Question 7.17: Cylinder with Nonuniform Density Determine the mass of the c...
Cylinder with Nonuniform Density
Determine the mass of the cylinder in Fig. 7.34 and the position of its center of mass if (a) it is homogeneous with mass density po ; (b) its density is given by the equation \rho = \rho _{o}(1 + x/L) .
Strategy
In (a), the mass of the cylinder is simply the product of its mass density and its volume and the center of mass is located at the centroid of its volume. In (b), the cylinder is nonhomogeneous and we must use Eqs. (7.24) and (7.26) to determine its mass and center of mass.
m = \int_{m}^{}{dm} = \int_{V}^{}{\rho dV} (7.24)
\overline{x} = \frac{\int_{V}^{}{\rho x dV} }{\int_{V}^{}{\rho dV} } , \overline{y} = \frac{\int_{V}^{}{\rho y dV} }{\int_{V}^{}{\rho dV} } , \overline{z} = \frac{\int_{V}^{}{\rho z dV} }{\int_{V}^{}{\rho dV} } (7.26)
