In the classic old-fashioned circular motion experiment, students whirl a rubber stopper in a horizontal circle above their heads. The stopper is tied to a string. The string is threaded through a hollow tube - that’s what the student holds to whirl the stopper - and attached at the other end to a mass hanger. The equipment for this experiment can be procured from your storeroom -- don’t buy the “discover centripetal force” kits that are now down to $20 from $39.

My class's version keeps the mass on the hanger constant. Students vary the radius

*r*of the circular motion in each trial, measuring that and the speed*v*of the stopper's circular motion. They plot a graph of*v*^{2}on the vertical axis and*r*on the horizontal axis, and take the slope.**Experimental tips:**I don't ever give handouts with procedural instructions. For this experiment, I demonstrate the acquisition of one data point, ask for a graph of

*v*

^{2}vs.

*r*, and shut my mouth. In that demonstration, though, students see rather involved and somewhat difficult technique:

- This is the one experiment all year where I insist on lab goggles. It's easy to get poked in the eye.
- The whirler should keep an eye on the hanger. If it's moving up or down, the stopper's speed isn't constant.
- The stopper's speed is measured by dividing the circle's circumference by the period. A decade ago, we measured ten periods with a stopwatch - I was never happy with the precision here, because it's
*tough*to keep constant speed for that long. But now, we use "hudl technique" or "coach's eye" apps. That is, we take a video of one cycle, and use frame-by-frame advancement to find the period to the nearest hundredth of a second. - The radius of the circular motion is measured with a meterstick immediately after the video is complete. That's a trickier measurement than you might think - students have to pay attention so that the string doesn't slip along the plastic tube.

**Analysis:**Since the hanger is in equilibrium and the rope is assumed to be horizontal, the tension

*T*in the rope is equal to the weight on the hanger. And that tension is then equal to

*mv*

^{2}/

*r*where

*m*is the mass of the stopper.*

*

*Not the mass of the hanging thing! When setting a net force equal to ma, the mass in question is the mass of the thing that has the acceleration. That's the stopper here.*
The goal is to use the slope of the graph to determine the stopper's mass. In my formulaic method of teaching graph linearization, students begin by writing the relevant equation and solving for the variable on the graph's vertical axis. Then, they write the equation for a line

*y = mx + b*. They circle the variables that match the*y*and*x*axis... what's left is the physical meaning of the graph's slope (or*y-*intercept, but that's usually zero).
We do this analysis in class, where I talk students through each step of the process above. I check their graph, check their slope calculation, make them communicate in writing every analysis step, and then verify that they have a reasonable mass for the stopper. Not the

*right*mass of the stopper - they're not allowed to measure that directly with a scale! No, I'm making sure that answers like "15 kg" or "1.5 kg" or "0.15 g" are exposed as unphysical - without even entering the lab, a person who knows what a kilogram is could reject these as possible masses of a bottle stopper. And I'm checking that everyone is using two digits throughout.***Not because we have to memorize rules of significant figures for the AP exam! No! Because (a) two digits is the best we're going to measure pretty much anything in a first year physics laboratory, (b) the precision of this particular experiment is limited by the two-digit radius and period measurements, and (c) most importantly, insisting on only two digits throughout the year on everything (rather than ignoring excess precision, or teaching those sig fig rules that no high school student understands) is one of the ways in which I help my students develop habits of mind relating numbers to physical reality.*

Once everyone finishes, I do use a scale to show the class the stopper's mass. I give a prize to whichever student(s) got closest. Since we started using frame-by-frame video on smartphones, our mass values have been not just reasonable, but often correct within 10% or so.

I've fallen back in love with this experiment now that the long road to getting the analysis right pays off with good results. I do mean long road. The analysis by itself might take more than an hour; data collection is 30-45 minutes for well trained students.