1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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


    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook