Center of Mass
* The center of mass is the location where all of the mass of the system could be considered to be located.
* For a solid body it is often possible to replace the entire mass of the body with a point mass equal to that of the body's mass. This point mass is located at the center of mass.
* For homogenous solid bodies that have a symmetrical shape, the center of mass is at the center of body's symmetry, its geometrical center.
* The center of mass is the point about which a solid will freely rotate if it is not constrained.
* For a solid body the center of mass is also the balance point. The body could be suspended from its center of mass and it would not rotate, i.e. not be out of balance.
* The center of mass of a solid body does not have to lie within the body. The center of mass of a hula-hoop is at its center where there is no hoop, just hula.
* The center of mass for a system of independently moving particles still has meaning and is useful in analyzing the interactions between the particles in the system.
System of N particles with total mass of M, Definition:
* This is the average position of the particles, weighted relative to the mass of each particle.
* The definition of the center of mass of a solid body can also be thought of as a weighted average of all the atoms out of which it is made. Since the atoms do not move (except for thermal vibration) relative to each we usually replace the sum with an integral because there are zillions of atoms in a solid.
Solid body with mass M, general definition:
Solid body of uniform density with mass M and volume V,
This equation is useful is determining the center of mass of objects like a cone. However, you will never have to use either of two above equations for solids in this class, since it will be obvious were the objects center of mass is located in all the solids you will encounter. None-the-less, it is useful to understand how one could figure out where the center of mass of an odd shaped body is located.