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Thermal Expansion Problem

  1. Nov 5, 2007 #1
    Problem
    Sphere has ring around it. At T=70C the diameter of the sphere is 0.05% larger than the diameter of the ring. At what temp will the ring be able to come off sphere?
    Attempt
    D_s=1.0005*D_r at 70C

    Sphere
    delta_V=beta*V_o*delta_T
    Ring
    delta_L=alpha*L_o*delta_T

    L=pi*D_r (circumference)
    delta_L=alpha*(pi*D_r,o)*delta_T

    Tried relating D_r and D_s with sphere volume eqn (V=4/3*pi*(D_s/2)^3) and circumference of ring eqn C=L_r=pi*D_r and I'm stuck...

    Temeperature where D_r=D_s is the point when ring can come off, but I don't know how to get there.
     
  2. jcsd
  3. Nov 5, 2007 #2
    use alpha for both because the size of sphere is proportional to the diameter. as the diameter gets larger or smaller the smaller the sphere's surface area gets. right? so you can use alpha for the shere and the right and make the lengths equal each other like you said.
     
    Last edited: Nov 5, 2007
  4. Nov 5, 2007 #3
    I just end up with too many unknowns.

    Both
    delta_L=alpha*L_o*delta_T
    L=pi*D
    Sphere
    Eqn 1
    pi*delta_D_s=alpha*(D_o,s)*delta_T
    Ring
    Eqn 2
    pi*delta_D_r=alpha*pi*(D_o,r)*delta_T
    Eqn3
    D_o,s=1.0005*D_o,r
    Eqn 4
    D_f,r=D_f,s

    Four eqns, 5 unknowns which are...
    D_o,s
    D_f,s
    D_o,r
    D_f,r
    T_f
     
  5. Nov 5, 2007 #4
    what material is the sphere and what material is the ring? What does that tell you about alpha value for each one? also on started out at what length in regards to the other?
    The change plus the initial value for each material should equal each other right?
     
  6. Nov 5, 2007 #5
    alphas don't matter, they are just from table. The only given info is that the diameter of the sphere is 0.05% larger than the diameter of the ring at the initial temp of 70C.

    "The change plus the initial value for each material should equal each other right?"

    The change in diameter of the sphere will be different than the change in diameter of the ring. If not, the ring would never come off. The sphere will shrink at a faster rate than the ring and so at some colder temp, when D_r=D_s, the ring can come off. I know the answer is 41C if that helps you work backwards.
     
  7. Nov 6, 2007 #6
    alpha's do matter, even if they are from a table. If they both had same materials, hence same alphas, that ring will never come off! Length initial of sphere equals .0005 times Length initial of ring added to length initial of ring. can you take it from here?
     
  8. Nov 6, 2007 #7
    You are repeating what I have already said…

    The alpha comment, duh…
    Your initial length sentence… my first post I stated that D_s=1.0005*D_r…

    And no, I can't take it from here because you have not told me anything I didn't already know.
     
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