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 527 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

Buoyany Force = Weight of displaced water - weight of the ball

F = (rho_w*g*V) - (rho_s*V*g)

F = (rho_w-rho_s)*V*g

V is the volume of the ball = (4/3)*pi*R^3

rho_w = 1000 kg/m^3

rho_s = 95 kg/m^3

then

527 =(1000-95)*9.8*(4/3)*3.142*(R^3)

R = 0.242 m

diamter is D = 2*R = 2*0.242 = 0.484 m

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