HW10_prob3_three mass points

HW10_prob3_three mass points - Problem Three mass points...

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Problem Three mass points The three mass points are located at the vertices of the triangle shown in the fi gure. The x , y and z variables are normalized by the factor a , so that X = x/a , Y = y/a , and Z = z/a . Z X Y We are supposed to fi nd the principal moments of inertia and a set of prin- cipal axes about the origin of the given body fi xed frame. We start by fi nding the elements of the inertia tensor in the given coordinate system. The general expression for the αβ component of the inertia tensor is I αβ = X i m i ( r 2 i δ αβ x x ) , (1) where the sum runs over all of the mass points in the body. For the body in question, there are three points with equal mass m , so we can write I αβ = m 3 X i =1 ( r 2 i δ αβ x x ) , (2) and the points can be numbered any way you please. It is straightforward to evaluate the di ff erent I αβ : I xx = m 3 X i =1 ( y 2 i + z 2 i ) = 11 ma 2 . (3) I yy = m 3 X i =1 ( x 2 i + z 2 i ) = 11 ma 2 , (4) I zz = m 3 X i =1 ( x 2 i + y 2 i ) = 20 ma 2 , (5) I xy = I yx = m 3 X i =1 ( x i y i ) = 4 ma 2 . (6) 1
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I xz = I zx = m 3 X i =1 ( x i z i ) = 0 , (7) I yz = I zy = m 3 X i =1 ( y i z i ) = 0 . (8) Thus, for the coordinate system in the fi
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