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Homework Help: Speed of metal melting through ice

  1. Nov 29, 2011 #1
    1. The problem statement, all variables and given/known data

    Question no. 4 in this document (there's a helpful picture, too):

    2. Relevant equations

    The Clausius-Clapeyron equation:

    [tex]\frac{\delta p}{\delta \tau}=\frac{l}{\tau ∆v},[/tex]

    where v is the volume per unit mass, i.e., the inverse of the density.

    3. The attempt at a solution

    The answer to question a) is simply the change in potential energy of the bar divided by time, which is


    Since the rate of energy transport through the steel bar is

    [tex]\frac{F}{bc}=\frac{Q}{\delta t},[/tex]

    where bc is the surface area of the bar, I think I should be able to set

    [tex]\frac{Q}{\delta t}=\frac{∆U}{∆t}[/tex]

    so that I now have


    From here on I'm pretty lost, though. It seems to me that since ∆z/∆t is the speed, the factor 2mg will end up in the denominator instead of in the numerator. Also, I don't really know what to do with the Clausius-Clapeyron equation. I can see that the term ∆v will eventually give me the fractions of the densities that you can see in the final expression, but it also contains a pressure term that I'm not quite sure I understand. Would that be the difference in pressure between the ice and the water? If so, should I perhaps restate it in terms of energy and volume, i.e., ∆U/∆v?
  2. jcsd
  3. Nov 30, 2011 #2
    No one?

    Perhaps I should state my answer to b) explicitly: the speed with which the bar sinks is


    Is this correct?
  4. Nov 30, 2011 #3
    I'd really need some help.

    Substituting [tex]∆\tau[/tex] in the last expression for

    [tex]\frac{\delta p ∆v\tau}{l}[/tex]

    (a rearrangement of the Clausius-Clapeyron equation, and using the fact that δT and [tex]∆\tau[/tex] are equal), I get

    [tex]v=\frac{\kappa bc\delta p \tau}{2mgal}(\frac{1}{\rho_i}-\frac{1}{\rho_w})[/tex]

    It kind of resembles the correct answer, but as I said, the term 2mg is in the wrong place, and I don't know what to do with δp.
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