Experiment: density of mystery fluid, and the audience for a lab writeup

Materials for the "density of unknown liquid" experiment

Fluid mechanics first entered the AP physics B course description in 2002.  That year, the laboratory question (#6, I don't have a legit link, but it's easy enough to look up) asked students to determine the density of an unknown liquid by submerging a mass on a spring into that liquid.  

My own classes did a version of that problem on their trimester exam.  The ones who got it wrong -- usually by conflating the density of the liquid with the density of the submerged mass -- did an exam correction on which they described the procedure correctly.  

And then this week, I had everyone actually, honest to goodness, do the experiment for themselves.  I've often suggested that the AP exams since 1996 provide a wonderful laboratory guide, if you can improvise a bit.  Pick a lab-based question, set it up with whatever equipment you have lying around, and there you have a college-level physics laboratory exercise.  I took my own advice and tried out this new experiment.

I gave each group a beaker of fluid and a mass.  (You can see the cubical masses in the picture -- they all are made of different materials.  I just dug them up in an old storeroom.  I have no idea where they came from.)  The groups were encouraged to pick the spring of their choice.  They could use any other equipment they wanted, including a balance scale; the only action I forbade was to directly measure the mass of the fluid in the beaker.

Now, most of my experiments, and many AP lab questions, call for a linear graph, a best-fit line to copious data on the graph, and interpretation of the physical meaning of the line's slope and intercept.  This particular experiment doesn't lend itself well to a graph, at least not the way it's presented on the 2002 exam.  So I came up with an alternative approach to the writeup.

I used a true mystery liquid rather than water.  I won't reveal what I used (because my students occasionally read this, and the writeup isn't due 'til the end of the week), but I don't even know its density right now.  We're going honestly double-blind here.

I'm asking each partnership to write up one typed page describing their results.  I give no specific instruction other than to imagine that they've been hired to determine this mystery density, and that their financial well-being depends on the quality and accuracy of their work.  They get one shot to impress their potential client with their writeup.

That "client" is Peter, the captain of our USIYPT research physics tournament team, and the rest of his class.  I will collect the typed pages with no names, and hand them to Peter and the research students.  They will rank the papers from best to not-so-best.*  I'll assign grades based on the rankings, and give a prize to the winners.

*Donald Trump would say "worst."  Why is it that teachers tend to get in trouble for such language, while Mr. Trump is lauded for his bluntness?

Sean measures the extension of the spring

Sure, this is a nice cutesy little game.  But there's a real message here.  Too often when students are asked to describe the results of an experiment, they use stilted, overly-formal language that stifles meaning in favor of big-arse words.*  I want them instead to write informally for an audience at their own level of physics.  A major obstacle to improved writing is the faceless audience.  High school students have never published anything; their understanding of a paper's audience is poor even in English class, let alone in a subject where they struggle both with the content *and* the writing skills.

*  "In this experiment, the experimentors carefully and consistently used a decimal-labeled wooden shaft to record precisely the extent to which the PASCO brand spring was extensively extended.  We ensured safety by wearing goggles and grounded electrical outlets." 

So, I put a face to the audience:  Peter.  Everyone knows Peter.  They talk to him in normal language.  They are not in fearful awe of him, but they are all quite clear that he and the research team have no use for incorrect physics.  (I wonder where they got that from...)

I've never done this particular exercise; but I have had students write with a named fellow student as their audience.  Their writing doesn't become perfect, but some of the filler gets filtered out.  And so we focus on the physics... which is what I want, anyway.

How many soda bottles in Brian's raft?

The question from Dec. 14:

Mr. Jacobs’ friend Brian Jackson saved two-liter soda bottles throughout his senior year of college.  During “Haverfest," he duct taped the bottles together to form a raft.  He then successfully floated himself out onto the duck pond.

Estimate how many bottles Brian used.  Explain your reasoning thoroughly and show all calculations for full credit.

While the majority does not always rule in physics, in this case "they" were right on.  My reasoning:

Call Brian 80 kg or so.  His weight is then 800 N.  That weight must be supported by the buoyant force, which is equal to the density of water times the displaced volume times g.  If each bottle is fully submerged, it displaces 2 L, or 0.002 cubic meters.  The buoyant force created by one bottle is then (1000 kg/m^3)(0.002 m^3)(10 N/kg) = 20 N.  To get to 800 N at 20 N per bottle, you'd need about 40 bottles.

What if Brian's not 80 kg?  Well, as I have to point out to people, 80 kg is a reasonable estimate for Brian, but college guys who drink soda are often heavier; and, in a recent development of Haverlore, I have discovered that Brian supported a second person on the raft as well.  Furthermore, even if Brian were 75 kg, 40 bottles would have to be nearly fully submerged, leaving essentially no safety margin, and getting Brian's feet* wet.  This is an order of magnitude estimate... why not double the estimate to 80 bottles or so?  Then the bottles are in the neighborhood of halfway underwater.  Brian can sit dry, he can bring a friend, he can eat at the COOP** all he wants; 80 bottles will support him.

