 | Rigid body dynamics: Encyclopedia II - Rigid body dynamics - Rigid body linear momentum
Rigid body dynamics - Rigid body linear momentum
The equation for particle linear momentum is
where:
- m is the particle's mass.
- v is the particle's velocity.
- fi is one of the N forces acting on the particle.
Assuming constant mass, this reduces to
To generalize, assume a body of finite mass and size is composed of such particles. There exist internal forces, acting between any two particles, and external forces, acting only on the outside of the mass. Each particle has:
- a mass dm.
- a position vector r.
Thus, the linear momentum equation of any given particle would look like this:
If the equation for each particle were added together, the internal forces would cancel out, since by Newton's third law, any such force would have opposite magnitudes on the two particles. Also, the left side would become an integral over the entire body, and the second derivative operator could come out of the integral, leaving
Letting M be the total mass, the left side can be multiplied and divided by M without changing the validity:
However, is the formula for the position of center of mass. Denoting this by rcm, the equation reduces to
Thus, linear momentum equations can be extended to rigid bodies by denoting that they describe the motion of the center of mass of the body.
 Adapted from the Wikipedia article "Rigid body linear momentum", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki |
