Monday, September 16, 2013

A Two-pulley Practicum

In the AP C: Mechanics course, we're working through some extensions to old models - non-constant acceleration as a more general application of CAPM principles, UFPM with non-constant forces, etc.

Even with constant acceleration and constant forces, there are some more subtle, but very powerful, techniques that we can't do the first time around. The first is the "look at the whole system" approach to force analysis. Instead of analyzing the half-Atwood machine, for example, as a hanging mass and a cart, and eliminating the tension algebraically, we decide that the net force accelerating the system is mg and the total mass of the system is m+M, and it's super easy to get the system's acceleration. I know that some folks do this in the first year, but I like having them draw those two FBDs and really puzzle out how the force sizes relate to each other, and I think that this is just a little too black-box (especially with the change in direction of the motion in the middle) for the first year. It's also really good algebra practice.

We'll do that today, but we'll also take a look at this situation:
This can be really tricky to analyze as either a separate or a combined system, but we can make an observation about the rope that's really helpful. When the cart moves a distance d to the right, the rope's downward motion is complicated by the presence of the pulley, but a little careful diagramming can show that half of that d will end up as a longer right-hand vertical rope, and half will end up as a longer left-hand vertical rope, so that the hanging mass m will only drop d/2. Using that, the acceleration of the hanging mass must be half of the acceleration of the cart, which allows us to really easily solve for the unknown acceleration.

This worked out very well as a practicum for me, using Pasco track, carts, and superpulleys (with 100g clamped into the jaws of the hanging one. The time's long enough (with a cart run of about a meter) to be timed pretty well with a stopwatch.
I think that this does a great job of emphasizing the importance of thinking during the problem-solving process and of adding a new twist to what students might think is a "completed" topic.

4 comments:

1. Great practical! There are some awesome variations on this in the MIT Relate Mechanics MOOC course HW problems that are real brain benders if you want to provide some extra gristle for the highly motivated.

1. Thanks - I took that one this summer, and took lots of problems from it, too :)

2. I love your blog and am really excited to explore more of it. I really like how you've integrated student work into your blog. This post has wonderful visuals and is very clearly laid out. I'm currently in a graduate certification program, taking a technology in the classroom course en route to become a high school biology/science teacher. How long have you been teaching? How has the integration of these physics software programs been in your classroom?

Keep posting!

1. Thanks for reading and for the kind words. I've been at it for about 10 years. The tech always has to serve the pedagogy, but I find that some polling software (can take more time than it's worth) and data acquisition software - Logger Pro - are the biggest bang and most versatile.