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.

Explanation / Answer

When child submerge the ball then
the forces on the ball will be
weight due to gravity downward(mg) , force applied downward (F), buoyancy force upward (FB)
let the diameter of ball say D , therefore the volume of ball V = (4Pi/3)R3  = (4/3)Pi*D3/8 = Pi*D3/6
Now we write the force equation
mg +F = FB
m = mass of ball = pfoam*V
FB = Pw*g*V
pfoam*gV +F = pw*g*V
(pw - pfoam)*gV = F
V = (554)/(9.81*(1000 - 95)) = 0.0624 m3
Pi*D3/6 = 0.0624
D = 0.492 m

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