key term - Centroid

Definition

The centroid is the geometric center of a plane figure or solid. It is the point at which the shape could be perfectly balanced on a pin.

5 Must Know Facts For Your Next Test

1. The centroid coordinates $(\bar{x}, \bar{y})$ can be found using integration formulas: $\bar{x} = \frac{1}{A} \int_{a}^{b} x f(x) \, dx$ and $\bar{y} = \frac{1}{A} \int_{a}^{b} \frac{1}{2}[f(x)]^2 \, dx$, where $A$ is the area.
2. The centroid of an object with uniform density coincides with its center of mass.
3. For composite shapes, the centroid can be found by dividing them into simpler shapes, finding each shape's centroid, and then using weighted averages.
4. In symmetrical objects, the centroid lies on the axis of symmetry.
5. When dealing with three-dimensional objects, centroids are calculated for all three coordinates: $(\bar{x}, \bar{y}, \bar{z})$.

Review Questions

Related terms

Moment:

The tendency of a force to rotate an object about an axis or pivot. In mathematics, it involves integrating over a distance multiplied by an area or volume element.

Center of Mass:

The point where the entire mass of an object can be thought to be concentrated. It coincides with the centroid if density is uniform.

Area Moment of Inertia: A geometrical property that measures how an area is distributed about an axis, used extensively in structural engineering and mechanics.