# Homework Help: Heating a steel plate with a hole

1. Jan 26, 2014

### Nikitin

1. The problem statement, all variables and given/known data
If a steel plate with a hole of 10 cm in diameter is heated 35 degrees kelvin, what will the new diameter be? $\alpha_L = 13 \cdot 10^{−6} K^{−1}$

2. Relevant equations
$\Delta L = \alpha_L \cdot \Delta T \cdot L$
3. The attempt at a solution
If I understand the length-expansion formula correctly, all the dimensions of the substance should change by a factor $\Delta L /L$ (why?).

Then the diameter will then increase in size to $L+\Delta L$? Uhm, how so? Considering there is a void inside the plate, and the rest of the plate is expanding, shouldn't the plate also expand into the hole? But according to the formula, the plate should expand in the same way as if the hole did not exist!

Can somebody explain how this works? I'm new to thermodynamics.

Last edited: Jan 26, 2014
2. Jan 26, 2014

### BvU

Consider the difference between a) making a ring from a steel rod, then heating, and
b) heating a rod of the exact same length, then making a ring of it.

3. Jan 26, 2014

### Nikitin

There is no difference.

But my problem here is, that according to the formula the plate from the OP will NOT expand inwards, despite there being a void there. It will only expand outwards. This makes no sense to me.

4. Jan 26, 2014

### BvU

As you say, there is no difference.
Like the rod, the inner ring around the hole wants to get longer. Going inward would make it shorter.

5. Jan 26, 2014

### Staff: Mentor

To expand on what BvU is saying, the circumference of the hole is one of the "lines" in the steel plate. Since all lines in the steel plate must increase in length by the same percentage, so must the circumference of the hole. If the circumference gets larger, so must the diameter of the hole.

Chet

6. Jan 27, 2014

### BvU

Sense, intuition, gut feeling... Very important in science. A matter of having the right mental model in your head. So let me try another tack:

Draw (or imagine) a 2D grid of 1 cm x 1 cm squares. Our simplest possible model for a solid. The grid points are the Fe atoms, the 1 cm connecting lines represent the distances between neighboring atoms. They are like springs at equilibrium: compress and the atoms repel each other more, expand and they attract more.

Cut out a circle around the center. Or color the grid points and strings within such a circle.

Now for some reason the equilibrium state of the springs shifts to a little bigger distance between atoms when the temperature goes up. The expansion coëfficiënt tells us how much.

In physics (and some other areas as well) it's nice to exaggerate 'a bit' and wonder what if...
So let's exaggerate the expansion coëfficiënt and the temperature rise and extend all connecting lines to 11 mm. Draw or imagine again. What happened to the circle ? Without any tension, shear, warping, whatever, everything one-dimensional got 10% bigger: diameter of the circle, circumference, you name it.

2D is too simple for reality, I hear someone mutter. Well, now it gets interesting. Use our imagination or do a lot of ("3D") drawing, doodling with steel balls and springs, or whatever.

Do you agree that a 10% increase in all connecting springs lets the diameter of a cut-out cylinder go up by 10% as well?

So why did your senses tempt you to think of a kind of swelling inwards? (And you are definitely not the only one. If physics were a democratic process, you'd probably win the elections).

Suppose, just suppose, there is a fixed layer above and a fixed layer below our 3D grid. (Fixed z)With the same springs vertically welded to 1 cm grid points between fixed layers and our springy 'solid'. Suppse we heat up this contraption - without the fixed layers giving in even the slightest bit - so much that all springs are 11 mm again. This time we get quite a bit of swelling inwards, just as you intuitively expected!

Only, that wasn't the situation described in the exercise....