Relationship between coefficients of linear and volume expansion

  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. ideasrule

    ideasrule 2,315
    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) get the idea.

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