thermal expansion of a metal plate

**Rohin.T.Narayan** · Mar22-05, 07:16 AM

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

**Andrew Mason** · Mar22-05, 09:37 AM

Originally Posted by Rohin.T.Narayan

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

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) $LaTeX Code: Asingle-quote= A + A\\alpha T$
(2) $LaTeX Code: Asingle-quote-rsingle-quote = (A-r) + (A-r)\\alpha T$

Subtracting (2) from (1):
$LaTeX Code: rsingle-quote = A + A\\alpha T - ((A-r) + (A-r)\\alpha T)$
$LaTeX Code: rsingle-quote = r + r\\alpha T$

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