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 ρw=1.000×103 kg/m3, the density of styrofoam is ρ foam
= 95.0 kg/m3, and the acceleration due to gravity is g = 9.81 m/s2.
To calculate the diameter (d) of the styrofoam ball, we need to use the concept of buoyancy.
The buoyant force (F_b) exerted on an object submerged in a fluid is equal to the weight of the fluid displaced by the object. In this case, the buoyant force will be equal to the weight of the styrofoam ball.
The weight of an object can be calculated using the formula:
Weight = Mass * Acceleration due to gravity
The mass of the styrofoam ball can be calculated using the formula:
Mass = Density * Volume
First, let's calculate the mass of the styrofoam ball. The density of styrofoam is given as ρ_foam = 95.0 kg/m^3. We can rearrange the formula for mass:
Mass = Density * Volume
Since the shape of the ball will not distort when it is forced under the water, the volume of the ball is the same as the volume of water it displaces. We can use the formula for volume:
Volume = (4/3) * π * (radius)^3
Now, let's calculate the volume of the ball by rearranging the volume formula:
Volume = (Mass_styrofoam) / (Density_water)
Finally, let's calculate the diameter (d) of the ball using the formula for volume:
Volume = (4/3) * π * (radius)^3
Volume = (4/3) * π * (diameter/2)^3
Substituting the value of volume calculated above:
(Mass_styrofoam) / (Density_water) = (4/3) * π * (diameter/2)^3
And rearranging the formula for diameter (d):
diameter = [(3 * (Mass_styrofoam) / (Density_water * π * 2^3)]^ (1/3)
Now we can plug in the given values and calculate the diameter:
To find the diameter of the styrofoam ball, we can use the principle of buoyancy. Buoyancy is the upward force exerted on an object immersed in a fluid, such as water, that opposes the force of gravity.
The buoyant force (Fb) can be calculated using the formula:
Fb = ρw * V * g
Where:
- ρw is the density of water
- V is the volume of the submerged object
- g is the acceleration due to gravity
The force needed to totally submerge the ball is given as 635 N. Since the submerged ball is in equilibrium, we can set the buoyant force equal to the force needed to submerge the ball:
Fb = 635 N
We can rearrange the formula for the buoyant force to solve for the volume (V) of the submerged ball:
V = Fb / (ρw * g)
Now we need to find the mass (m) of the ball. The mass can be calculated using the formula:
m = ρfoam * V
Where:
- ρfoam is the density of Styrofoam
- V is the volume of the submerged ball
Substituting the equation for V into the equation for mass, we get:
m = ρfoam * (Fb / (ρw * g))
Finally, we can calculate the diameter (d) of the ball using the formula:
V = (4/3) * π * (d/2)^3
Rearranging the formula to solve for d:
d = (2 * (3 * V / (4 * π))^(1/3)
Now we have all the necessary formulas to calculate the diameter of the styrofoam ball. Let's substitute the given values into the equations and calculate the diameter.
635 N. = Wt. of water displaced.
M*g = 635.
M = 635/g = 635/9.81 = 64.73 kg = Mass of water displaced.
64.73kg * 1m^3/1000kg = 0.0647 m^3 = Vol. of water displaced.
0.0647 m^3 = Vol. of ball submerged.
V = (4/3) *pi *r^3 = 0.0647.
4.2r^3 = 0.0647.
r = 0.25 m.
d = 2r = 0.50 meters.