Bookkeeping for angular momentum in sphere/dumbbell collision

[hkyriazi]

Simple problem, but I'm pulling my hair out trying to understand the way angular momentum is handled.

NOTE: this is not a homework problem. I'm simply using a simple example to try to understand the concepts involved.

Object 1, a sphere of mass M, moves with velocity V in the +Z direction (upward) and collides perpendicularly (and perfectly elastically) onto one end of object 2, a horizontally oriented, stationary dumbbell of total mass M. Thus, particle 1 hits another sphere of mass M/2. (For simplicity, we'll assume that no mass is contained in the connecting rod between the ends of the dumbbell, and that the spheres are very tiny compared to the length of the connecting rod.)

The result is that object 1 slows to V/3, the dumbbell's center of mass (CoM) now moves in the +Z direction with speed 2V/3, and twirls, with the ends of the dumbbell also moving at 2V/3 with respect to its CoM.

The energy and linear momentum conservations work out fine. It's the accounting for angular momentum that bothers me.

It seemed to me that the angular momentum in the twirling of the dumbbell came out of nowhere: we went from purely linear motion to a combination of linear and angular motion. But I was told here, a few years ago, that one could/should consider that object 1, the sphere, has an initial angular momentum about the dumbbell's CoM of MVr, where r is the distance from the dumbbell's CoM to each sphere on each end (where object 1 impacts the dumbbell); r is essentially the "radius" of the dumbbell.

That was fine as far as it went, but beyond that is where I get lost. Let's say that object 1 contacts the horizontal object 2 on its left side. In that case, we'd have a clockwise rotation, and angular momentum vector pointing away from us, into the plane of our paper (or computer screen). That impact would result in a clockwise rotation of the dumbbell, with a similar angular momentum vector. Fine so far. But it's the accounting for the relative motion of the CoMs of objects 1 and 2 after the collision that throws me off.

For this "imaginary" angular momentum, where we're considering the two objects as a connected system, I assume what we want to do is consider object 1's motion around object 2's CoM. At the instant prior to contact, this equaled MVr. But after collision, object 2 moves up and away from object 1 at V/3, i.e., its angular momentum vector now points in the opposite direction (toward us rather than away from us). The dumbbell's angular momentum of twirling is 2/3 MVr.

The initial angular momentum, MVr, does not equal the sum of its two post-collision parts: -1/3 MVr + 2/3 MVr.

Somehow, I've got the sign of that "relational" ("imaginary") angular momentum wrong (the "-1/3 MVr" part).

But, I don't see why it should be positive.

 

[hkyriazi]

No responses yet. Is this question too stupid, or too subtle? Too verbose? Improperly framed? Should I give all of my momentum and energy calculations, showing how I got the final velocities? I didn't think that was necessary, because the problem I'm having is more conceptual.

Let me try to frame it differently.

I can see the logic in examining the relative motion of the centers of mass of any two objects we're considering. If those centers of mass are on a collision course, there can never be additional angular momentum added to either of the objects. If they're offset, then if the objects collide, they could have their net individual angular momentum changed.

But how do we account for that same relative motion of the centers of mass once the objects have passed their point of closest approach ("r" in the original problem), and are now moving away from each other? Do we mentally reverse their direction, bring them back together, and say "they must've passed this way before, and if no collision occurred, their centers of mass must've had such-and-such a relative angular motion"? What's the logic involved?

In the problem, the relative motions of the two objects were reversed, i.e., 1 was approaching 2 on 2's left side, and after the collision, 1 was moving in the opposite direction, still on 2's left side. That would make me think that the absolute value of the angular momentum given to object 2 would have to be greater than what existed initially (MVr), but it's actually less (2/3 MVr). Where am I going wrong?

 

[hkyriazi]

Never mind, guys. I found out at myphysicslab.com that one must use a fixed point: "Angular momentum is measured with regard to a particular point in space, for example the origin."