Thermal expansion of a metal plate

  • #1
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
 

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  • #2
Andrew Mason
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Rohin.T.Narayan said:
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)[tex]A'= A + A\alpha T [/tex]
(2)[tex]A'-r' = (A-r) + (A-r)\alpha T [/tex]

Subtracting (2) from (1):
[tex]r' = A + A\alpha T - ((A-r) + (A-r)\alpha T) [/tex]
[tex]r' = r + r\alpha T [/tex]

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
 
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