Fluids for AP Physics 1: buoyant force demonstration and/or lab

I have an aluminum cylinder here.  I hang the cylinder from a string, and attach the top end of the string to a force probe.*  The probe reads 1.1 N.  

*Or a spring scale.  This particular experiment can be done with 1960s equipment.

Next, I am planning to keep the cylinder attached to the force probe, but submerge the cylinder completely in a beaker of water, without the cylinder touching the bottom of the beaker.  What will be the reading in the force probe when the cylinder is submerged?

This is a force problem.  Even though I might be doing this demonstration during the new AP Physics 1 fluids unit, it's still a force problem.  And thus the starting point is a free body diagram, regardless of the exact question being asked.

The free body for the cylinder includes an upward tension T, an upward buoyant force Fb, and a downward force of the earth mg.  The cylinder will be hanging in equilibrium, so up and down forces balance: T + Fb = mg.  I'm looking for the reading in the force probe, which is the tension in the string.  Solving for tension gives T = mg - Fb.  

In this case, we already know mg, the weight of the cylinder, because of the initial force probe reading before we submerged the cylinder: mg = 1.1 N.

The buoyant force on a submerged object is equal to the weight of the displaced fluid.  This is written mathematically by the equation Fb = (density of fluid)(volume submerged)(g).  I write words rather than variables here because it's so easy to get the wrong density, or the wrong volume.  Generally, density times volume gives mass, and mg gives weight.  The mass of the displaced fluid is the density of the fluid times the volume of the displaced fluid.  

Well, we know the density of water: 1000 kg/m^3.  (See the previous post for a brief digression about "as much as you can hug.")

But how can we figure out the volume of this cylinder?  I ask the class for ideas.  There's no one right answer; and this creative experimental brainstorming is exactly the kind of practice that can help students approach AP Physics lab questions.

Idea 1: It's a cylinder, which has volume equal to the area of the base times the height.  So take a ruler and measure the diameter (and thus the radius) of the base; measure the height.  The volume is pi*r^2*h.  Excellent.  Any other thoughts?

Idea 2: Use water displacement.  Pour water into a narrow graduated cylinder.  Look at the initial volume reading when the cylinder isn't submerged; look at the final volume reading when the cylinder is fully submerged.  Subtract those volumes to get the volume of the cylinder.  Great.  Any further thoughts?  Anyone?  

Idea 3: You said it was an aluminum cylinder.  We can look up the density of aluminum; we know the mass by dividing the cylinder's weight by g, meaning the mass is 110 g.  The cylinder's volume is its mass divided by its density.  Fantastic!  Any OTHER ideas?  No?  

My idea: I look at the cylinder, or at least the part of the cylinder that was facing away from the audience.  "It says right here on the cylinder, written clearly in permanent marker, "42 mL".  Okay, okay, it's a trick.  Ha ha ha.  The point has been not to get the right answer, but to have this particular conversation.  I've got the accurate volume pre-measured (I pre-measured using the water displacement approach), the same way a cooking show will have a soufflé pre-baked so the audience doesn't have to sit through the, um, less exciting parts of the cooking process.

I use google to find that 42 mL is equal to 4.2x10^-5 m^3.

And now we can do the buoyant force calculation.  The buoyant force is (1000 kg/m^3)(4.2 x 10^-5)(10 N/kg) = 0.42 N.  The reading on the scale will be (1.1 N - 0.42 N) which should be about 0.7 N.  

Sure enough, I press collect on the app connected to my force probe, submerge the cylinder... and the reading drops from 1.1 N to 0.7 N.  

Physics works.

Extensions:  So many great questions you can ask.  Does submerging halfway down make the buoyant force greater than, less than, or equal to 0.21 N?  

How would the scale reading change if the cylinder touched the bottom of the beaker?

What if you put the beaker on a platform scale and submerged the cylinder, without allowing it to touch the bottom of the beaker?  What would the platform scale read?  This one's rather complex.  I analyze this question here.