One of my favorite questions from the 2014 released AP Physics 1 practice exam involves angular momentum in a collision between a sphere and a pivoted rod. Teachers can check out the question (I think) at this AP classroom link. (If that doesn't work, search the AP classroom question bank for "A thin rod of length d".)
In this question, a sphere bounces off of a pivoted rod. The rod rotates after collision, and the sphere rebounds with a given *linear* momentum. The question asks for the change in the rod's angular momentum as a result of the collision.
Some folks have tried to understand the solution by combining linear and angular momentum. But that's not correct! It's never allowed to combine linear and angular momentum in the same equation - you've gotta either conserve one or the other. Linear momentum never transfers to angular momentum (nor vice versa).
This particular AP question explicitly asks about angular momentum. Therefore, even though the problem gives the linear momentum of the sphere as pf and pi before and after collision, we've got to use conservation of just angular momentum about the pivot.
Angular momentum of an extended object is Iw. That's what we'd use for the rod, if we needed it. For a point object, angular momentum is mvr, where r is the distance of closest approach. In this problem, the variable d is the closest the ball ever can get to the pivot, so d is the distance of closest approach.
Now, the change in angular momentum of the rod is equal to the change in angular momentum of the ball - that's what angular momentum conservation means! The ball's change in angular momentum is (mvid) - (-mvfd), because the ball rebounds - it changes direction, represented by the negative sign mathematically. The negative signs cancel to make a positive sign. Here the masses m and speeds vi and vf aren't given, but the linear momentum is: mvi *is* pi, and mvf *is* pf. That's where we get pfd + pid as the answer!
Hey Greg,
ReplyDeleteYou said the ball's change in angular momentum is (mvid) - (-mvfd), but isn't change in angular momentum final minus initial? I got -mvfd - mvid which simplifies to -(pf + pi)d. Now, the problem wanted the magnitude which means I can ignore the negative symbol I got and then our answers would be the same. Thoughts?
You're right, the change in angular momentum should be (mvfd)-(-mvid). Or what you said is fine, too - depending on which direction you call "negative." But the change should be stated as final - initial.
ReplyDeleteHi Greg, how well does your 2-day conceptual physics class align with AMTA modeling? Thanks.
ReplyDeleteWell, while I do things that are in the spirit of modeling, and other things that are “bastard modeling” or “inverse modeling”, I am definitely not a Modeler. Pure modeling is great, but it’s not exactly my style. Come to the Conceptual physics summer institute - you’ll see!
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