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04 January 2013

That dang cart-and-pulley experiment

As we introduce Newton's second law, the obvious experiment is to measure the net force on an object, and the object's acceleration; a plot of net force vs. acceleration has a slope equal to the object's mass.  Easier said than done.

The trick is providing a truly constant force, one that can be easily varied and easily measured.  Most introductory classes default to the cart on a track attached to a hanging mass, as shown in the picture: easily varied, easily measured.  The motion detector can measure the acceleration of the cart as it's speeding up (or slowing down).  

Problem is, though, the hanging weight is NOT the net force on the cart.  If you just add and subtract mass from the mass hanger, the a plot of hanging weight vs. acceleration will not be linear.  Two experimental solutions come to mind:

(1) Measure the tension in the string, which is the net force on the cart.  You can put attach a Vernier force probe to the cart and to the string; the reading on the probe is the tension in the string.  As the cart is accelerating, you'll see the probe reading drop from a value equal to that of the hanging weight to something slightly smaller.  Use that on the vertical axis, and the acceleration from the motion detector on the horizontal; the slope will be the mass of the cart.

I do this demonstration in honors/AP physics, in which we predict the tension in the string and the acceleration of the cart.  However, this is a complicated way to do the experiment.  The easy way is...

(2)  The hanging weight is the net force of the entire set of objects including the cart and the hanging stuff.  So, the trick is to add about 100 g of masses in small denominations to the top of the cart.  Then, when changing the hanging weight, just redistribute mass from the top of the cart to the mass hanger.  Now the mass of the cart-and-hanger system is remaining constant.  So the plot of hanging weight vs. acceleration will have a slope equal to the total mass of the cart plus the 100 g.

Recognize that the distinction between treating the cart-hanger system as a unit, and treating the cart and hanger separately, is very subtle and difficult for even top level introductory students.  I don't even want to broach the subject with my freshmen, who are doing this experiment next week.  But that's okay... I don't have to tell them WHY they're keeping all the mass either on the cart or on the hanger.  If anyone asks, I'll explain that we need to keep the total mass of the moving stuff constant.*

* The super-smart-and-curious freshmen who would have demanded and understood a deeper explanation have long since been siphoned into an honors section, where they will learn all of the relevant subtleties.  

With the setup in the picture, my freshmen will be able to graph the hanging weight and the acceleration of the cart.  Interestingly, our new Vernier Labquest 2 is giving me trustworthy acceleration vs. time graphs!  This clever new device will spit out an average acceleration with reasonable ease.  I'll set up the carts ahead of time -- half the class will get the 250 g PASCO cart with 150 g of added mass... the other half will get a silver 500 g PASCO cart with 100 g of added mass.  We'll all use the same set of prepared axes, so we'll all see at a glance that the setup with larger mass produces a steeper line on the Fnet vs. a graph.

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