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Thread: Another Newton's Law of Cooling/Warming

  1. #1
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    Another Newton's Law of Cooling/Warming

    A small metal bar, whose initial temp was 20C, is dropped into a large container of boiling water. How long will it take the bar to reach 90C if it is known that its temp increases 2 degrees in 1 second? How long will it take it to reach 98C?

    With this problem, I'm not sure where to even begin, other than with the equation

    dT/dt = k(T-Tm)

    What is the initial temperature, 20C at T(0)? How does 2 degrees in 1 second play into it? How do I find k? The book gives no examples, so I'm not sure how to even start.

    Thanks in advance,
    --Hank

  2. #2
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    Re: Another Newton's Law of Cooling/Warming

    Quote Originally Posted by hank
    A small metal bar, whose initial temp was 20C, is dropped into a large container of boiling water. How long will it take the bar to reach 90C if it is known that its temp increases 2 degrees in 1 second? How long will it take it to reach 98C?

    With this problem, I'm not sure where to even begin, other than with the equation

    dT/dt = k(T-Tm)

    What is the initial temperature, 20C at T(0)? How does 2 degrees in 1 second play into it? How do I find k? The book gives no examples, so I'm not sure how to even start.

    Thanks in advance,
    --Hank
    Please check the problem carefully for accuracy. By any cance did the problem say

    "...that its temp initially increases 2 degrees in 1 second..."
    “... mathematics is only the art of saying the same thing in different words” - B. Russell

  3. #3
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    Re: Another Newton's Law of Cooling/Warming

    Quote Originally Posted by Subhotosh Khan
    Please check the problem carefully for accuracy. By any cance did the problem say

    "...that its temp initially increases 2 degrees in 1 second..."
    Thanks for getting back to me.
    I double checked the problem, and what I typed is correct.
    Perhaps the book made a typo?

    The answers given in the back of the book are 82.1s and 145.7s.

  4. #4

    Re: Another Newton's Law of Cooling/Warming

    i have the same problem.
    I think I solved it, but the answer isn't exact.
    I keep getting my rate as 0.025
    I know that's the problem but I can't seem to find another rate.
    If i modify the rate to 0.02531 it works out well.
    But I have not found a way to get THAT rate.
    thx

  5. #5
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    Re: Another Newton's Law of Cooling/Warming

    However, if the problem is correctly stated (as posted), then

    dT/dt = 2 = constant

    so

    T = T[sub:34llk5uc]o[/sub:34llk5uc] + 2 *t

    T = 20 + 2*t

    However, that is highly unlikely
    “... mathematics is only the art of saying the same thing in different words” - B. Russell

  6. #6

    Re: Another Newton's Law of Cooling/Warming

    the problem is stated correctly.

    here is what I did, (again there is something wrong with my rate, what should I do differently?)
    dT/dt = k(T-A)
    2 = k(20 -100)
    k = -0.025

    *it is heating up instead of cooling.*

    T(t)= T + (A-T)e^(kt)
    T(t)= 100 + (20 -100)e^(kt)
    T(0)= 100 -80
    T(0)= 20
    (this makes sense thus far, the temperature of the metal bar when t=0 is 20)

    *for the first temperature*
    90= 100 + (20 -100)e^(-0.025t)
    -10= -80e^(-0.025t)
    Ln(1/8)= -0.025t
    t = 83.17 s (answer is supposed to be 82.1s)

    *for the second temperature*
    98= 100 + (20 -100)e^(-0.025t)
    -2=-80e^(-0.025t)
    0.025=e^(-0.025t)
    Ln(0.025)= -0.025t
    t = 147.55 s (answer is supposed to be 145 s)

    The error has to be with my rate (k). If I change 0.025 to 0.02531 it works out to the correct answer.
    How can I obtain this different rate?
    any help is appreciated.

  7. #7
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    Re: Another Newton's Law of Cooling/Warming

    Quote Originally Posted by sifusith
    the problem is stated correctly.

    here is what I did, (again there is something wrong with my rate, what should I do differently?)
    dT/dt = k(T-A)
    2 = k(20 -100)<<< You are assuming that the initial rate is 2 °C/min - the problem does not say so.
    “... mathematics is only the art of saying the same thing in different words” - B. Russell

  8. #8

    Re: Another Newton's Law of Cooling/Warming

    Quote Originally Posted by hank
    A small metal bar, whose initial temp was 20C, is dropped into a large container of boiling water. How long will it take the bar to reach 90C if its temp increases 2 degrees in 1 second?
    Am I misinterpreting the given condition?
    I don't know how to find the rate.
    Thank you in advance.

  9. #9
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    Re: Another Newton's Law of Cooling/Warming

    Your interpretation would be correct - if the problem said:

    How long will it take the bar to reach 90C if initially its temp increases 2 degrees in 1 second
    “... mathematics is only the art of saying the same thing in different words” - B. Russell

  10. #10

    Re: Another Newton's Law of Cooling/Warming

    ok, I see.
    but I still don't know how to find the rate.
    what do I use the 2C in 1 sec for?
    how should I go about finding the rate?

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