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Spherical raindrop, mass, radius, and time

  1. Sep 18, 2015 #1
    1. The problem statement, all variables and given/known data

    Separate variables and integrate to find an expression for r(t), given r0 at t=0

    2. Relevant equations

    M=ρ(4/3)πr3, thus V=(4/3)πr3

    dM/dt=Cr3 where C is a constant

    3. The attempt at a solution

    ∫dM=∫Cr3dt

    M+constant=??

    I have no idea how to integrate r because it's a function of t but we're not given the function. I don't think that integrating the left side this way will be very helpful either. Any advice would be much appreciated!
     
  2. jcsd
  3. Sep 18, 2015 #2

    SteamKing

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    You want to express M in terms of r. Once you do that, then dM(r) / dt = Cr3, and you can separate the variables to get r on one side and dt on the other.
     
  4. Sep 19, 2015 #3
    Ok... I guess I don't really understand how to do that. Can I do that using just the M equation, or do I need something else?
     
  5. Sep 19, 2015 #4

    SteamKing

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    You gave an expression for the mass of the raindrop in terms of the radius right there in Section 2 of the template. It's the equation M = ...
     
  6. Sep 19, 2015 #5
    Alright, so I have M=ρ(4/3)πr3. What exactly do I do with this? Take the derivative and set it equal to the other? Sorry, I'm sure I'm missing something really obvious.
     
  7. Sep 19, 2015 #6

    SteamKing

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    This problem is about solving a differential equation by separation of variables. Have you studied how to do this yet?
     
  8. Sep 19, 2015 #7
    Yes, but a few years ago so maybe I've forgotten something key. So I have C*r*dt=ρ*4*π*dr. Is that correct?
     
  9. Sep 19, 2015 #8

    SteamKing

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    Not quite.

    Presumably, as the rain drop falls, it gets bigger as it collects more moisture; therefore, the radius of the drop grows as time passes.

    Since the drop is spherical, M(t) = ρ(4/3)π*[r(t)]3, where M and r are written as functions of time.

    You are also given the condition that the change in the mass of the drop, dM(t)/dt, at any given time is proportional to the cube of the radius of the drop, or dM(t)/dt = Cr3. At t = 0, r(t) = r(0) = r0.

    You should check your differentiation of M(t) w.r.t. time. You might have to use the chain rule here, since M is a function of r, but r is a function of t.
     
  10. Sep 19, 2015 #9
    I got it!!! Can't believe I forgot about the chain rule. Thank you so much for your help!
     
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