# Conservation of linear momentum applied to rotating systems? (with picture)

1. Dec 17, 2011

### lillybeans

1. The problem statement, all variables and given/known data

I've encountered problems like this: A bullet with velocity v strikes a stick (intially at rest) at a distance d from the center of mass, then the bullet sticks to it, and the bullet-stick system rotates about the center of mass. But they ask me to find weird things like the speed of center of mass (v in the equation below), which requires me to apply conservation of LINEAR momentum.

Question: Isn't the center of mass MOTIONLESS if the bullet-rod system is rotating? Although it IS possible to calculate the linear velocity of the center of mass after the collision, why is it even relevant in the case of a rotating system?

If the bullet had striked the rod AT the rod's center of mass, then sure, this question makes sense, as the bullet-stick moves forward with that linear velocity, and conservation of linear momentum is relevant. But if the bullet strike anywhere AWAY from the stick's center of mass, the system will rotate and the center of mass will NOT move. So, WHAT DOES THIS NON-ZERO VALUE I GOT FOR THE LINEAR VELOCITY OF CENTER OF MASS REPRESENT IN THE CASE OF A ROTATING SYSTEM?

Lilly

Last edited: Dec 17, 2011
2. Dec 17, 2011

### ehild

The centre of mass will move after the impact if it is not fixed. As I understood the problem the stick is free. Any in-plane motion of a rigid body consist of a translation and a rotation about the CM. Both the linear momentum and the angular momentum are conserved in the impact.
Put a ruler on the table and give it a push. It will move away, but at the same time it will rotate, too. Do you really think that the CM will move if the bullet strikes the stick exactly at the CM, but will not move at all if the bullet hits it 0.1mm away from the CM?

ehild

3. Dec 17, 2011

### lillybeans

Thank you very much! Very clear explanation!

4. Dec 17, 2011

### ehild

You are welcome.

ehild