A solid aluminum sphere has a radius of 2.28 m and a temperature of 89.1 °C. The sphere is then completely immersed in a pool of water whose temperature is 11.9 °C. The sphere cools, while the water temperature remains nearly at 11.9 °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 11.9 °C. Use Archimedes' principle to find the magnitude of the difference between the weights.

To find the magnitude of the difference between the weights of the solid aluminum sphere before and after cooling in water, we can use Archimedes' principle, which states that the buoyant force on an object submerged in a fluid is equal to the weight of the fluid displaced by the object.

Step 1: Find the volume of the aluminum sphere.
The volume of a sphere is given by the formula V = (4/3)πr^3, where r is the radius of the sphere. In this case, the radius is 2.28 m.
V = (4/3)π(2.28)^3 = 52.482 m^3.

Step 2: Find the weight of the water displaced by the sphere.
The density of water is approximately 1000 kg/m^3. So, the mass of the water displaced is given by m = ρV, where ρ is the density of water and V is the volume of the sphere.
m = (1000 kg/m^3)(52.482 m^3) = 52482 kg.

Step 3: Find the weight of the sphere in air.
The weight of the sphere in air can be found using the formula W = mg, where m is the mass of the sphere and g is the acceleration due to gravity, which is approximately 9.8 m/s^2.
W = (mass of sphere)(g) = (density of aluminum)(volume of sphere)(g).
The density of aluminum is approximately 2700 kg/m^3.
W = (2700 kg/m^3)(52.482 m^3)(9.8 m/s^2) = 1348545.8424 N.

Step 4: Find the weight of the sphere in water at 11.9 °C.
The sphere cools in the water, which causes a decrease in volume. The new volume and weight of the sphere can be found using the same steps as before, but with the density of aluminum at the lower temperature. The density of aluminum decreases with temperature, but the change in volume is small enough that we can assume the density change is negligible for this calculation. So, we can use the given density of aluminum.

Step 5: Calculate the difference in weight.
The magnitude of the difference between the weights is equal to the weight in air minus the weight in water at the lower temperature.
Difference = Weight in air - Weight in water at 11.9°C.
Difference = 1348545.8424 N - Weight in water at 11.9°C.