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## 06 July 2014

### Physics 2 Argument: What happens to the water level? Answer in the comments.

In the picture you see a 200 g mass floating inside a very light cup.  The water is inside a large beaker.  The water level is marked on the side of the beaker.

I'm going to remove the 200 g mass from the cup and place it back in the water on the bottom of the beaker.  How does the new water level compare to the marked level?  Give your reasoning in the comments.

In the most recent post, I explained about arguments and the new AP Physics 1 and 2 courses.  We absolutely must get our students discussing physics, arguing about physics, making errors and catching errors.  I personally think in terms of "physics fights", the ritual debates over research-style problems that underlie the US Invitational Young Physicists Tournament.

In order to argue about physics, first we've gotta have something specific to argue about.  We need problems that are simple enough to be accessible to first-year physics students, problems that are within the scope of AP Physics 1 and 2; but also, these problems must be complex enough to, well, produce reasonable and legitimate disagreement about physics. Arguments can't be artificial.  If you present a ridiculously bogus line of reasoning to your class, they not only won't engage, they won't even buy in to the necessary process of discussion.

My primary piece of advice is to be flexible in your teaching.  When a problem organically provokes a good discussion, go with it!  Engage that authentic argument until it's resolved, even if you hadn't anticipated it.  Similarly, if a problem that you expected to be tough gets the whole class nodding their heads in agreement, just move on.

Last week at my Mahopac, NY institute, the problem above provoked an unplanned but long and deep discussion.  I heard four different lines of reasoning from the participating physics teachers.  Five minutes of talking amongst themselves failed to resolve anything.  So we kept talking.  It was tough for some of the teachers to articulate their reasoning; others articulated clearly, but didn't convince their colleagues.  I'm an old debate coach... I noticed how a bunch of teachers knew that an argument was incorrect, but couldn't address precisely why it was incorrect; all they could do was reiterate their own argument.  (In debate we call that a failure to "clash.")

Of course, what I love about physics over debate is that once the discussion petered out, we just did the experiment.  Nature is the ultimate judge, not nine political appointees.

So what's your answer?  Please post a comment with your line of reasoning.  Don't be afraid to be wrong -- all Jacobs Physics readers are teammates, we all love each other like brothers and sisters.  The only justification I don't want -- for now -- is "I did the experiment, and this is what happened."  Comments are moderated so we don't get linkspam, so it'll take a few hours for me to post them.

GCJ

1. Answer: the new water level is lower.

The displaced water for the floating cup is 200ml (corresponding to the 200g). This displaced water has to fit between the beaker and the cup, so the water rises by 200ml divided by the difference in cross-sectional area of the beaker and the cup.

The displaced water for the mass is the volume of the mass, or 200ml, whichever is less. If the density of the mass is greater than water, then the mass will remain on the bottom of the beaker displacing less water than it did when it was floating. This smaller volume is divided by the full cross-sectional area of the beaker, making the water rise even less.

If the density of the mass is less than water, then it will bob up to the surface when you let it go, and the displaced water will be 200ml. The rise will be 200ml divided by the difference in cross-sectional area of the beaker and the mass, which is a larger area than the difference between the beaker and the cup, since the mass fit inside the cup. So again the rise is less.

2. The water level should be less as the mass on its own would have a smaller displacement than when it is in the cup. This seems similar to the 2013 AP B problem with the anchor and sailboat. (Although that problem went in a different direction)

3. Answer: When the mass is moved to the bottom of the beaker, the water level will go down.

Rationale: Archimede's Principle says that the buoyant force is proportional to the weight of the displaced fluid. When the mass is in the cup, there is a buoyant force equal to the weight of the mass (2N). When the mass is moved to the bottom of the beaker, there is now a normal force acting on it. This decreases the buoyant force (less than 2N), so the amount of displaced fluid decreases, meaning the water level lowers.

4. It takes less water to stand in a pool than to float in it. Normal force to the rescue!