Fluids for AP Physics 1: pressure in a static column, demonstration and/or lab

Fluid mechanics is trading places.  Since 2015, fluids has been part of the P2 curriculum, for the 25,000 or so students who take that exam.  But next year, fluids moves over to P1 and its 150,000 students.  So now's probably a good time to share some thoughts about teaching fluids.  

I do recommend coming to an AP Summer Institute, where you can see me do this and other fluids demonstrations live!  You can see my APSI schedule in the left sidebar.

Pressure in a static column

When a tank of fluid is not moving, the pressure anywhere in the fluid is given by P =P0 + ρgy. Here P0 represents the pressure at the surface* of the fluid, ρ is the fluid density, and y is the depth below the surface at the position where you're measuring pressure.  

*Yes, this is *usually* atmospheric pressure... but not always.  Consider one fluid on top of another.  The pressure in the bottom layer is the pressure at *its surface* plus ρgy.

I have one of them giant graduated cylinders filled nearly to the top with water.  My pressure sensor is connected to my ipad under the document camera; I set the Vernier Graphical Analysis app to produce a graph of pressure vs. time.

I attach a long tube to the pressure sensor.  First, I read the pressure when the tube is NOT submerged - this is P0.  The sensor generally reads in kPa; we want a reading in Pa, where 1 Pa is equal to 1 newton per square meter.

Next, I announce that I will predict the pressure sensor reading when the tube is submerged to the very bottom of the container.  The relevant equation is P =P0 + ρgy.  What additional information do we need?

We have the surface pressure - usually around 102,000 Pa, but that varies by weather and especially altitude.  We can measure the depth y with a meterstick - this is usually about 18 cm, i.e. 0.18 m.  We know the gravitational field g to be 10 N/kg.  

What about the density of water?  

AP Physics 1 does not require unit conversions!  The number of times that a student is even asked to give a numerical answer to anything is minimal.  So for this in-class problem where conversions are necessary, I do the conversions in my head and state the result; or I use google ("convert 102 kPa to Pa").  De-emphasize number crunching, and you'll un-de-emphasize concepts.  :-)

That said, thought experiments are well within the scope of AP Physics 1.  Students don't hesitate to volunteer that the density of water is "one."  One what?  The standard units of density are kilograms per cubic meter.  So one kilogram per cubic meter for water, right?

How big is a cubic meter, anyway?  That question provokes several seconds of pure puzzlement, followed by students waving their hands about a meter apart.  "It's as much as you can possibly hug," I say.  Close enough.

So, consider a tank filled with a cubic meter of water - as big a tank as you can possibly hug.  Can you lift that tank?  Oh.  No way.  Well, you can lift a kilogram without trouble - here, catch this 2 pound thing.  That huggable tank doesn't weigh 2 pounds.  No, it weighs a ton.  The density of water is a THOUSAND kilograms per cubic meter.

Back to the calculation.

The gauge pressure at the bottom of the cylinder - gauge pressure means just the ρgy bit, ignoring P0 - is (1000 kg/m^3)(10 N/kg)(0.18 m).  That's 1800 Pa, or 1.8 kPa.  

Thus, the pressure sensor reading should increase from about 102 kPa to 104 kPa.  The actual sensor has a precision in the neighborhood of +/- 0.2 kPa, so this difference will be pretty much dead on when you send the tubing to the bottom.

Follow-up questions include "what's the gauge pressure halfway down?   Greater than, less than, or equal to 0.9 kPa?  Answer?  Equal to: because in ρgy, y is in the numerator and neither squared nor square-rooted, halving y halves the gauge pressure, too.  Sure enough, submerging the tube halfway verifies this prediction.

Physics works.

Make this a laboratory investigation!

Put pressure on the vertical axis, depth on the horizontal.  The slope of the graph should be ρg, and the y-intercept should be atmospheric pressure.  You can use the slope to determine the density of water.  Or, use a different fluid like vegetable oil, and you'll determine the density of the mystery fluid!