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A cylinder with cross-section area a floats with its long axis vertical in a liquid of density rho.

Part A
Pressing down on the cylinder pushes it deeper into the liquid. Find an expression for the force needed to push the cylinder distance z deeper into the liquid and hold it there.
Part B
A 5.0-cm-diameter cylinder floats in water. How much work must be done to push the cylinder 17 cm deeper into the water?

User Fabrizio Regini
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1 Answer

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19 votes

Final answer:

The force needed to push a cylinder deeper into liquid is equal to the weight of the displaced fluid, given by ρgAz. To push a 5.0-cm-diameter cylinder 17 cm deeper into the water, calculate the cylinder's cross-sectional area, the force required, and finally, the work by multiplying the force by the displacement.

Step-by-step explanation:

The force needed to push a cylinder distance z deeper into a liquid and hold it there can be found by considering the weight of the fluid displaced by the submerged part of the cylinder. The weight of the displaced fluid is equal to the buoyant force acting upon the cylinder. According to Archimedes' principle, the buoyant force, FB, is equal to the weight of the displaced fluid, which is ρgAz, where ρ is the density of the fluid, g is the acceleration due to gravity, and A is the cross-sectional area of the cylinder.

For Part B, the work done in pushing the cylinder down is equal to the force times the distance moved in the direction of the force. If the cylinder has a diameter of 5.0 cm, we find its cross-sectional area and calculate the force for the specified distance of 17 cm. The work done is then:

  • Find the cross-sectional area A = π(d/2)2 with d being the diameter of the cylinder.
  • Calculate the force needed F = ρgAz for water (ρ = density of water).
  • Calculate work done as W = Fz.

User Othman Benchekroun
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