Center-of-Mass
When an object is thrown into the air, different parts of the object can follow complicated paths if the object spins as it travels. However, the center-of-mass of the object will always follow a parabolic trajectory through the air.
The center-of-mass is the point that moves as though all the mass is concentrated there.
The center-of-mass of an object, or a collection of objects, can be found using:
| Xcom |
= |
| x1m1 +x2m2 + ...
|  |
| m1 + m2 + ...
|
|
That tells you the x-coordinate of the center of mass. The y-coordinate and z-coordinate can be found from equivalent expressions.
If you have mass distributed in some way over an object instead of having a number of discrete masses, it's generally necessary to integrate to find the center of mass. The integral equation looks like:
| Xcom |
= |
| ∫
x dm
| |
| ∫
dm
|
|
= |
| ∫
x dm
| |
| M
|
|
where M is the total mass of the object |