Question 5.8: Using the theorems of Pappus-Guldinus, determine (a) the cen...
Using the theorems of Pappus-Guldinus, determine (a) the centroid of a semicircular area and (b) the centroid of a semicircular arc. Recall that the volume and the surface area of a sphere are \frac{4}{3}\pi r^3 and 4πr², respectively.
STRATEGY: The volume of a sphere is equal to the product of the area of a semicircle and the distance traveled by the centroid of the semicircle in one revolution about the x axis. Given the volume, you can determine the distance traveled by the centroid and thus the distance of the centroid from the axis. Similarly, the area of a sphere is equal to the product of the length of the generating semicircle and the distance traveled by its centroid in one revolution. You can use this to find the location of the centroid of the arc.
MODELING: Draw diagrams of the semicircular area and the semicircular arc (Fig. 1) and label the important geometries.
ANALYSIS: Set up the equalities described in the theorems of Pappus-Guldinus and solve for the location of the centroid.
\begin{matrix} V=2\pi \bar{y}A && \frac{4}{3}\pi r^3=2\pi \bar{y}\left(\frac{1}{2} \pi r^2\right) && \bar{y}=\frac{4r}{3 \pi} \\ \\ A=2 \pi \bar{y}L && 4 \pi r^2=2 \pi \bar{y}(\pi r) && \bar{y}=\frac{2r}{\pi} \end{matrix}
REFLECT and THINK: Observe that this result matches those given for these cases in Fig. 5.8.
