Question 9.SP.10: For the homogeneous rectangular prism shown, determine the m...
For the homogeneous rectangular prism shown, determine the moment of inertia with respect to the z axis.
STRATEGY: You can approach this problem by choosing a differential element of mass perpendicular to the long axis of the prism; find its moment of inertia with respect to a centroidal axis parallel to the z axis; and then apply the parallel-axis theorem.
MODELING and ANALYSIS: Choose as the differential element of mass the thin slab shown in Fig. 1. Then
dm = ρbc dx
Referring to Sec. 9.5C, the moment of inertia of the element with respect to the z′ axis is
d I_{z^{\prime}}=\frac{1}{12} b^{2} d mApplying the parallel-axis theorem, you can obtain the mass moment of inertia of the slab with respect to the z axis.
d I_{z}=d I_{z^{\prime}}+x^{2} d m=\frac{1}{12} b^{2} d m+x^{2} d m=\left(\frac{1}{12} b^{2}+x^{2}\right) \rho b c d xIntegrating from x = 0 to x = a gives you
I_{z}=\int d I_{z}=\int_{0}^{a}\left(\frac{1}{12} b^{2}+x^{2}\right) \rho b c d x=\rho a b c\left(\frac{1}{12} b^{2}+\frac{1}{3} a^{2}\right)Since the total mass of the prism is m = ρabc, you can write
I_{z}=m\left(\frac{1}{12} b^{2}+\frac{1}{3} a^{2}\right) \quad I_{z}=\frac{1}{12} m\left(4 a^{2}+b^{2}\right)REFLECT and THINK: Note that if the prism is thin, b is small com-pared to a, and the expression for I_{z} reduces to \frac{1}{3} m a^{2}, which is the result obtained in Sample Prob. 9.9 when L = a.
