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, we need to calculate the buoyant force acting on the aluminum sphere in water using Archimedes' principle.

Archimedes' principle states that the buoyant force acting on an object submerged in a fluid is equal to the weight of the fluid displaced by the object. The buoyant force can be calculated using the formula:

Buoyant force = weight of displaced fluid

In this scenario, the water exerts an upward buoyant force on the submerged sphere, counteracting the downward force due to gravity.

To calculate the weight of the aluminum sphere in water (before it begins to cool), we need to determine the buoyant force. The buoyant force is equal to the weight of the water displaced by the sphere.

The volume of a sphere can be calculated using the formula:

Volume = (4/3) * π * radius^3

Given that the radius of the solid aluminum sphere is 2.28 m, we can calculate its volume.

Volume of aluminum sphere = (4/3) * π * (2.28 m)^3

Once we have the volume of the sphere, we can calculate the weight of the aluminum sphere in air using the density of aluminum. The density of aluminum is typically around 2700 kg/m³.

Weight of aluminum sphere in air = Volume of aluminum sphere * Density of aluminum * Acceleration due to gravity

Gravity accelerates objects on Earth at approximately 9.8 m/s².

Now, to calculate the weight of the displaced water, we use the same volume and the density of water. The density of water is approximately 1000 kg/m³.

Weight of displaced water = Volume of aluminum sphere * Density of water * Acceleration due to gravity

Finally, to find the magnitude of the difference between the weights, we subtract the weight of the displaced water from the weight of the aluminum sphere in air.

Magnitude of difference between weights = Weight of aluminum sphere in air - Weight of displaced water

Calculating these values will give you the desired magnitude of the difference between the weights.