The principal moments of inertia of a uniform plate are I 1 , I 2 >I 1 , and I 3

Question

The principal moments of inertia of a uniform plate are I1, I2>I1, and I3=I1+I2. Choose a coordinate system with the origin at the cm of the plate. The plate rotates with an angular velocity ? about an axis that makes an angle ? with the plane of the plate such that at time t=0, ?1(t=0)=?cos ?, ?2(t=0)=0, and ?3(t=0)=?sin ?. The rotation is force free, and I1/I2 = cos 2?.

a) Show that Euler's equations require that ?12+?22=?2cos2? at all times.

b) Show that ?2 and ?3 are related by the equation ?32 = ?2 sin2?

Explanation / Answer

Note that the integral required to obtain this result is given in appendix E of the text.

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