Center of Parallel Forces and Center of Gravity

The center of parallel forces is the point through which the line of action of the resultant of a parallel-force system passes when the relative positions of the force application points remain unchanged. The center of gravity is an important physical application of this concept.

Coordinates of the center of parallel forces

For parallel forces $F_i$ acting along a common direction, the center coordinates follow from moment equivalence. For example, $x_C=\sum F_i x_i/\sum F_i$ and $y_C=\sum F_i y_i/\sum F_i$, provided the denominator is nonzero. Oppositely directed forces are included with their algebraic signs.

Center of gravity

The gravitational forces acting on the particles of a body are effectively parallel in a uniform gravitational field. The point of application of their resultant is the center of gravity. For a homogeneous body it coincides with the center of mass, and for a homogeneous thin plate it coincides with the geometric centroid of its area.

Composite plane areas

For a plane area composed of simple parts with areas $A_i$ and known centroid coordinates $(x_i,y_i)$, the centroid coordinates are:

$$x_C=\frac{\sum A_i x_i}{\sum A_i},\qquad y_C=\frac{\sum A_i y_i}{\sum A_i}.$$

Holes are treated as negative areas. All component coordinates must be measured in the same coordinate system.

This method is especially convenient for sections that can be decomposed into rectangles, triangles, circles, and other simple regions. Holes and cutouts are assigned negative areas.

Using symmetry

If a homogeneous area has an axis of symmetry, its centroid lies on that axis. With two axes of symmetry, the centroid is at their intersection. Symmetry can therefore determine one or both coordinates without calculation.

Example

An area consists of two rectangles with $A_1=6$ cm² and $A_2=4$ cm² whose centroid coordinates are $x_1=2$ cm and $x_2=7$ cm. Then $x_C=(6\cdot2+4\cdot7)/(6+4)=4$ cm.

Common mistakes

• mixing coordinates measured from different origins;
• failing to treat holes as negative areas;
• assuming the center of gravity of a nonhomogeneous body is its geometric volume centroid;
• using an arithmetic mean of coordinates instead of a weighted mean.