A thin uniform rectangular plate of mass has sides along and along

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

Invariance, Ch Index notation and Cartesian tensors

Problem

A thin uniform rectangular plate of mass

M

has sides

a

along

x

and

2a

along

y

. It lies in the

z=0

plane with one corner at the origin, so that it covers

0\le x\le a

and

0\le y\le 2a

, and it turns about that corner. ((a)) Compute the inertia tensor about the corner from

I_{ij}=\int\left(r^2\delta_{ij}-x_ix_j\right)\mathrm dm

. ((b)) Solve the characteristic equation for the three principal moments and find the principal axes. Give the angle that the in-plane axis of smallest moment makes with the

x

axis, and say whether that axis lies along the diagonal of the plate through the corner. ((c)) The plate is spun about its long edge, the

y

axis. Find the angle between

\mathbf L

and

\boldsymbol\omega

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