Question

A child playing in a swimming pool realizes that it is easy to push a small inflated ball under the surface of the water while a large ball requires a lot of force. The child happens to have a styrofoam ball (this way the shape of the ball will not distort when it is forced under the surface) which is being forced under the surface of the water. If the child needs to supply 635 N to totally submerge the ball, calculate the diameter of the ball. The density of water is ?w = 1.000 g/cm3, the density of styrofoam is ?foam = 0.0950 g/cm3 and the acceleration due to gravity is g = 9.81 m/s2.

Answer 1

The problem requires calculating the diameter of a styrofoam ball based on the buoyant force needed to submerge it, using Archimedes' Principle and the known densities of styrofoam and water, along with gravitational acceleration.

Step-by-step explanation:

The student's question deals with calculating the diameter of a styrofoam ball that requires 635 N to submerge it in water, using the concept of buoyancy from Archimedes' Principle. To submerge the styrofoam ball, the child must counteract the buoyant force, which is equal to the weight of the water displaced by the ball. By using the given densities for water (1.000 g/cm3) and styrofoam (0.0950 g/cm3), and the gravitational acceleration (9.81 m/s2), we can find the volume of water displaced, which is the same as the volume of the ball. A force of 635 N corresponds to a mass m = F/g, where F is the force (635 N) and g is the acceleration due to gravity. We convert the density of water to SI units to calculate the mass (and thus, the volume) of water displaced. Once we have the volume V, we can find the diameter D of the ball using the formula for the volume of a sphere, V = (4/3) * π * (D/2)3. The detailed computation would involve algebraic manipulation of these equations to solve for the diameter D.