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Relationship between coefficients of linear and volume expansion

  1. Jan 24, 2010 #1
    1. The problem statement, all variables and given/known data

    If a solid material is in the form of a block rather than a rod, its volume will grow larger when it is heated, and a coefficient of volume expansion beta defined by
    [tex]\beta = \frac{{{V_2} - {V_1}}}{{{V_1}\left( {{t_2} - {t_1}} \right)}}[/tex]
    may be quoted. Here [tex]{V_1}[/tex] and [tex]{V_2}[/tex] are the initial and final volumes of the block, and [tex]{t_1}[/tex] and [tex]{t_2}[/tex] are the initial and final temperatures. Find the relation between the coefficients [tex]\alpha[/tex] and [tex]\beta[/tex].


    2. Relevant equations

    [tex]\alpha = \frac{{{L_2} - {L_1}}}{{{L_1}\left( {{t_2} - {t_1}} \right)}}[/tex]


    3. The attempt at a solution

    I'm assuming I need to set [tex]{V_1} = {L_1}{W_1}{H_1}[/tex] and [tex]{V_2} = {L_2}{W_2}{H_2}[/tex]

    and attempt to extract [tex]\frac{{{L_2} - {L_1}}}{{{L_1}\left( {{t_2} - {t_1}} \right)}}[/tex] from [tex]\frac{{{L_2}{W_2}{H_2} - {L_1}{W_1}{H_1}}}{{{L_1}{W_1}{H_1}\left( {{t_2} - {t_1}} \right)}}[/tex]

    I've only gotten so far:

    [tex]{W_1}{H_1}B = \frac{{{L_2}{W_2}{H_2} - {L_1}{W_1}{H_1}}}{{{L_1}\left( {{t_2} - {t_1}} \right)}}[/tex]

    but I can't figure out the rest of the algebraic manipulation.

    Is this possible, or am I going about the problem incorrectly?
     
  2. jcsd
  3. Jan 24, 2010 #2

    ideasrule

    User Avatar
    Homework Helper

    Now, express L2, W2, and H2 in terms of L1, W1, and H1. Remember that the linear expansion equation, [tex]\alpha = \frac{{{L_2} - {L_1}}}{{{L_1}\left( {{t_2} - {t_1}} \right)}}[/tex], applies for the width and height too.

    A less messy way to do this problem is to write the linear expansion equation as Lf=Li(1+alpha*delta-T). Then LWH=Li(1+alpha*delta-T)*W*(1+alpha*delta-T)...you get the idea.

    That step is correct algebraically, but it gets you farther from the solution.
     
  4. Jan 24, 2010 #3
    You helped me see that I was just over-thinking the problem--I got it figured out. Thank you.
     
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