The center-of-mass and the geometrical middle point (also referred to as the center or focus) are related concepts but differ in their underlying principles and applications.
The center-of-mass is a concept from physics and refers to the average position of mass in a system. It takes into account both the masses and their respective distances from a reference point. The center-of-mass is calculated by considering the mass of each object or particle in a system and multiplying it by its position vector, and then summing up these values and dividing by the total mass of the system. The resulting vector represents the center-of-mass of the system. The center-of-mass is often used to analyze the overall motion and stability of objects or systems, especially in physics and engineering.
On the other hand, the geometrical middle point or center refers to the point that lies equidistant from the boundary or extremities of a geometric shape. It is determined solely by the geometric properties of the shape, such as its size, symmetry, and dimensions. For example, in a regular triangle, the geometrical center would be the intersection point of the medians, which are the line segments connecting each vertex to the midpoint of the opposite side. The geometrical center is often used in various disciplines like geometry, architecture, and design to establish balance, symmetry, or aesthetic considerations.
To illustrate the difference between these concepts, let's consider a simple example. Imagine a see-saw or a teeter-totter with two people sitting on either end. The center-of-mass of the system is the point where the see-saw would balance perfectly if it were placed on a fulcrum. It takes into account the masses of the individuals and their distances from the fulcrum. On the other hand, the geometrical center of the see-saw would be the exact middle point along its length, regardless of the masses of the individuals. This point would be equidistant from both ends of the see-saw.
In summary, the center-of-mass is a physics concept that considers mass and its distribution in a system, while the geometrical center or middle point is a geometric property that defines a point equidistant from the boundaries of a shape.