2.II.17B

Methods | Part IB, 2004

Let Iij(P)I_{i j}(P) be the moment-of-inertia tensor of a rigid body relative to the point PP. If GG is the centre of mass of the body and the vector GPG P has components XiX_{i}, show that

Iij(P)=Iij(G)+M(XkXkδijXiXj),I_{i j}(P)=I_{i j}(G)+M\left(X_{k} X_{k} \delta_{i j}-X_{i} X_{j}\right),

where MM is the mass of the body.

Consider a cube of uniform density and side 2a2 a, with centre at the origin. Find the inertia tensor about the centre of mass, and thence about the corner P=(a,a,a)P=(a, a, a).

Find the eigenvectors and eigenvalues of Iij(P)I_{i j}(P).

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