*smallest*applied force to keep it moving at constant speed?

I asked this of my freshmen last week, as we come to a close in our Newton's second law unit. We've done a large number of this type of problem, explaining the multi-step logic involved in justifying the answer. This is

**not easy**for a physics student of any level -- rather than simply applying a single fact or equation, this seemingly simple question requires students to make a number of connections.

An answer I'd expect:

Moving at constant speed means zero acceleration and net force, so the applied force is equal to the friction force. The friction force is Ff = μFn. The normal force on each block is the same, because with no vertical speed change Fn = weight, and all weigh 100 N. So by the equation the smallest Ff takes the smallest μ -- that's copper.

I had students grade each others' work on this question to a five point rubric:

**5 points:**

¨
1 point for use of the
equation

*F*_{f}=*mF*_{n}
¨
1 point for

*recognizing*that the normal force is the same for each
¨
1 point for

*explaining why*the normal force is the same for each (i.e. vertical net force or acceleration is zero)
¨
1 point for correct
arrows OR several false calculations showing that we want the smallest

*m*
¨
1 point for the correct
answer

(That fourth point is phrased in language my freshmen understand and have seen before... for other physics teachers I'd say "1 point for showing how the equation supports finding the smallest μ.")

Now, you might come up with a bobzillion other valid rubrics. I didn't give an explicit point for pointing out that the applied force must equal the friction force, for example; on another day I might have. Five points for a single question on a problem set is a lot. My students are expected to do four to seven problems of this type each night; I don't expect a thesis, just a few sentences using facts and equations which hit the important logical progression. By the time I or classmates have looked at every problem most nights, plus all the similar in-class work, each student gets the feedback he needs to improve his understanding.

That is, each student gets that feedback

**if he looks at his graded work and gives a crap why it's right or wrong.**That's the holy grail of high school teaching, of course... leading our horses to water AND making them drink. We need to deal positively and appropriately with the significant but frustrating population who keep making the same mistake over and over again, never listen to or care about feedback, but tell the universe with hangdog eyes how impossible physics is.

So that's why I have students grade each others' work. Not only do they see the statements I'm looking for to award credit, but they have to read carefully for themselves to see whether someone else's paper meets the appropriate criteria.

**I didn't "go over" this problem set**-- because I would have helped just the one of the fifteen students in the class who was paying attention while I talked.

**I didn't just post the solution -**- because I would have helped the 0.3 of the fifteen students in the class who would take the time to look at the posted solution. (And even then, I would have helped only five percent of the students who looked at the solution, because only that five percent looked more carefully than at the answer.)

**I made the students grade each others' work**, and I made them responsible for grading correctly. The questions I answered in class were about how to apply the rubric. Occasionally I would get a content question; my answer in those cases was carefully followed. When I required corrections to the problem set, the students did excellent work, much better than if I had graded the set and passed it back.

Making students grade is one of the best ways to foster collaboration and a team atmosphere. They realize quickly that correct answers aren't a matter of my opinion, that points aren't awarded randomly or arbitrarily. The teacher is not the opponent -- rather,the class is working together to understand this mysterious natural world. And I'll use every trick in my repertoire to foster such a positive and useful attitude toward physics class.

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