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

[lillybeans]

Homework Statement

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?

Thanks in advance,

Lilly

 

[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

 

[lillybeans]

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

Thank you very much! Very clear explanation!

 

[ehild]

You are welcome.

ehild

 

[asteriatic]

Dear Lilly,

You are correct in your understanding that the center of mass of a rotating system will remain motionless if the bullet strikes the rod at its center of mass. However, in the case where the bullet strikes the rod at a distance from the center of mass, the system will experience both rotational and translational motion.

In order to fully understand the motion of this system, we must use the concept of conservation of linear momentum. This principle states that the total linear momentum of a system remains constant unless acted upon by an external force. In this case, the bullet and rod are initially separate objects with their own individual linear momenta. When they collide, their momenta become combined and the system will experience both rotational and translational motion.

The non-zero value you have obtained for the linear velocity of the center of mass represents the translational motion of the system as a whole. This value is relevant because it allows us to understand the overall motion of the system, rather than just the individual motions of the bullet and rod.

I hope this helps clarify the relevance of conservation of linear momentum in the case of rotating systems. Keep up the good work in your studies of physics!

Best,