5.3 Composite bodies and Pappus-Guldinus theorems

Centroids and Centers of Gravity

Composite Bodies and Their Centroids

A composite body is a complex shape built from simpler shapes you already know: rectangles, triangles, semicircles, etc. Instead of integrating over the entire complex shape, you find the centroid by treating each simple piece separately and then combining the results with a weighted average.

The centroid formulas for a composite body are:

$\bar{x} = \frac{\sum A_i \bar{x}_i}{\sum A_i}$

$\bar{y} = \frac{\sum A_i \bar{y}_i}{\sum A_i}$

where $A_i$ is the area of each component shape, and $\bar{x}_i$, $\bar{y}_i$ are the centroid coordinates of that component (values you can look up in standard tables for common shapes).

A few things to keep in mind:

• Every component must be referenced to the same coordinate system. Pick an origin (usually a corner or axis of symmetry) and stick with it.
• Holes and cutouts count as negative areas. If your shape is a rectangle with a circular hole, include the circle with a negative $A_i$ in your summation.
• The centroid of each simple shape is measured from your chosen origin, not from the shape's own corner.

Centers of Gravity in Composite Bodies

For objects with uniform density, the center of gravity is the same point as the centroid. The center of gravity is where the body would balance if you could support it at a single point.

If density varies between components, you'd replace area $A_i$ with weight $W_i$ in the formulas:

$\bar{x} = \frac{\sum W_i \bar{x}_i}{\sum W_i}$

For 3D composite bodies, you use volumes $V_i$ instead of areas (or weights, if density isn't uniform), and you add a $\bar{z}$ equation.

Step-by-step process for composite centroid/center of gravity problems:

1. Sketch the composite body and break it into simple shapes (including any holes as negative shapes).
2. Set up a coordinate system with a clearly defined origin.
3. For each component, record its area $A_i$ and the coordinates of its centroid $(\bar{x}_i, \bar{y}_i)$ relative to your origin.
4. Organize everything in a table. Columns for shape, $A_i$, $\bar{x}_i$, $\bar{y}_i$, $A_i \bar{x}_i$, and $A_i \bar{y}_i$ make the arithmetic much cleaner.
5. Sum the columns and apply the centroid formulas.
6. Sanity check: the result should fall within the physical boundary of the shape. If it doesn't, look for a sign error or a misplaced centroid coordinate.

Pappus-Guldinus Theorem for Surface Area

Pappus-Guldinus First Theorem

This theorem gives you the surface area of a solid of revolution without integration. The idea: when a plane curve revolves around an external axis, the surface it sweeps out has an area equal to the curve's arc length multiplied by the distance its centroid travels.

$SA = 2\pi \bar{r} \cdot L$

• $L$ = arc length of the generating curve
• $\bar{r}$ = perpendicular distance from the centroid of that curve to the axis of revolution
• $2\pi \bar{r}$ = the circumferential distance the centroid travels in one full revolution

The generating curve must not cross the axis of revolution (otherwise the theorem doesn't apply directly).

Applying the First Theorem

1. Identify the generating curve (the line or curve that sweeps out the surface) and the axis of revolution.
2. Calculate the arc length $L$ of the generating curve. For straight segments, this is just the length. For curves, you may need the arc length formula or a table value.
3. Find $\bar{r}$, the distance from the centroid of the generating curve to the axis of revolution. For composite curves, use the weighted-average centroid approach with arc lengths as weights.
4. Plug into $SA = 2\pi \bar{r} \cdot L$.

Example: A straight line segment of length 5 m sits 3 m away from (and parallel to) an axis. Revolving it produces a cylindrical surface with area $SA = 2\pi(3)(5) = 30\pi \approx 94.2 \text{ m}^2$. You can verify this matches the standard cylinder lateral area formula $2\pi r h$.

Pappus-Guldinus Theorem for Volume

Pappus-Guldinus Second Theorem

This theorem gives you the volume of a solid of revolution. When a plane region revolves around an external axis, the volume it generates equals the region's area multiplied by the distance its centroid travels.

$V = 2\pi \bar{r} \cdot A$

• $A$ = area of the generating plane region
• $\bar{r}$ = perpendicular distance from the centroid of that region to the axis of revolution

Again, the generating region must not cross the axis.

Applying the Second Theorem

1. Identify the generating plane region (the 2D shape being revolved) and the axis of revolution.
2. Calculate the area $A$ of the generating region.
3. Find $\bar{r}$, the distance from the region's centroid to the axis of revolution.
4. Plug into $V = 2\pi \bar{r} \cdot A$.

Example: A semicircle of radius $R = 5$ m revolves around its diameter. The area of the semicircle is $A = \frac{1}{2}\pi(5)^2 = \frac{25\pi}{2}$ m². The centroid of a semicircle sits at $\bar{r} = \frac{4R}{3\pi} = \frac{4(5)}{3\pi} = \frac{20}{3\pi}$ m from the diameter. So:

$V = 2\pi \left(\frac{20}{3\pi}\right)\left(\frac{25\pi}{2}\right) = \frac{500\pi}{3} \approx 523.6 \text{ m}^3$

This matches the sphere volume formula $\frac{4}{3}\pi R^3 = \frac{4}{3}\pi(125) = \frac{500\pi}{3}$, which confirms the theorem works.

Applying Pappus-Guldinus Theorems to Composite Bodies

When the generating curve or region is itself a composite shape, combine the composite centroid technique with the Pappus-Guldinus theorems.

For surface area with a composite generating curve: find the overall centroid of the curve using arc-length weighting:

$\bar{r} = \frac{\sum L_i \bar{r}_i}{\sum L_i}$

Then apply $SA = 2\pi \bar{r} \cdot L_{\text{total}}$.

For volume with a composite generating region: find the overall centroid of the region using area weighting:

$\bar{r} = \frac{\sum A_i \bar{r}_i}{\sum A_i}$

Then apply $V = 2\pi \bar{r} \cdot A_{\text{total}}$.

Example: A composite region consists of a rectangle (4 m × 2 m) with a semicircle (radius 2 m) attached to one of its short sides. This region revolves around the opposite short side of the rectangle (4 m away from the semicircle). To find the volume of the resulting solid:

1. Calculate each component's area and centroid distance from the axis of revolution.
2. Find the composite centroid distance $\bar{r}$ using the area-weighted formula.
3. Apply $V = 2\pi \bar{r} \cdot A_{\text{total}}$.

The key advantage of Pappus-Guldinus is that you never need to set up an integral. As long as you know the area (or arc length) and centroid location of the generating shape, you can find the volume (or surface area) directly. This is especially useful on exams where time is limited.