Question

Consider the motion of a rigid body with principal moments of inertia I < I<I,, in absence of external forces and torques (i.c., a free rigid body). Assume the body is a rectangular figure of width W, height H and length L (i.e., a book), with H<W<L, as shown in the figure. The angular velocity vector of the rigid body, in the body system, is (, . The conserved energy of the top is E, and the conserved angular momentum vector is L, which has magnitude L and, in the body system, has components L L. L2, Ls) (Io, 102, 13o) x3 Prove that for a given energy E, the value for the angular momentum has minimum and maximum values 2E1/ <し2 < 2E13 . (Hint: write expressions for 2E, , 2E, and L, in terms of oi, 2, os.)

Answer 1

Given, L*_1 = 2 (1/2 I^2_1 w^2_1 + 1/2 I^2_2 w^2_2 + 1/2 I^2_3 w^2_3) 2E I_1 = (1/2 I_1 w^2_1 + 1/2 w^2_2 + 1/2 I_3 w^2_3) 2I_1 2E I_3 = (1/2 I_1 w^2_1 + 1/2 I_2 w^2_2 + 1/2 I_3 w^2_3) 2 I_3 from (1), L^2 = (1/2 I_1 w^2_1 middot I_1 + 1/2 I_2 w^2_2 middot I_2 + 1/2 I_3 w^2_3 middot I_3) 2 AS, E is constant, so contribution to energy E from each rational direction must be constant so, 1/2 I w^2_1, 1/2 I_2 w^2_2, 1/2 I_3 w^2_3 will have fixed value. So, for max L^2_1 = 2 (1/2 I_1 w^2_1 middot I_3 + 1/2 I_2 w^2_2 middot I_3 + 1/2 I_3 w^2_3 middot I_3) 2 middot E middot I_3 \r\n and min L^2 = 2 (1/2 I_1 w^2_1 middot I_1 + 1/2 I_2 w^2_2 I_1 + 1/2 I_3 w^2_3 middot I_1) 2 middot E I_1 Thus, 2E I_1 < L^2_1 < 2E I_3