Centre of gravity and centroids
Every body behaves as if its whole weight acted at a single point, the centre of gravity. For shapes, the equivalent geometric point is the centroid.
Centre of gravity
The centre of gravity (CG) is the point where the resultant of all the gravitational forces on a body acts. A body supported at its CG balances in any orientation. For bodies small compared with Earth, the CG coincides with the centre of mass, the average location of the body's mass.
Centroids
The centroid is the geometric centre of a shape, depending only on geometry. For a body of uniform density, the centroid, centre of mass, and centre of gravity all coincide. Engineers find centroids of beam cross-sections, plates, and volumes.
Formulas
For an area divided into small pieces, the centroid coordinates are weighted averages: \(\bar{x} = \frac{\int x\,dA}{\int dA}\) and \(\bar{y} = \frac{\int y\,dA}{\int dA}\), or for a set of parts, \(\bar{x} = \frac{\sum \tilde{x}_iA_i}{\sum A_i}\), where \(\tilde{x}_i\) is each part's own centroid. The numerator is called the first moment of area.
Using symmetry
If a shape has an axis of symmetry, its centroid lies on that axis; with two axes, it lies at their intersection. Rectangles, circles, and I-beams therefore have centroids at their geometric centres without any calculation.
Standard shapes
A rectangle's centroid is at its centre. A triangle's centroid is one third of the height above its base, where its medians meet. A semicircle of radius r has its centroid \(\frac{4r}{3\pi} \approx 0.424r\) from its straight edge, and a quarter circle's centroid lies \(\frac{4r}{3\pi}\) from each straight edge.
Finding a CG experimentally
Hang an irregular flat object from a point and draw a vertical line down from the point; hang it from another point and repeat. The CG lies where the lines cross. Aircraft are weighed on three scales, one under each wheel, to locate their CG, which must stay within strict limits for safe flight.
Stability
An object resting on a base is stable as long as the vertical line through its CG falls inside the base. Lowering the CG or widening the base improves stability; racing cars sit low, and loaded trucks with high CGs are more likely to roll over on bends.
Worked example
A uniform triangular sign with a 1.2 m base and 0.9 m height hangs with its base horizontal at the bottom. Its centroid is 0.9 / 3 = 0.3 m above the base, on the median from the top vertex. Bolting the support at 0.3 m above the base midpoint lets it hang without twisting.
Key takeaways
The centre of gravity is where a body's weight effectively acts; for small bodies it equals the centre of mass, and for uniform bodies the centroid. Centroids are weighted averages: the first moment of area divided by total area. Symmetry axes contain the centroid. A triangle's centroid is a third of the height above its base, and a semicircle's is 4r/(3 pi) from its diameter. Hanging tests locate CGs, aircraft weighing keeps them in limits, and stability requires the CG to stay above the base.
Flashcard
Where weight effectively acts
Check your understanding
Where is the centroid of a triangle relative to its base?