A uniform rectangular block of mass kg has edge lengths m along

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

Action and Symmetry, Ch The inertia tensor, principal axes and Euler angles

Problem

A uniform rectangular block of mass M=2.4M = 2.4 kg has edge lengths a=0.30a = 0.30 m along xx, b=0.20b = 0.20 m along yy and c=0.10c = 0.10 m along zz, with its centre of mass at the origin and its edges parallel to the axes. ((a)) Write down the inertia tensor about the centre of mass, using symmetry to say which entries vanish. ((b)) Use the tensor parallel-axis theorem to find the inertia tensor about the point on the zz axis a distance c/2c/2 from the centre (the mid-point of one face). ((c)) State the principal moments and principal axes at that point, and explain from the form of the shift vector d\mathbf{d} why no further diagonalisation is needed. ((d)) The block turns with Euler-angle rates ϕ˙=1.5\dot\phi = 1.5 rad/s, θ˙=0.4\dot\theta = 0.4 rad/s, ψ˙=8.0\dot\psi = 8.0 rad/s (3-1-3 convention) at θ=40.0∘\theta = 40.0^\circ, ψ=25.0∘\psi = 25.0^\circ, with body axes 1, 2, 3 taken along x,y,zx,y,z. Find ω1\omega_1, ω2\omega_2, ω3\omega_3.

Browse the problem library