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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 5.00×102 N to totally submerge the ball, calculate the diameter d 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.

Respuesta :

boffeemadrid

Answer:

0.47556 m

Explanation:

[tex]F_B[/tex] = Child supplying force = 500 N

W = Weight of foam

F = Force on air

g = Acceleration due to gravity = 9.81 m/s²

By balancing the forces we get

[tex]F+W=F_B\\\Rightarrow F_B=500+W\\\Rightarrow F_B-W=500\\\Rightarrow \dfrac{4}{3}\pi R^3(\rho_{a}-\rho_{f})g=500\\\Rightarrow R=(\dfrac{500\times 3}{4\pi (\rho_{a}-\rho_{f})g})^{\dfrac{1}{3}}\\\Rightarrow R=(\dfrac{500\times 3}{4\pi (1000-95)\times 9.81})^{\dfrac{1}{3}}\\\Rightarrow R=0.23778\ m[/tex]

The diameter of the ball is [tex]2\times 0.23778=0.47556\ m[/tex]

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