COSMOS Chapter 9 Solution 177 - Edubirdie

University:
Michigan State University
Course:
ERG 160 | Success in Science, Technology, Engineering, and Mathematics
Academic year:

2019
Views:

245

Pages:

1
Author:

Caroline Mcfarland

COSMOS: Complete Online Solutions Manual Organization System Chapter 9, Solution 177. (a) From Figure 9.28 Ix = 1 2 ma 2 I y = Iz = 1 m 3a 2 + L2 12 ( ) Now observe that symmetry implies I xy = I yz = I zx = 0 Using Equation (9.48), the equation of the ellipsoid of inertia is then I x x 2 + I y y 2 + I z z 2 = 1: 1 2 2 1 1 ma x + m 3a 2 + L2 y 2 + m 3a 2 + L2 = 1 2 12 12 ( ) ( ) For the ellipsoid to be a sphere, the coefficients must be equal. Therefore, 1 2 1 ma = m 3a 2 + L2 2 12 ( ) or a L = 1 3 (b) Using Fig. 9.28 and the parallel-axis theorem Have I x′ = 1 2 ma 2 I y′ = I z′ = 2 1 7 2 L 1 m 3a 2 + L2 + m   = m  a 2 + L  12 4 4 48     ( ) Now observe that symmetry implies I x′y′ = I y′z′ = I z′x′ = 0 From Part a it then immediately follows that 1 2 7 2 1 ma = m  a 2 + L  2 4 48   or a L = 7 12

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