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Relationship between coefficients of linear and volume expansion 
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#1
Jan2410, 05:11 PM

P: 2

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
Jan2410, 08:29 PM

HW Helper
P: 2,323

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


#3
Jan2410, 11:10 PM

P: 2

You helped me see that I was just overthinking the problemI got it figured out. Thank you.



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