Three point masses

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

Three point masses, each of mass

m

, sit at the vertices of an equilateral triangle of side

s

lying in the

xy

-plane, with the origin at the centroid (so each vertex is a distance

R=s/\sqrt3

from the origin). ((a)) Write the inertia tensor about the origin using

I_{ij}=\sum_a m\,(r_a^2\delta_{ij}-x_i^{(a)}x_j^{(a)})

, and use the three-fold symmetry of the arrangement to argue that

I_{xx}=I_{yy}

and that every off-diagonal component vanishes. ((b)) Evaluate

I_{xx}

I_{yy}

and

I_{zz}

in terms of

m

and

s

, and check the result against

I_{ii}=2\sum_a m r_a^2

. ((c)) Compare the principal moment for rotation about the axis perpendicular to the plane with those for rotation about axes lying in the plane.

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