A rigid dumbbell consists of two 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

A rigid dumbbell consists of two point masses, each of mass m=2 kgm = 2\,\mathrm{kg}, joined by a light rigid rod along the xx axis. One mass sits at (0.5,0,0) m(0.5, 0, 0)\,\mathrm m and the other at (−0.5,0,0) m(-0.5, 0, 0)\,\mathrm m, both measured from the midpoint of the rod. The inertia tensor of a set of point masses about the origin is Iij=∑ama(ra2δij−xi(a)xj(a))I_{ij} = \sum_a m_a(r_a^2\delta_{ij} - x_i^{(a)}x_j^{(a)}), with ra2=xk(a)xk(a)r_a^2 = x_k^{(a)}x_k^{(a)}. ((a)) Compute IijI_{ij} for this dumbbell about its midpoint. ((b)) State the principal moments of inertia and the axes to which they belong, without solving a characteristic equation.

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