# Thermal expansion of a metal plate

1. Mar 22, 2005

### Rohin.T.Narayan

Hi guys ,I am stuck up with the following problem,can you please give me a solution.

When a metal plate with a circular hole at its centre, is heated, definitely along with the areal expansion of the plate the diametre of the circular hole also increases .But can you give a mathematical proof for this using the differential equations of coefficients of expansions

2. Mar 22, 2005

### Andrew Mason

You don't need to worry about differential equations. It is a linear expansion.

The sides of the plate are A and B. Divide it into quarters. The length of the sides with the quarter hole taken out of them are A-r and B-r. Heat it up. The A side and holed A side will be:

(1)$$A'= A + A\alpha T$$
(2)$$A'-r' = (A-r) + (A-r)\alpha T$$

Subtracting (2) from (1):
$$r' = A + A\alpha T - ((A-r) + (A-r)\alpha T)$$
$$r' = r + r\alpha T$$

Do the same thing for the B and B-r sides.

So the hole radius increases at the same linear rate as the metal.

AM

Last edited: Mar 22, 2005