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 whereas a large ball requires a lot of force. The child happens to have a styrofoam ball (the shape of the ball will not distort when it is forced under the surface), which he forces under the surface of the water. If the child needs to supply 635 N to totally submerge the ball, calculate the diameter D of the ball. The density of water is rhow=1*10^3 kg/m^3, the density of the styrofoam is rhofoam= 95 kg/m^3, and the acceleration due to gravity is g=9.81 m/s^2.

Options:
a. 0.157 m
b. 0.202 m
c. 0.180 m
d. 0.134 m

Answer 1

The diameter of the styrofoam ball can be calculated using Archimedes' principle. By equating the buoyant force to the weight of the ball and solving for the volume of the ball, we can then use the formula for the volume of a sphere to find the diameter.

Step-by-step explanation:

To calculate the diameter of the styrofoam ball, we can use Archimedes' principle. The buoyant force acting on the ball is equal to the weight of the water displaced by the ball. The formula to calculate the buoyant force is:

Buoyant force = Weight of water displaced = Volume of ball * Density of water * Acceleration due to gravity

Since the ball is forced to be totally submerged, the buoyant force is equal to the weight of the ball. Therefore, the equation becomes:

Weight of ball = Volume of ball * Density of water * Acceleration due to gravity

We can rearrange the equation to solve for the volume of the ball:

Volume of ball = Weight of ball / (Density of water * Acceleration due to gravity)

Given that the weight of the ball is 635 N, the density of water is 1*10^3 kg/m^3, and the acceleration due to gravity is 9.81 m/s^2, we can substitute these values into the equation:

Volume of ball = 635 N / (1*10^3 kg/m^3 * 9.81 m/s^2)

Now, we can calculate the volume of the ball:

Volume of ball = 0.0646 m^3

The density of the styrofoam is given as 95 kg/m^3. We can use the formula for the volume of a sphere to find the diameter of the ball:

Volume of ball = (4/3) * pi * (Diameter/2)^3

Substituting the known values and solving for the diameter:

0.0646 m^3 = (4/3) * pi * (Diameter/2)^3

Diameter/2 = (0.0646 m^3 * 3 / (4 * pi))^(1/3)

Diameter = 2 * ((0.0646 m^3 * 3 / (4 * pi))^(1/3))

Calculating the diameter gives us approximately 0.202 m. Therefore, option b. 0.202 m is the correct answer.