(Products of inertia) A rigid body with density is placed wi
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(Products of inertia) A rigid body with density is placed with its centre of mass at the origin and occupies a region R
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(Products of inertia) A rigid body with density is placed with its centre of mass at the origin and occupies a region R of 3-space. Suppose the six second moments Pxx, Pyy, Pzz, Pxy, Pxz, and Pyz are all known, where Pxx D ZZZ R x2 dV; Pxy D ZZZ R xy dV; : (There exist tables giving these six moments for bodies of many standard shapes. They are called products of inertia.) Show how to express the moment of inertia of the body about any axis through the origin in terms of these six second moments. (If this result is combined with that of Exercise 37, the moment of inertia about any axis can be found.)
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