A uniform cube of mass and side has

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Mechanics & Dynamical Systems · Rotation, rigid bodies and accelerated frames

Invariance, Ch General bases, the metric and tensors in spacetime

Problem

A uniform cube of mass

M = 1.5\,\mathrm{kg}

and side

L = 0.20\,\mathrm{m}

has, about its centre and referred to orthonormal axes

\mathbf{e}_1,\mathbf{e}_2,\mathbf{e}_3

along its edges, the inertia tensor

I_{ij} = \tfrac16 ML^2\,\delta_{ij}

, a result held from the mechanics course. A new, non-orthogonal basis is defined by

\mathbf{e}'_1 = \mathbf{e}_1

\mathbf{e}'_2 = \mathbf{e}_1+\mathbf{e}_2

\mathbf{e}'_3 = \mathbf{e}_3

, so that

\mathbf{e}'_i = A^j{}_i\mathbf{e}_j

with

A^1{}_1=1

A^2{}_1=0

A^1{}_2=1

A^2{}_2=1

A^3{}_3=1

, all other entries zero. ((a)) Using the covariant transformation law

I'_{ij} = A^k{}_iA^l{}_jI_{kl}

, find

I'_{11}

I'_{12}

and

I'_{22}

(the

33

component is unaffected). ((b)) Find the metric

g'_{ij} = \mathbf{e}'_i\cdot\mathbf{e}'_j

of the new basis, invert its

1

–

2

block, and use

g'^{ij}

to raise both indices,

I'^{ij} = g'^{ik}g'^{jl}I'_{kl}

, obtaining

I'^{11}

I'^{12}

and

I'^{22}

Browse the problem library