* Or more likely, his tuckus

** The yummy snack bar... it used to be too expensive for me, but now I find out that students can make their parents pay for the COOP as part of these newfangled meal plans.  Ach, and nowadays students can access email from their rooms, too.

What about the weight of the soda bottles themselves?  Some students will tell me "the soda bottles are of negligible weight."  Okay, but are they?  What's the evidence?  

Some students found that an empty bottle has mass about 40-50 g, for a weight of about half a newton or so. That means that each bottle will only support 19.5 N of Brian rather than the 20 N previously conjectured.  

Does that mean, as some say, that the proper answer is "41 bottles?"  No, certainly not.  As discussed in the previous paragraph, the uncertainty in Brian's mass, and in just how much of the raft is submerged, far outweighs this 3% change due to the weight of the bottle.  The answer is still somewhere around 80 bottles, or better yet, some dozens of bottles. 

Soda Raft Question

And if any soda company would give me money, I'd use their
brand name in the problem statement. :-)

The following is a true story.  I use it as a problem in static fluids every year.  This year I assigned it when I was absent -- an upcoming post explains how I assigned the problem in class.  

For now, though, look at the poll at the left of the blog -- vote for your estimate!

1. Mr. Jacobs’ friend Brian Jackson saved two-liter soda bottles throughout his senior year of college.  During “Haverfest," he duct taped the bottles together to form a raft.  He then successfully floated himself out onto the duck pond.

Estimate how many bottles Brian used.  Explain your reasoning thoroughly and show all calculations for full credit.

Buoyant Force problem and demonstration with a crazy answer

A fluid mechanics problem I often assign is based on a problem from, I think, one of the older Serway editions.  It shows a beaker full of oil sitting on a platform scale.  A bar of iron is suspended in the oil from a rope which is attached to a spring scale.  The problem asks:

A 1.0 kg beaker containing 2.0 kg of oil (density = 916 kg/m3) rests on a platform scale. A 2.0 kg block of iron is suspended from a spring scale and is completely submerged in the oil.

(a) Which scale reading should be larger (or should they be the same)? Explain conceptually.

(b) When the iron is in equilibrium, what is the reading in the spring scale?

(c) When the iron is in equilibrium, what is the reading on the platform scale?

Most everyone gets the idea that the platform scale reads a bigger force -- after all, even without considering anything tricky (like fluid mechanics), the spring scale seems to read just the 20 N weight of the iron, while the platform scale seems to read the 30 N weight of the beaker/oil.  A bit more logic with buoyant forces convinces the students that the spring scale must read LESS THAN 20 N, because of the upward buoyant force on the iron.  No problem.

 

Part (b) is similar to a demonstration from class, and numerous example and practice problems in texts.  They know to draw a free body, calculate the buoyant force using Archimides' principle, and use the free body to calculate the tension in the string connected to the scale.  The only halfway tricky part is finding the volume of the iron, which is easily done once the density of iron is looked up.  The buoyant force is about 2 N in this case.

 

Part (c) is the part that causes trouble.  Most of the class, at least initially, says that the reading on the platform scale is just 30 N -- the weight of the oil plus the weight of the beaker.  Others get the right answer of 32 N, but for crazy reasons.  Some come to the conclusion that since the oil "lost" the 2 N buoyant force, that we must return these 2 N to the oil through the reading on the platform scale by conservation of force.  Others simply draw the buoyant force acting down directly on the beaker.  Many make no argument whatsoever, but just add in 2 N, presumably because their friends told them to and they couldn't quite explain it. 

 

I'm glad that so many students have the physics instincts to recognize that 30 N can't be right.  A few will say they made a lucky guess, but I consider such a guess good physics intuition.  However, only a very few students get the justification for why the platform scale reads 32 N.  Do you know?

 

It's Newton's Third Law.

 

The buoyant force is the upward force of the oil on the iron.  Therefore, there must be a downward force of the iron on the oil.  When we consider the oil-beaker system, the downward forces sum to 32 N, including the weights of the oil and beaker, and the third law companion force to the buoyant force.

 

Do you believe me?

 

My students don't, at least not if they didn't get the answer right in the first place. So I set up a similar situation.  The picture at the top (credit to Frederic Lamontagne, WFS class of '11, for the photography) shows a beaker containing a submerged aluminum weight, just like in the problem.  When I remove the weight from the water, the reading in the spring scale increases, but the balance scale goes out of balance!  I have to rebalance the scale to make up for the removal of that downward force of the aluminum on the water.  Since the spring scale reading increased by 0.2 N, I had to add about 20 g to the balance scale reading.  Physics works.