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HW12 prob3 rod pivoting

HW12 prob3 rod pivoting - Problem Kinetic Energy of a Rod(a...

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Problem: KineticEnergyo faRod (a) First we need to f nd the principal moments of the rod. This can be done with the origin either at the end of the rod (at the pivot) or at the center of the rod (the COM). There is a conceptual advantage with the latter approach, so we’ll take that route. Start with the general expression for the αβ component of the inertia tensor ( α , β = x , y , z ) I αβ = Z ρ ( r )( r 2 δ αβ x α x β ) d 3 r. (1) Note that r is measured from the origin of the body frame coordinate system, r 2 = x 2 + y 2 + z 2 ,and x x = x , x y = y , etc. For this rod, the linear mass density λ is constant, and its mass M is M =2 lλ, (2) where 2 l is the length of the rod. Given the geometry of the object, it is straightforward to use cartesian coordinates to do the required integrations. Let the rod lie along the z axis, so that x = y =0 along the length of the rod. The density can then be written as ρ ( r )= λδ ( x ) δ ( y ) ,u s ingD i racde l ta functions to constrain the x and y values. The easiest moment to evaluate is I zz , I = λ
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HW12 prob3 rod pivoting - Problem Kinetic Energy of a Rod(a...

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