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18 February 2010

New lab: resistance of a mystery resistor

My AP labs generally follow a predictable pattern:  We take a large set of data, make a graph that's curved, manipulate the graph's axes to form a straight-line, and then use the slope of that line to determine some value associated with the experiment.  AP exam questions frequently follow this same approach.

For example, my first experiment of the year asks students to attach a cart to a spring scale, then to put the cart on an incline.  We measure the force parallel to the incline plane that holds up the cart; we plot that force as a function of the angle of the incline, giving a curve.  Next, we plot the force we measured vs. the sine of the incline angle; this gives a straight line.  Students show that the slope of the line is equal to the weight of the cart, and they divide this weight by g to find the cart's mass.

This process goes on again and again -- make a straight line, and relate the slope to a constant quantity.  I know I'm doing my job when students start to groan about going through the same process again and again.

BUT:  The slope of a stright-line graph isn't always the most meaningful of that line's attributes.  Many folks were shocked in 2007 when problem 6 (check out the link, page 11) required the use not of the slope, but of the y-intercept of an experimental graph.  The problem involved use of the thin lens equation, 1/f = 1/di + 1/do.  The y-axis was 1/di, the x-axis was 1/do, so the y-intercept became the recipricol of the lens's focal length.

I do that experiment in the spring.  But I want to give the class some experience with using the y-intercept of an experimental graph now, and we haven't dealt with lenses yet. I've used the pressure in a static column equation, P=Po + ρgh before -- use a vernier probe to measure the absolute pressure in a water-filled container as a function of depth.  The slope of the pressure - depth graph is ρg, while the y-intercept is the surface pressure Po.  This year, for a variety of reasons, I wanted an experiment less reliant on computer data collection and more complex in its analysis.

The equivalent resistance of parallel resistors is given by 1/Req = 1/R1 + 1/R2.  Huh... This equation  is quite similar in mathematical form to the thin lens equation.  I have the equipment to measure the equivalent resistance of a parallel combination (i.e. an ohmmeter); I have several "variable resistance boxes," pictured above, which allow the resistance to be varied through a very wide range.  So why not use this equation as my "y-intercept training?"

I took a bunch of resistors in the 5-100 kΩ range, put them on breadboards, and called them the "mystery resistors."  I showed the class how to put the variable resistance boxes in parallel with the mystery resistors, and how to measure the equivalent resistance with the meter.  I initially asked them to graph the equivalent resistance as a function of the box resistance -- this gave a curve. 

Next, I asked them to plot 1/Req on the vertical axis, and the recipricol of the box resistance on the horizontal axis.  This produced a line.  Of course, everyone knew by now to draw a best-fit line; but most folks reflexively took the slope of that line.  It was only after I made them identify the y-axis, x-axis, slope, and intercept using the equation of a line (y =mx +b) that they recognized to use the y-intercept -- the inverse of the y-intercept is the resistance of the mystery resistor.

I will scan in some sample data below, once some folks turn in their labs.  It didn't occur to me until later, and it never occurred to any student that I know of, that the original Req vs. Rbox graph could be used directly to find the mystery resistance:  draw the assymptote as the box resistance gets very large... then read off the vertical axis to get the mystery resistor.  You can tell everyone WHY that works in the comments; I might ask that question as a quiz someday.


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