I decided to do a lab out of order of the textbook today. There were two reasons for this:

- I had just gotten a bunch of physics toys that I had ordered from Arbor Scientific.
- I wanted the students to start working on designing a water-bottle rocket simulation and experiment.

One of the toys I had bought was an “Elasticity of Gases” demo. This simple device has a large syringe with a plug that you can screw into the tip, and a pair of sturdy fiberboard disks to act as a stand and a load platform.

I had the students pour water into a bucket, weigh bucket, then balance it on the platform, reading the volume in the syringe.

Here is the data as they recorded it:

# Pressure-volume relation for air in a cylindrical plunger # The cylinder has a cross-sectional area 6.64 square cm # Mass of applied weight (kg), Volume of gas (cubic cm) 0.0 50 1.56 42 2.49 38 3.39 35 5.385 29

I then challenged their data, since pressure times volume is supposed to be a constant. It took a bit more nudging than I had expected for them to realize that there was atmospheric pressure at the beginning. Then one of the students did a quick fit with gnuplot and got that everything fit fairly well.

We tried measuring the cross-sectional area of the syringe two ways: measuring the inside diameter of the syringe with calipers and measuring the scale on the side of the syringe. The calipers produced a substantially smaller estimate of the cross-sectional area, probably because the calipers couldn’t reach past the bump on the end of the syringe that makes it difficult to pull the plunger all the way out. We ended up using the estimate from measuring the scale on the side of the syringe.

I redid the gnuplot script to use proper metric units:

set title "volume vs. pressure" set xlabel "applied pressure (N/m^2)" set ylabel "volume (ml)" set key top right area=6.64E-4 # cross-sectional area of cylinder in m^2, # estimated from measuring the length of the scale on # the syringe: 7.53 cm for 50 ml g = 9.7995 # local gravitational field in N/kg # according to the Wolfram Alpha's gravitational widget # http://www.wolframalpha.com/widgets/view.jsp?id=d34e8683df527e3555153d979bcda9cf press(mass) = mass*g / area # pressure in N/m^2 (Pascals) b=1e4 a=50*b fit a/(x+b) 'press-vol.gnudat' using (press($1)):2 via a,b # The fit got # a = 5.66175e+06 +/- 1.398e+05 (2.47%) # b = 112736 +/- 3338 (2.961%) platform = 0.150 # platform weight 150g d=102800 # barometric pressure of 102.8kPa measured at # http://www.wunderground.com/weatherstation/WXDailyHistory.asp?ID=KCASANTA133 fit c/(x+press(platform)+d) 'press-vol.gnudat' using (press($1)):2 via c f=1e4 e=50*f fit e/(x+press(platform)+f) 'press-vol.gnudat' using (press($1)):2 via e,f plot 'press-vol.gnudat' using (press($1)):2 title "data", \ a/(x+b) title "fit PV and barometric pressure", \ c/(x+press(platform)+d) title "fit PV, barometer=102.8kPa, 150g platform"

Gnuplot’s fit was

a = 5.66175e+06 +/- 1.398e+05 (2.47%)

b = 112736 +/- 3338 (2.961%)

so the initial pressure in the cylinder is estimated as 112.7kPa (the ± 3.3kPa is bogus, as that just reflects the error in the fit, not the error in the measurements—the volume measurements were probably about ±1ml, which would be a 2–3% variation). According to a local weather station the barometric pressure at the time was 102.8 kPa, so our correction of 112.7kPa was about 10% too high—perhaps some of the problem was due to the weight of the platform, which could have increased the pressure so that the 50ml corresponded to a pressure a little higher than atmospheric. Adding the force for the 150g platform reduced the estimate of barometric pressure to 110.5kPa, which is still 7.5% too high. It might have helped to have more measurements, averaging out some of the noise, to get better estimates of atmospheric pressure. (Masses around 500g, 1kg, 4kg, and 6kg would have been useful to add, but we ran out of time.)

The pressure vs. volume theory is not until Chapter 13 (Gases and Engines) and we’re still on Chapter 11 (Angular Momentum). But I really want the students to design and execute a soda-bottle-rocket lab. I even bought a medium-priced launcher from Arbor Scientific, instead of having them use the much cheaper friction-fit launcher I’ve had for the past 5–10 years, so they could get more repeatable launch pressures. I considered making my own trigger-release launcher, but at $26 the launcher was cheap enough not to entice me to build my own. (I probably could have done it for under $10, but I don’t have much time over the next 2 weeks.)

We haven’t done the full ideal gas law—we’ve left temperature out of the equation for now. We’ll get to that when we do Chapter 13. But the adiabatic simplification (constant temperature) is good enough for now.

I don’t believe that they can come up with reasonable equations for the force expelling water from the bottle and for the mass of the water still in the bottle, unless they can model the volume of the air and the pressure in the bottle as they change with time. I think that this will most likely have to be a computational model, rather than an analytic one, as too many things are functions of time. Maybe everything simplifies so that analytic solutions are reasonable, but I suspect not.