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Homework Help: Find the magnitude of the difference between the weights (temperature question)

  1. Dec 1, 2006 #1
    1. The problem statement, all variables and given/known data
    A solid aluminum sphere has a radius of 2.34 m and a temperature of 89.0 °C. The sphere is then completely immersed in a pool of water whose temperature is 22.3 °C. The sphere cools, while the water temperature remains nearly at 22.3 °C, because the pool is very large. The sphere is weighed in the water immediately after being submerged (before it begins to cool) and then again after cooling to 22.3 °C. Use Archimedes' principle to find the magnitude of the difference between the weights.


    2. Relevant equations
    Weight of sphere in water before cooling W1 = W - ρ * g * V1 ρ = density of water
    where W = actual weight of sphere and V1 the volume before cooling
    Also, W2 = W - ρ * g * V2 Then W2 - W1 = -(V2 - V1 ) * ρ * g
    V1 = 4/3 * π * r^3 V2 = 4/3 * π * (r + Δr)^3
    Expanding (r + Δr)^3 and dropping any terms with Δr^3 or Δr^2 since these are very small
    V2 = 4/3 * π * (r^3 + 3 * r * Δr)
    V2 - V1 = 4 * π * r^2 * Δr
    Δr = k * r * ΔT where k = coefficient of linear expansion for Al and ΔT the temperature change

    This gives V2 - V1 = 4 * π * r^3 * k * ΔT
    W2 - W1 = -4 * π * r^3 * k * ρ * g * ΔT


    3. The attempt at a solution

    i get -2420.678 and thats not right. any suggestions?
     
  2. jcsd
  3. Dec 1, 2006 #2

    OlderDan

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    Science Advisor
    Homework Helper

    You seem to have the right idea. I did not verify your number, but the problem asks for the magnitude of the difference in weights, which should be a positive number.

    Besides the sign, you appear to have a problem with your ΔV expression, but I think you fixed it. ΔV should be the area of the sphere times Δr. In one place you have just an rΔr instead of r²Δr. I think you used the correct expression in your calculation.
     
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