A uniform thin lamina has the shape of an equilateral triangle
PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Mechanics & Dynamical Systems · Rotation, rigid bodies and accelerated frames
Action and Symmetry, Ch The inertia tensor, principal axes and Euler angles
Problem
A uniform thin lamina has the shape of an equilateral triangle. It lies in a plane, with its centroid at the origin. It has a threefold rotation symmetry (a rotation by 120∘ about the axis perpendicular to the lamina through the centroid leaves it unchanged) and three mirror planes, each containing that perpendicular axis and one of the triangle's medians. Take axis 3 along the perpendicular through the centroid, and axes 1, 2 as any orthogonal pair lying in the plane of the lamina through the centroid.
((a)) Explain, from the symmetry alone, why axis 3 is a principal axis, and why axes 1 and 2 may be chosen with I1=I2 even though no single mirror plane forces this.
((b)) The lamina's mass is M=0.24kg and its moment about axis 3 is I3=3.0×10−4kgm2. Using the perpendicular-axis relation for a lamina, I3=I1+I2, find I1 and I2, and classify the lamina as a spherical, symmetric or asymmetric top.
((c)) Sketch the lamina's inertia ellipsoid: give its semi-axes along axes 1, 2 and 3, and name the axis along which it is shortest.