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Heat and increasing a rod's length

  1. Apr 3, 2015 #1
    1. The problem statement, all variables and given/known data
    A 1.0m long rod of metal has a diameter of 0.75 cm. This metal has a coefficient of linear expansion α = 4.8 × 10−5 1/K , a density of 9.7 × 103 kg/ m3 , and a heat capacity of 390 J/K . How much energy needs to be added as heat to increase the length of the rod by 7.5 × 10−3 m?
    2. Relevant equations
    V0=πr2h
    ΔV=βV0ΔT
    m=vρ
    3. The attempt at a solution
    V0=πr2]h
    =π(0.0075/2)2(1)
    =4.42×10-5

    V0=πr2h
    =π(7.5×10-3+1)(0.0075/2)2
    =4.45×10-5

    ΔV=βV0ΔT
    ΔT=ΔV/(3αV0)
    =(4.45×10-5-4.42×10-5)/(3(4.42×10-5))

    m=vρ where do I go from here? Because the the mass would always be changing
     
  2. jcsd
  3. Apr 3, 2015 #2

    NascentOxygen

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    Staff: Mentor

    The mass is fixed. Probably the density will change.
     
  4. Apr 3, 2015 #3
    So you would do M=∫ρdV to find the mass?
     
  5. Apr 3, 2015 #4

    NascentOxygen

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    Staff: Mentor

    You know the rod's dimensions and its density, so can determine mass. This mass doesn't change when you apply heat to expand it.

    I'm wondering why you involve ΔV when all that is required is ΔL? I could be wrong, but it seems that you are making the problem more difficult than is intended.
     
  6. Apr 3, 2015 #5
    Ok thanks! I'll try it with just change in length but doesn't the volume change since the length is changing?
     
  7. Apr 3, 2015 #6

    NascentOxygen

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    Staff: Mentor

    Certainly volume will change when length and diameter change. But the problem specs centre on the change in length.
     
  8. Apr 3, 2015 #7
    So I should use this volume:
    V0=πr2h
    =π(0.0075/2)2(1)
    =4.42×10-5
    ....because I'm not getting the right answer
     
  9. Apr 3, 2015 #8
    You are attacking the problem backwards. How much of a temperature increase do you need to increase the length by the desired amount? What is the mass of the rod?

    Chet
     
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