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07 December 2011

Laboratory quiz question: pressure in a static column

A primary laboratory skill, one that is frequently tested on the AP exams, is determination of the physical meaning of the slope and intercept of a linear graph.  My own approach to such a question is to solve the relevant equation for the vertical axis of the graph, then to identify the variable representing the horizontal axis.  Anything multiplying this variable is the slope of the best-fit line; anything added to this term is the y-intercept.  We religiously go through this process of identifying the slope and intercept of a straight-line graph in every laboratory activity.

However, just doing laboratory work isn't enough to develop this skill.  In a 90 minute lab period or a lab report, a large subset of students will parrot their friends' answers or my suggestions without sufficient understanding.

So then, how do I check for "sufficient" understanding?  I give quiz and test questions that ask directly about the physical meaning of graphs that the class hasn't seen before.  For example, a recent "justify your answer" question showed a graph of weight on the vertical axis, and mass on the horizontal; what is the physical meaning of the slope of that graph?

It was instructive to read the justifications.  Most folks got that the slope is g, the gravitational field.  The stronger students recognized the relevant equation weight = mg; since weight is on the vertical and mass on the horizontal, whatever is multiplying m must be the slope.

The weaker students, though, got the correct answer reasoning from the units of the axes.  The vertical axis, they said, "was" newtons.  The horizontal axis "was" kilograms.  Since we've shown that g has units N/kg, the slope must be g.

I've got to force these weaker students to get away from the crutch of using units to determine a slope's meaning.  While such an approach is better than nothing, often the units of the slope won't obviously match any known quantity; or, a factor of 1/2 or 2π will be missed.  It's not like the method I'm proposing (of first writing the relevant equation) is too difficult for anyone.*

* The correct method does require remembering or looking up the correct equation, though, which is sometimes an obstacle; but that's a separate issue.

Below is a quiz that will help practice the skill of identifying the physical meaning of a slope.  Note that, by this point in the year, if we just graphed GAUGE pressure vs. depth, most of my class would have little trouble seeing that the slope is ρg.  The addition of the Po term in the equation for pressure in a static column causes difficulty.

1.    In the laboratory, you are given a tall graduated cylinder full of fluid, along with a pressure probe which reads absolute pressure.  You submerge the pressure probe in the fluid and record the reading in the probe P at various depths d below the surface.  The pressure at the surface is 1.01 x 105 Pa.

A graph is made of P on the vertical axis and d on the horizontal axis.

(a)    Is the graph linear, or curved? 

o  Linear
o  Curved


(b)   If the graph is linear, explain how the density of the fluid r could be determined from the best-fit line.  If the graph is curved, explain what quantities could be graphed in order to produce a linear graph from which the fluid density r could be determined.

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