards reconciling the nine vs. three numbers dilemma is that the nine is really six ( I = m δ x x x i i, k, i, k ( = m δ x x x = I So the only unique numbers are i, k,, i i k I11, I1, I13, I, I3, I33 If the rigid body is a continuous distribution of matter rather than a collection of discrete points, the I i are found by: ( ( r δ = I ρ x x x dv i i k i V k
Angular Momentum The general definition of angular momentum is: = L r p The value depends on our choice of origin. One can choose any point, but two choices make things easy: 1. If a point in the body is fixed (in some inertial reference frame, choose that point as the origin. If there is no fixed point, use the center of mass as the origin In other words, try to choose an origin about which the body is rotating! For such an origin, we know that v =ω r, so that: ( L= r m v = m r ω r
Here again we find that a vector identity is useful: A ( B A = A B A( A B Using this, ( L = m rω r r ω or, in terms of Cartesian components: L = m ω x x x ( ω i i, k, i, k = m ωδ x x x ( ω i, k, i, k = ω m δ x x x = ω I i, k, i, k i The inertia tensor appears again!
In tensor notation, we write this as: L= I ω This implies something quite interesting: Despite all you ve heard about L = Iω, the L and ω vectors are not parallel in general Example: dumbbell rotating about non-symmetric axis: m L z ω L TOT m L 1 Take CM as the origin The ω vector is directed upwards Le ( ( L= L + L = r p + r p 1 1 z Note also that L is not constant direction changes as dumbbell rotates. External torque must be applied to cause this motion.
Inertia Tensor in Different Coordinate Systems So far I ve called the numbers I i a tensor, but have provided no definition of what a tensor is To provide one, consider what happens to the angular momentum if one rotates the coordinate system Since L is a vector, its components must transform according to: L = λ L By the same token: And we also know that: k m ω = λ ω k k mk k kl l l m L = I ω Rotation matrix (defined in first week of class
So we must have: λ L = I λ ω mk m kl k m l λ λ L = λ I λ ω ik mk m ik kl k k m k l ( I λλ L = λλ ω ik mk m ik k kl mk, kl, We learned before that the rotation matrix is orthogonal, so that: λλ = δ ik mk im Using this, the above equation becomes: k = ( I ( λλ I ω δ L λ λ ω im m ik k kl mk, kl, L = i ik k kl k, l
But we also know that L = I ω i i The only way it can all work out is for I and I to have the following relationship: I = λ λ I i ik k kl kl, Any collection of numbers that follow this rule for rotations is called a tensor The number of indices determines the rank of the tensor Moment of inertia is a second-rank tensor In general, a tensor of arbitrary rank will transform according to: T = λ λ λ λ T abcd... ai b ck dl ikl... i,, k, l...
This means that a first-rank tensor transforms as: T = λ T In other words, it s a vector! a ai i i For second-rank tensors, the transformation is the same as matrix mulitplication: t I = λi λ Another property of orthogonal matrics is that their transpose is equal to their inverse. Using this, we can write: 1 I = λi λ Any transformation of this type is called a similarity transformation
More On Angular Momentum Returning to the definition of L i, L = I ω i i 1 we can multiply this by ω and sum over i: i 1 1 ω L = I ωω = T i i i i i i, In other words: 1 1 T rot = ω L = ω I ω So we see that a tensor times a vector gives a vector, while a tensor times two vectors gives a scalar rot