Centers of gravity in non-uniform fields
The Gravizentrum or the center of gravity of a body called the mean of all positions, weighted by the gravitational force acting on each point.
- A homogeneous gravitational field ( for example, near the surface) the Gravizentrum with the center of mass of the body is consistent. Therefore, both terms are often referred to as undifferentiated focus.
- In the general case of inhomogeneous gravitational fields ( the third case mentioned below) are different Gravizentrum and center of mass. Which of the two points is called a "center of gravity ", it is dependent on the author.
On closer inspection, the concept of center of gravity as the center of gravity a more complex structure, as one of the intuitive perception ago - under simplified conditions such as constant gravity and uniform density - expected.
- Given a homogeneous density and homogeneous gravity ( gravitational acceleration ) can be the total focus of a collection from the weighted sum of the focal points of all subsystems determine:
- X - position vector
- V - volume
- G - total weight
- For bodies with inhomogeneous density, for example, are irregularly shaped, and constant gravitational field of the overall center of gravity is calculated as the first moment of the distribution function of the density of a cluster in space, normalized to the total weight:
- In addition, the gravitational field is inhomogeneous () so integrated not on the density (mass ) but more than the specific gravity:
Magnitude of the deviation of the center of mass and Gravizentrum
The gravitational acceleration at the surface is on the upper side of a one meter cube of 3.10 -7 smaller than the acceleration of gravity at the lower side of the cube. This means that the center of mass and are Gravizentrum about 1.5 microns apart. The deviations are so small that they are of no practical importance for technical applications.
In astrophysics, however, the difference be important because of far greater dimensions.