I've tried several setups through the years for students to model Newton's 2nd law. Qualitatively (balanced vs. unbalanced, direction on "unbalance" is the same as the direction of acceleration) getting the concept should come first - I have a previous post on that here, but when it comes time for modeling acceleration's dependence on force and mass, the setup can be tricky. Some that I've tried and/or seen:

- Pulling a cart with springs, measuring acceleration with a motion detector; it involves a lot of performance (keeping the spring stretch constant) time and practice that I'd rather have them spend on the analysis, especially with my small number of class days
- Half Atwood machine: you can easily vary the hanging mass to change the force exerted on the cart, but you can't equate the hanging weight to the tension, and you can't solve for it unless you know N's 2nd already. You can change the mass of the cart, but then you're changing the tension.
- Half Atwood machine, analyzed as one system: it's procedurally easier, but mystifying for students. Looking at the whole system means that your vector directions will have to be changed because of the pulley, which seems mysterious to students that haven't done that much force analysis. Keeping the weight of the hanging mass constant while adding mass to the cart is easy, but then they really need to record the total mass of the system, which is a bit conceptually tricky so early on. Keeping the total mass of the system constant when you are changing the hanging mass is similarly black-box for them at this point. It's an elegant setup from our point of view, but doesn't ultimately make much sense to most of them at this point in their physics careers

Here's what I used this year:

Half Atwood machine, with a force sensor screwed to the cart (the string's tied to the sensor). A motion detector helps them find acceleration (from the slope of the v vs t graph):

- The system being analyzed here is just the cart. The force probe measures the tension directly, so no complex analysis or tricky conceptual arguments need to be made. About halfway through, it's good to stop them and ask them to compare the hanging weight to the tension force reading and to explain the discrepancy conceptually.
- Varying the force exerted on the cart just means varying the hanging mass, and varying the mass of the cart is simple, too - no mysteriously motivate shuffle of masses back and forth. The experimental design is completely transparent to them.
- You might want to have them split into two factions: some groups investigate how acceleration varies with force and others investigate the dependence on cart mass. They then present whiteboards and the class can determine the combined model from the two partial models. Don't underestimate the reasoning leading from the two individual models to the combined model.

Everything seemed to go quite smoothly (that last conversation's still a bit difficult - I need a better angle on that, because it happens frequently), and my students this year have become stronger conceptually and computationally than with the setups that I've tried in the past, while moving at a faster pace. This one seemed to work quite well.

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