Motion of the Center of Mass
The center of mass of a system of objects obeys Newton’s second law. Two common examples might
illustrate the point:
(1) Imagine that an astronaut on a spacewalk throws a rope around a small asteroid, and then pulls the
asteroid toward him. Where will the asteroid and the astronaut collide?
Answer: at the center of mass. Since no forces acted except due to the astronaut and asteroid, the
center of mass must have no acceleration. The center of mass started at rest, and stays at rest, all the
way until the objects collide.
(2) A toy rocket is in projectile motion, so that it is on track to land 30 m from its launch point. While in
the air, the rocket explodes into two identical pieces, one of which lands 35 m from the launch point.
Where does the first piece land?
Answer: 25 m from the launch point. Since the only external force acting on the rocket is gravity,
the center of mass must stay in projectile motion, and must land 30 m from the launch point. The two
pieces are of equal mass, so if one is 5 m beyond the center of mass’s landing point, the other piece
must be 5 m short of that point.Finding the Center of Mass
Usually the location of the center of mass (cm) is pretty obvious ... the formal equation for the cm of
several objects is
Mxcm = m 1 x 1 + m 2 x 2 + ...Multiply the mass of each object by its position, and divide by the total mass M , and voila, you have the
position of the center of mass. What this tells you is that the cm of several equal-mass objects is right in
between them; if one mass is heavier than the others, the cm is closer to the heavy mass.
Very rarely, you might have to find the center of mass of a continuous body (like a baseball bat) using
calculus. The formula is
Do not use this equation unless (a) you have plenty of extra time to spend, and (b) you know exactly
what you’re doing. In the highly unlikely event it’s necessary to use this equation to find a center of mass,
you will usually be better off just guessing at the answer and moving on to the rest of the problem. (If you
want to find out how to do such a problem thoroughly, consult your textbook. This is not something worth
reviewing if you don’t know how to do it already.)
Elastic and Inelastic Collisions
This brings us to a couple of definitions.