Find the magnitude of the difference between the weights (temperature question)

[physics1007]

Homework Statement

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.

Homework 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

The Attempt at a Solution

i get -2420.678 and that's not right. any suggestions?

 

[OlderDan]

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.