Oh, the barbell

Last week, we tackled this excellent "Figuring Physics" problem by Hewitt (as posted in TPT):

I love this problem - it helps students differentiate between translational and rotational motion.

It's also a great example of a chain of reasoning that's not too difficult to lay out, but which has a critical choice to be made right at the beginning. This choice bifurcates the reasoning tree into "the real answer" and "bizarro world".  One's true, the other's the opposite, and... it all relies on getting one little piece correct.  Here are the dueling trees:


Now it's time to make the call: is the bottom of the bar moving forwards or backwards relative to the track?

If it wasn't moving at all - like it wasn't as the dumbbell rolled at the beginning, and like it won't be after the friction sets the rotation "right" again - then the bar would be rolling without slipping, which is a super-important concept for students to have a handle on.

The most amazing thing about rolling without slipping is that the total velocity of the contact point of the wheel is zero!  Here's a great picture of a wheel that's rolling without slipping, taken by Archan Baldev Luhar of Medfield High School, for the AAPT Photo Contest (any Tatnall students interested?!):

 

Great photo!  The v due to the rotation is the same as the translational v, but in different directions at the top and the bottom (and zero in the middle).  Excuse my lame graphic:

This means that the top moves at 2v, the hub at v, and the bottom doesn't move at all!

This is an excellent entryway into how friction works with wheels, both static and kinetic.  The direction of the static friction on a wheel can be hard for students to get, but it's just resisting the potential slippage of the tire (just like static friction always does), so you just need to figure out how the wheel would slip if there weren't any friction, and the static friction force is the opposite direction.  Neat, but a bit beside the point.

Here, we're looking at kinetic friction.  What's happening is that either the bar of the dumbbell is rotating too fast for the translational speed (which means that the bottom is moving backwards, relative to the track) or that the bar is rotating too slowly for the translational speed (which means that the bottom of the bar is moving forwards relative to the track).  It starts with the "correct" rotational speed, but that's the speed for the larger radius of the weight, not for the bar.

This brings us back to the question at hand, the choice that will send us down one path or the other (BTW, our dueling trees did narrow it down to a choice between C and D, so we already know more than we did at the beginning!), the measurement that creates two parallel bits of the multiverse, etc.

Is the bar rotating too fast or too slowly for the translational speed?

How fast does something rotate if it's rolling without slipping?  The rotational speed must be (you might have to click through to the post to read the LaTeX if you're on a reader):

We can easily determine the rotational speeds that the barbell already has and that it'll need to roll without slipping when the bar is on the track:

Here, r is the bar's radius and R is the larger radius of the weights.

It's easy to compare them with my favorite tool, the ratio:

If the needed angular velocity of the bar is greater than it currently has as it rolls on the weights,  we can say that the bar is rotating too slowly when it hits the track.  This means that the bottom of the bar is moving forwards (the forwards translational velocity is not canceled by the "rotational part" of the velocity), which puts us on tree #2!

As von Braun tells us, one experiment is worth a thousand expert opinions, so... (click through for video)

 The reason that I make a big deal about experiment here is that we all talked our way through the long chains of reasoning, and were smugly satisfied that the first chain was correct.  Sometimes, it really does pay to write a little something down.

In particular, it really helps to write things down when there's a symmetrical comparison to be made - it's really easy to get things backwards!

The reason that I bring up the experiment being worth so much is that the kids (bless their hearts) recognized that they hadn't quite wrapped their heads around the situation, and leapt to "let's do it!"  A quick field trip to the fitness room while others set up the track, and we were off.

The explanation came quickly, and the understanding was much more solid (and correct!).

Doing is learning.

Oh yeah - it works backwards, too: now the bar speeds up and the rotation slows! (click through for video):