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Homework Help: Thermal expansion of a metal plate

  1. Mar 22, 2005 #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
  2. jcsd
  3. Mar 22, 2005 #2

    Andrew Mason

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    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.

    Last edited: Mar 22, 2005
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