A hardback volume of mass is modelled

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 hardback volume of mass M=1.0 kgM = 1.0\,\mathrm{kg} is modelled, for a rough estimate, as a uniform rectangular block of length a=0.25 ma = 0.25\,\mathrm{m}, width b=0.20 mb = 0.20\,\mathrm{m} and thickness c=0.030 mc = 0.030\,\mathrm{m}, with principal axes 1,2,31,2,3 along a,b,ca,b,c through the centre of mass. Integrating the definition of the inertia tensor over the block gives I1=M(b2+c2)/12I_1 = M(b^2+c^2)/12, I2=M(a2+c2)/12I_2 = M(a^2+c^2)/12 and I3=M(a2+b2)/12I_3 = M(a^2+b^2)/12. The volume is spun about an axis ω\boldsymbol\omega that lies in the 11-33 plane, tilted by 10∘10^\circ from the long axis (axis 11, the axis of smallest moment of inertia) towards the thickness axis (axis 33). ((a)) Estimate the two relevant principal moments of inertia I1I_1 and I3I_3. ((b)) Estimate the angle between the angular momentum L\mathbf L and ω\boldsymbol\omega.

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