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Chapter 10: Q13P (page 517)
are both axial vectors.
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Show by the quotient rule (Section 3 ) that in is a -rank tensor.
Point masses 1 at (1, 1, -2) and 2 at (1, 1, 1).
Find the inertia tensor about the origin for a mass of uniform density =1, inside the part of the unit sphere where and find the principal moments of inertia and the principal axes. Note that this is similar to Example 5 but the mass is both above and below the plane. Warning hint: This time don’t make the assumptions about symmetry that we did in Example 5.
If P and S are -rank tensors, show that coefficients are needed to write each component of P as a linear combination of the components of S. Show that is the number of components in a -rank tensor. If the components of the -rank tensor are , then equation gives the components of P in terms of the components of S. If P and S are both symmetric, show that we need only 36different non-zero components in . Hint: Consider the number of different components in P and S when they are symmetric. Comment: The stress and strain tensors can both be shown to be symmetric. Further symmetry reduces the 36components of C in (7.5)to 21or less.
Show that the fourth expression in (3.1) is equal to . By equations (2.6) and (2.10) , show that , so
Compare this with equation (2.12) to show thatis a Cartesian vector. Hint: Watch the summation indices carefully and if it helps, put back the summation signs or write sums out in detail as in (3.1) until you get used to summation convention.
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