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

2024
Views:

75

Pages:

2
Author:

Dixie Summers

COSMOS: Complete Online Solutions Manual Organization System Chapter 9, Solution 175. (a) Using Figure 9.28 and the parallel-axis theorem have at point A. I x′ = I y′ 1 m b2 + c2 12 ( ) 2 1 1 a m a2 + c2 + m   = m 4a 2 + c 2 = 12 12 2 ( I z′ = ) ( 2 1 1 a m a 2 + b2 + m   = m 4a 2 + b 2 12 12 2 ( ) ( ) ) Now observe that symmetry implies I x′y′ = I y′z′ = I z′x′ = 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 or 1 1 m b2 + c 2 x2 + m 4a + c 12 12 ( ) ( )y + 1 m 4a + b 12 ( )z =1 For the ellipsoid to be a sphere, the coefficients must be equal. Therefore, 1 1 1 m b2 + c2 = m 4a 2 + c 2 = m 4a 2 + b 2 12 12 12 ( ) ( ) ( b 2 + c 2 = 4a 2 + c 2 Then b 2 + c 2 = 4a 2 + b 2 and ) or b = 2 a or c = 2 a (b) Using Figure 9.28 and the parallel-axis theorem, we have at point B I x′′ = I y′′ 2 1 1 c m b2 + c 2 + m   = m b 2 + 4c 2 12 2 12   ( ) 2 ( 1 1 c m a2 + c2 + m   = m a 2 + 4c 2 = 12 12 2 I z′′ = ( 1 m a 2 + b2 12 ( ) ( ) ) ) continued COSMOS: Complete Online Solutions Manual Organization System Now observe that symmetry implies I x′′y′′ = I y′′z′′ = I z′′x′′ = 0 From part a it then immediately follows that 1 1 1 m b 2 + 4c 2 = m a 2 + 4c 2 = m a2 + b2 12 12 12 ( ) ( ) ( ) Then b 2 + 4c 2 = a 2 + 4c 2 or b = 1 a and b 2 + 4c 2 = a 2 + b 2 or c 1 = a 2