Question 5.10: Determine the location of the center of gravity of the homog...
Determine the location of the center of gravity of the homogeneous body of revolution shown that was obtained by joining a hemisphere and a cylinder and carving out a cone.
STRATEGY: The body is homogeneous, so the center of gravity coincides with the centroid. Since the body was formed from a composite of three simple shapes, you can find the centroid of each shape and combine them using Eq. (5.20).
\bar{X}\sum{V}=\sum{\bar{x}}V \quad \quad \quad \bar{Y}\sum{V}=\sum{\bar{y}}V \quad \quad \quad \bar{Z}\sum{V}=\sum{\bar{z}}V \quad \quad \quad \quad \quad (5.20)
MODELING: Because of symmetry, the center of gravity lies on the x axis. As shown in Fig. 1, the body is formed by adding a hemisphere to a cylinder and then subtracting a cone. Find the volume and the abscissa of the centroid of each of these components from Fig. 5.19 and enter them in a table (below). Then you can determine the total volume of the body and the first moment of its volume with respect to the yz plane.
ANALYSIS: Note that the location of the centroid of the hemisphere is negative because it lies to the left of the origin.
| Component | Volume, mm³ | \bar{\pmb{x}}, \pmb{\text{mm}} | \bar{\pmb{x}}\pmb{V}, \pmb{\text{mm}^4} |
| Hemisphere | \frac{1}{2}\frac{4 \pi}{3}(60)^3 = 0.4524 \times 10^6 | -22.5 | -10.18 \times 10^6 |
| Cylinder | \pi(60)^2(100)=1.1310 \times 10^6 | +50 | +56.55 \times 10^6 |
| Cone | -\frac{\pi}{3}(60)^2 (100)=-0.3770 \times 10^6 | +75 | -28.28 \times 10^6 |
| \sum{V}=1.206 \times 10^6 | \sum{\bar{x}}V=+18.09\times 10^6 |
Thus,
\bar{X}\sum{V}=\sum{\bar{x}}V \quad \quad \quad \bar{X}(1.206 \times 10^6 \text{ mm}^3)=18.09 \times 10^6 \text{ mm}^4\bar{X}=15 \text{ mm}
REFLECT and THINK: Adding the hemisphere and subtracting the cone have the effect of shifting the centroid of the composite shape to the left of that for the cylinder (50 mm). However, because the first moment of volume for the cylinder is larger than for the hemisphere and cone combined, you should expect the centroid for the composite to still be in the positive x domain. Thus, as a rough visual check, the result of 115 mm is reasonable.
