Center of Mass - Center of Gravity

Center of Gravity

Some applications, which require the location of the center of mass, use the term "center of gravity". Examples are calculations involving aircraft, vehicles and vessels. In these cases the use of the term "center of gravity" relies upon the implicit assumption of the near earth gravitational field. In such applications "center of gravity" is defined as the point where the weight of a body can be considered to be acting. As a practical matter, the slight difference which arise between the position of the center of gravity and center of mass are disregarded.

In physics the benefits of using the center of mass to model a mass distribution can be seen by considering the resultant of the gravity forces on a continuous body. Consider a body of volume V with density ρ(r) at each point r in the volume. In a parallel gravity field the force f at each point r is given by,

where dm is the mass at the point r, g is the acceleration of gravity, and k is a unit vector defining the vertical direction. Choose a reference point R in the volume and compute the resultant force and torque at this point,

and

If the reference point R is chosen so that it is the center of mass, then

which means the resultant torque T=0. Because the resultant torque is zero the body will move as though it is a particle with its mass concentrated at the center of mass.

For this choice of reference point the torque associated with the gravity forces balance, and the weight of the body can be considered to be concentrated at the center of mass. In this case, the center of mass is known as the center of gravity.

The center of mass of a body with an axis of symmetry and constant density must lie on this axis. Thus, the center of mass of a circular cylinder of constant density has its center of mass on the axis of the cylinder. In the same way, the center of mass of a spherically symmetric body of constant density is at the center of the sphere. In general, for any symmetry of a body, its center of mass will be a fixed point of that symmetry.