232_Problem CHAPTER 9

232_Problem CHAPTER 9 - PROBLEM 9.162 Thin aluminum wire of...

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PROBLEM 9.162 Thin aluminum wire of uniform diameter is used to form the figure shown. Denoting by m the mass per unit length of wire, determine the products of inertia , xy I , yz I and zx I of the wire figure. SOLUTION First compute the mass of each component. Have m mL m L L  ==   Then 15 1 1 22 mmm R m R ππ = ( ) 24 2 1 RR 322 mmR m R Now observe that because of symmetry the centroidal products of inertia, ,, x yy z I I ′′ and , zx I ′ ′ of components 2 and 4 are zero and ( ) ( ) ( ) ( ) 1 13 3 00 xy yz II = = ( ) ( ) 5 5 0 = = Also 12 2 34 45 0 xx y y y z z = = = Using the parallel-axis theorem [Equations (9.47)], it follows that x zz x I for components 2 and 4.
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