thermal expansion of a metal plate

[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

[Andrew Mason]

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 expansionsYou 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