Centroids and centers of gravity

Up to now, every body's weight has been treated as a single force acting at a known point: the middle of a uniform beam, the center of a crate. Real objects are rarely so simple. A crane with its boom, counterweight and cab, a T-shaped concrete beam, a truck with a load, a ship, a dam: each has weight spread through it, and to write its equilibrium equations you must know where the resultant of all that weight acts. That point is the center of gravity. For areas and shapes, the corresponding geometric point is the centroid, and it appears again and again in later engineering courses: the neutral axis of a beam passes through the centroid of its cross-section (Lesson 11), the resultant of a distributed load acts at the centroid of the load diagram (Lesson 7), and the stability of vehicles, cranes and structures depends on the height of their center of gravity. This lesson defines these points, shows how to find them by integration and, much more often in practice, by dividing a shape into simple parts, and introduces the theorems of Pappus for areas and volumes of revolution.

Center of gravity, center of mass and centroid

A body is made of a huge number of particles, each pulled down by its own small weight. These parallel forces can be replaced by one resultant equal to the total weight W, acting at the center of gravity G, the point about which the weights of all the particles produce zero net moment. Applying the moment equivalence of Lesson 3 gives:

Three related points are worth distinguishing:

• The center of gravity uses weight. In a uniform gravity field, which is accurate for anything smaller than a mountain range, it coincides with the center of mass, which uses mass (dm instead of dW).
• The center of mass matters in dynamics, where it is the point whose motion follows Newton's second law for the whole body.
• The centroid is purely geometric: the average position of a line, area or volume. For a body made of one uniform material, the center of gravity is at the centroid of its volume.

For an area A in the x-y plane, the centroid C is at:

The integrals ∫x dA and ∫y dA are the first moments of the area about the y and x axes. If an area has an axis of symmetry, the centroid lies on it, because every element on one side is matched by one on the other with an equal and opposite moment. A rectangle's centroid is at its center, a circle's at its center, and a symmetric I-beam's at mid-depth.

Centroids by integration

For shapes bounded by simple curves, choose a convenient element dA, usually a thin strip, write its area and the coordinates of its own centroid, and integrate.

Composite areas

Most engineering shapes are made of rectangles, triangles, circles and other simple pieces whose centroids are known. Then the integrals become sums. Divide the shape into parts, find each part's area and centroid, and combine:

A hole or cut-out is treated as a part with negative area. Working in a table, with columns for each part's area, centroid coordinates and first moments, keeps the arithmetic organized and easy to check.

Centers of gravity of composite bodies

The same method works in three dimensions and for bodies made of different materials: replace the areas by weights (or masses) and the centroids by each part's center of gravity.

This is how engineers locate the center of gravity of a loaded truck, a crane, an aircraft or a ship. Aircraft are loaded to keep the center of gravity within a certified envelope on every flight, because a center of gravity outside it can make an aircraft uncontrollable; incorrect loading has caused fatal crashes.

Stability and the height of the center of gravity

The higher a body's center of gravity, the smaller the tilt or sideways force needed to tip it. A simple measure for vehicles is the static stability factor, SSF = T/(2h), where T is the track width (the distance between the left and right wheels) and h is the height of the center of gravity. It is the sideways acceleration, in g, at which a rigid vehicle would begin to tip on a flat surface.

The same idea governs ships, where the center of gravity must stay below a point called the metacentre; cranes and aerial work platforms, which are tested on slopes; tall storage racks and shelving, which must be anchored; and precast concrete panels and steel beams during lifting, where the lifting points must be placed relative to the center of gravity so the piece hangs level.

The theorems of Pappus–Guldinus

Two theorems, known since Pappus of Alexandria in the fourth century, give areas and volumes of bodies of revolution from centroids. When a plane curve or area is rotated about an axis in its plane that does not cross it:

• The surface area generated by rotating a curve equals the curve's length times the distance traveled by its centroid: \(A = 2\pi\bar{r}L\).
• The volume generated by rotating an area equals the area times the distance traveled by its centroid: \(V = 2\pi\bar{r}A\).

Here \(\bar{r}\) is the distance from the axis to the centroid of the curve or area. The theorems save integration for tanks, domes, pipes, rings, pulleys and other axisymmetric parts.

Practical notes

Finding centroids by hand is mostly an exercise in organization. Steel section tables list the centroid (and the moments of inertia of Lesson 11) of every standard shape, and CAD programs report the centroid and mass properties of any modeled part. Engineers still need the hand method to check software, to deal with built-up sections and unusual shapes, and to understand what the numbers mean. Always ask which way a result should move when part of a body is added or removed, and confirm that the centroid lies within the body's overall extent and on any axis of symmetry.

Lesson 7 uses centroids to replace distributed loads and fluid pressures by single resultant forces, and Lesson 11 builds on the same first moments to find second moments, the moments of inertia that control how beams bend.