Center Gravity

Overview

Engineering reference data for Center Gravity in mechanics.

Key Formulas

Newton's Second Law

$F = ma$

Force = mass × acceleration.

Work

$W = Fd\cos\theta$

Work = force × displacement × cos(angle).

Kinetic Energy

$E_k = \frac{1}{2}mv^2$

Energy of motion.

Potential Energy

$E_p = mgh$

Gravitational potential energy.

Variables

Geometric Shapes

For common geometric shapes, the center of gravity (centroid) can be determined by specific formulas.

Triangle:
The center of gravity is located at the intersection of the medians (lines from a vertex to the midpoint of the opposite side). Its distance from the base is: $a = \frac{h}{3}$

Trapezoid:
The centroid can be found by decomposing the trapezoid into two triangles. It lies at the intersection of the line connecting the centroids of these two triangles and the line segment connecting the midpoints of the parallel sides.

Perimeter of a Triangle:
The center of gravity for the perimeter (the triangle's boundary) is the center of its inscribed circle. The distance d from the side of length a is: $d = \frac{h (b + c)}{2 (a + b + c)}$

System of Bodies

The center of gravity for a system composed of two distinct bodies can be calculated based on their weights and the distance between them.

For two bodies with weights or masses P and Q, separated by a distance a, the distances from each body's individual center of gravity to the overall system's center of gravity are: $b = \frac{Q \cdot a}{P + Q}$ $c = \frac{P \cdot a}{P + Q}$ Where:

• b is the distance from the center of mass P to the overall centroid.
• c is the distance from the center of mass Q to the overall centroid.

Perimeter

For the perimeter of a triangle ABC, the center of gravity is at the center of the inscribed circle, located at the midpoints of the triangle's sides. The distance d from the vertex is calculated as:

$$d = \frac{h \cdot (b + c)}{2 \cdot (a + b + c)}$$

Trapezoid

The center of gravity of a trapezoid can be found by dividing it into two triangles. The center of gravity lies at the intersection of the line connecting the centers of the two triangles and the trapezoid's middle line (CD).

General Properties

The center of gravity (CG) is the point at which the entire weight of a body or system can be considered to act. For a body in a uniform gravitational field, it coincides with the center of mass.

Key properties:

• Balance Point: A body will balance perfectly if supported at its CG.
• Symmetry: For homogeneous bodies with geometric symmetry (circles, rectangles, symmetric triangles), the CG is at the geometric center.
• Suspension: If suspended from any point, the CG lies directly below the point of suspension.
• Multiple Bodies: The CG of a system of bodies is a weighted average of the individual CGs, where the weights are the individual masses or gravitational forces (weights).

References

• Original Source