PROBLEM 9.164 For the homogeneous tetrahedron of mass mshown, (a) determine by direct integration the product of inertia ,zxI(b) deduce yzIand xyIfrom the results obtained in part a. SOLUTION (a) First divide the tetrahedron into a series of thin vertical slices of thickness dzas shown. Now 1azxzaacc=−+ =−and 1bzyzbb−The mass dmof the slab is 211122zdmdVxydzabdzcρρρ===−Then 23003cczczmdmabdzab−=−−∫∫16abc=Now zxz xEL ELdIdIz x dm′′=+where ( )0symmetryzxdI=and 133ELELzzzxxac=−Then 2032czxzxzzIdIz
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