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02 April 2015

Bar Chart exercise: x and y momentum, angular momentum, and energy

I first became explicitly aware of the qualitative bar chart representation for energy conservation TIPERS book.  I know some of the newer textbooks, such as Etkina, use it as well.  Qualitative bar charts are a great way to cover energy conservation rigorously yet conceptually, without equations getting in the way of understanding.  They are simple enough to use... One bar per type of energy in the system, with three major columns:  Initial state, final state, and "work done by external forces".  The bars in the "initial" column plus the bars in the "work done by external forces" column must add to the bars in the "final" column.

I like the bar chart approach enough that I tried it for impulse-momentum problems, too.  Instead of one bar per type of energy, the approach becomes one bar per object within the system.  The columns become initial state, final state, and "impulse provided by external forces".  Again we can add bars: initial plus impulse equals final.  

In particular, the bar chart approach helps tremendously with the meaning of a negative sign.  Students can see quickly whether or not the bars add together properly; they can see why an impulse must be negative, and translate that to the direction that the external force is pushing.  Or, they can see why, mathematically, the work done on a system must be negative, and relate that to the direction of force and motion.  When I use equations only, negative signs tend to slip through the cracks.

(As an aside, I certainly didn't use qualitative bar charts EXCLUSIVELY.  I covered impulse and energy the same way I always have, but once we were comfortable with the basic concepts and ideas, I introduced the bar charts as a way of reviewing, and as a way of justifying conceptual questions without a bunch of equations.)

Now that we're at the end of the year, it's time to put everything together.  All those bar charts blend together in students' minds.  They know that everything in the universe is conserved, of course -- ask them, and they'll tell you that "it's the same before and after."  But ask them what "it" is, or why "it" is conserved, or how you can show that "it" is the same, and you get confused folks.

In order to review all of the conservation laws, I created a set of bar chart exercises.  For each, I posed a situation, with a clear definition of a system, an initial state, and a final state.  Then I asked for four separate bar charts: one for horizontal momentum, one for vertical momentum, one for angular momentum, and one for energy.  An example is at this google docs link to the first bar chart exercise .  

For each chart, I asked for a justification.  I did the first exercise in class, with the students coming to me after they finish each chart.  In that justification, I was primarily looking for a discussion of the external forces, external torques, or work done by external forces; they need to tell me what specific force external to the system was acting, and why it did or did not cause a torque or do work.

Note that this is a DIFFICULT exercise.  For all those who pooh-pooh when I talk about rigorous physics without mathematics, try this with your class and see how they do.  Questions like this are the first step toward being able to do mathematical physics.  If you can't set up an equation properly, then all the math skill in the world won't help you get complicated problems right.

Try the problem in the link, which is based on a situation in an old AP Physics C exam.  A future post will likely discuss the solutions, along with some student misconceptions.  Let me know if you have any questions or improvements.


  1. I looked at the linked problem. AP's version of the next logical question is what if he throws it tangentially. But, AP locks the disk with a pole so it only rotated. My question is, what if the ball is thrown tangentially but the disk is free to rotate and slide on the ice (no friction): would both linear and angular momentum be conserved? My initial inclination is to say yes, but wouldn't the final linear momentum (and thus final linear KE) of the disk be the same as in the radial throw? Except now with a tangential throw, the disk is also spinning. So the system will have gained more total KE (due to the work done in throwing the ball) in the tangential case than the radial case. This would mean that the work required to throw a ball with initial velocity v_0 would depend on the direction you throw it. That does not sit well with me.

  2. I see the discomfort. The problem you describe with the disk spinning AND translating is one of the next exercises. The majority of the students agreed with you that since energy is directionless, the work required to throw the ball must be the same in either case.

    What's missing is that the initial velocity v_0 is WITH RESPECT TO THE GROUND. Sure, it will take the same work done by your muscles to change the ball's speed relative to you by the same amount. That's not what's happening here, though. The disk "recoils". And the "recoil" is different whether it is fixed, rotateable, or both rotateable and translateable. (If those are real words.)

    I prefer to think in terms of the ball-person-disk system, and the kinetic energy gained by the system -- all that kinetic energy must be provided by the work done by your muscles. However, the "recoil" issue and recognizing that v_0 is with respect to ground is a good alternate conceptual explanation to why more work is done by your muscles in the tangential case.