A sphere of radius 25 cm rests at the bottom of a pool containing fresh water, completely submerged. If the upward normal force on the sphere from the bottom of the pool is 1,025 N, what is the mass of the sphere?

Sure, let's solve this step by step:

1. First, we need to understand that the upward normal force is equal to the weight of the water displaced by the sphere. This is according to Archimedes' principle which states that the buoyant force (upward force) on an object is equal to the weight of the fluid displaced by the object.

2. The weight of the water displaced can be calculated using the formula: Weight = Volume * Density * g, where g is the acceleration due to gravity (approximately 9.8 m/s^2).

3. The volume of the sphere can be calculated using the formula: Volume = 4/3 * π * r^3, where r is the radius of the sphere. Substituting r = 25 cm or 0.25 m, we get Volume = 4/3 * π * (0.25 m)^3 = 0.06545 m^3.

4. The density of fresh water is approximately 1000 kg/m^3. So, the weight of the water displaced is: Weight = 0.06545 m^3 * 1000 kg/m^3 * 9.8 m/s^2 = 640.41 N.

5. However, the problem states that the upward normal force is 1,025 N. This means that the weight of the sphere is the weight of the water displaced plus the upward normal force: Weight_sphere = 640.41 N + 1,025 N = 1,665.41 N.

6. Finally, we can find the mass of the sphere by rearranging the formula for weight: Weight = mass * g. So, mass = Weight / g = 1,665.41 N / 9.8 m/s^2 = 170 kg.

So, the mass of the sphere is approximately 170 kg.