# Does angular momentum exist around the moving center of mass of a released swinging rigid body?

I am trying to understand angular momentum about the center of mass of a centripetally accelerating object. My scenario has a meter stick being swung horizontally from the end of a rigid rod, somewhat similar to how an Olympic discus thrower spins around to throw the discus. The rod uses a vise clamp to attach to the center of mass of the meter stick, and I am assuming that the meter stick is infinitely thin. There is also a switch that can be pressed to open the clamp and release the meter stick.

My three questions relate to the diagram that I have attached:

1. Does the concept of angular momentum around the center of mass of the meter stick even make sense, as the center of mass is undergoing centripetal acceleration? I see that the rod is changing its orientation but I don't know if that means that the rod has angular momentum about its center of mass. In this scenario, the center of mass is accelerating, and thus a reference frame moving with the center of mass would be a non-inertial reference frame.

2. If there is angular momentum about the center of mass of the meter stick, how was the angular momentum generated? My intuition is that no torque around the center of mass is generated throughout the swing since the radius arm would be zero, so I don't understand how the angular momentum could have gotten started.

3. If the rod is allowed to detach from the meter stick, will the meter stick spin about its center of mass in its now straight-line trajectory?

• Unless the thin rod can apply a torque on the stick it would not rotate at all and always point towards the same direction. So to answer you need to specify the details of the attachment point. – John Alexiou Mar 22 at 0:04
• @JohnAlexiou, I am asking about the case where the rod attaches to the ruler using something like a vise clamp. In a different scenario, if the rod was attached to the rule with a frictionless peg that was free to rotate, there is now another degree of freedom, the initial rotation state around the peg before the rod is swung. So the rigid clamp scenario that I'm asking about is more constrained. – lamplamp Mar 22 at 2:37

So the instant of release the problem changes and is now just a free-moving body with certain initial conditions. Just before the release, the rod has rotational velocity equaling the orbital rotation, linear (tangential) velocity.

After the release, there are no external forces/torques acting so the body continues with the center of mass on a straight line maintaining the forward momentum and continues to spin about the center of mass maintaining the rotational momentum.

Have in mind Newton's laws of motion, which relate the change in momentum to the forces acting on a body and change in angular momentum (about the center of mass) to the torques acting on the body (about the center of mass). And for clarification, any rotation about the center of mass does not affect the linear momentum of the body. Linear momentum is a function of the motion of the center of mass only.

Given the above description, I am going to try to answer your questions

1. Momentum isn't conserved for objects that are attached to the ground in general because the attachment can transfer forces and/or torques to the body. In the context of Newton's laws, the change in momentum isn't zero when forces/torques are present.

2. The angular momentum of the rod has, was generated by the torque transmitted through the connection.

3. Yes, the center of mass will fly in a straight line, and the body will (keep) also rotate about the center of mass.