235_Problem CHAPTER 9

235_Problem CHAPTER 9 - PROBLEM 9.164 For the homogeneous...

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PROBLEM 9.164 For the homogeneous tetrahedron of mass m shown, ( a ) determine by direct integration the product of inertia , zx I ( b ) deduce yz I and xy I from the results obtained in part a . SOLUTION ( a ) First divide the tetrahedron into a series of thin vertical slices of thickness dz as shown. Now 1 az xz a a cc  =− + =   and 1 bz yz b b The mass dm of the slab is 2 11 1 22 z dm dV xydz ab dz c ρρ ρ == = Then 23 0 0 3 c c zc z md m a b d z a b = −− ∫∫ 1 6 abc = Now zx z x EL EL dI dI z x dm ′′ =+ where ( ) 0 symmetry zx dI = and 1 33 EL EL z zzx x a c = Then 2 0 32 c zx zx zz I dI z
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