A child playing in a swimming pool realizes that it is easy to push a small infl

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 554 N to totally submerge the ball, calculate the diameter of the ball. The density of water is w = 1.000 × 103 kg/m3, the density of styrofoam is foam = 95.0 kg/m3 and the acceleration due to gravity is g = 9.81 m/s2.

answer in meters

Explanation / Answer

given,

force needed to submerge the ball = 554 N

density of water = 1000 kg/m^3

density of styroform = 95 kg/m^3

The 554 N (F, the buoyancy) is the weight of the displaced water minus the weight of the ball itself:

F = (w - foam) * V * g

V = 4/3 pi R^3

V = 4/3 pi (D/2)^3

V = 1/6 pi D^3

So

F = (w - foam) * pi D^3 / 6 * g

D^3 = 6 F / ( pi*g*(w - foam))

D = ( 6 F / ( pi*g*(w - foam)) )^(1/3)

D = ( 6 * 554 / ( pi * 9.8 * (1000 - 95)) )^(1/3)

D = 0.492 m

the diameter of the ball = 0.492 m

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