Mail Time: Hole in an expanding steel plate

Joe Konieczny, who is now at the Walker School in Atlanta, writes in with a question about thermal expansion.

Hey Greg, this is my first year actually teaching thermodynamics to an AP class.  In previous years I just haven't had time (unless we had a Saturday school crash course in thermo) and it is also my weakest area of understanding in physics.

Joe has a class this year with some previous background in physics, so he can move more quickly than he did at other schools.  I've always made a different choice -- I think there's enough material that students of modest ability can handle to justify making thermodynamics a priority.  When I'm pressed for time, I ditch atomic physics, or perhaps induced EMF or magnetic fields created by wires.  But that's me.

  I asked my students a question I thought I understood, but it turns out I don't!  See the question below:

A square steel plate with sides of length 1.00 m has a hole in its center 0.100 m in diameter.  If the entire plate is heated to such a temperature that its sides become 1.01 m long, the diameter of the hole will be

(A)0.090 m         

(B)0.099 m         

(C)0.100 m         

(D)0.101 m        

(E)0.110 m

The answer is D.  The dimensions of the plate increase by 1%, so the diameter must increase by 1% as well.  Is that the correct explanation?

Yes.  By the thermal expansion equation, all of the steel everywhere expands by the same fraction, ΔL/L which equals αΔT.  

  My students expected that the hole would "expand", which to them meant expanding "in", and therefore the diameter would decrease by 1% to 0.099 m.  What is the error in that explanation?  Thanks.  

Yeah, that's a classic question.  The hole expands outward, too.  Three ways to think of it:

(1) Consider a long, thin rod bent into a circle.  Thermal length expansion means that heating the rod expands the rod's length.  Well, the length of the rod is the circumference of the circle.  Making the circumference larger also increases the circle's radius by 2πr.  Increasing a circle's circumference certainly doesn't *decrease* its diameter.

(2) Imagine that instead of cutting a hole, you just drew a circle on the steel.  Would the circle expand or contract?  Of course it would expand!  Turns out, it doesn't matter if you cut that material out, the principle is the same.

(3) I keep a blob of silly putty in the back of the room just for the purpose of answering this question.  Draw a circle on the silly putty.  Then stretch the silly putty out in all directions.  The circle... expands.  Then do the same thing with a cut-out circle in the silly putty.  The circle gets bigger, in an obvious manner that you can watch happen.  Even if students aren't buying the first two abstract arguments above, they'll quit arguing when they stretch the silly putty.