Mathematical Methods for Physics and Engineering - Matematica.NET

26.12 PHYSICAL APPLICATIONS OF TENSORSsection 26.7, since J and ω are vectors). The tensor is called the inertia tensor at Oof the assembly and depends only on the distribution of masses in the assemblyand not upon the direction or magnitude of ω.A more realistic situation obtains if a continuous rigid body is considered. Inthis case, m (α) must be replaced everywhere by ρ(r) dx dy dz and all summationsby integrations over the volume of the body. Written out in full in Cartesians,the inertia tensor for a continuous body would have the form⎛ ∫(y 2 + z 2 )ρdV − ∫ xyρ dV − ∫ ⎞xzρ dVI =[I ij ]= ⎝ − ∫ ∫xyρ dV (z 2 + x 2 )ρdV − ∫ yzρ dV− ∫ xzρ dV − ∫ ⎠ ,∫yzρ dV (x 2 + y 2 )ρdVwhere ρ = ρ(x, y, z) is the mass distribution and dV stands for dx dy dz; theintegrals are to be taken over the whole body. The diagonal elements of thistensor are called the moments of inertia and the off-diagonal elements without theminus signs are known as the products of inertia.◮Show that the kinetic energy of the rotating system is given by T = 1 2 I jlω j ω l .By an argument parallel to that already made for J, the kinetic energy is given by∑T = 1 m ( (α) ṙ (α) · ṙ (α))2= 1 2= 1 2= 1 2α∑αm (α) ɛ ijk ω j x (α)kɛ ilmω l x (α)m∑m (α) (δ jl δ km − δ jm δ kl )x (α)kx(α) m ω j ω lα∑ [ (m (α) δ ) ]jl r(α) 2− x(α)x (α) ω j ω lα= 1 I 2 jlω j ω l .Alternatively, since J j = I jl ω l we may write the kinetic energy of the rotating system asT = 1 J 2 jω j . ◭The above example shows that the kinetic energy of the rotating body can beexpressed as a scalar obtained by twice contracting ω with the inertia tensor. Italso shows that the moment of inertia of the body about a line given by the unitvector ˆn is I jlˆn j ˆn l (or ˆn T Iˆn in matrix form).Since I (≡ I jl ) is a real symmetric second-order tensor, it has associated with itthree mutually perpendicular directions that are its principal axes and have thefollowing properties (proved in chapter 8):(i) with each axis is associated a principal moment of inertia λ µ , µ =1, 2, 3;(ii) when the rotation of the body is about one of these axes, the angularvelocity and the angular momentum are parallel and given byJ = Iω = λ µ ω,i.e. ω is an eigenvector of I with eigenvalue λ µ ;951jl