A cylinder with cross-section area A floats when it is partially submerged (dept
ID: 1311323 • Letter: A
Question
A cylinder with cross-section area A floats when it is partially submerged (depth h) in a liquid of density rho liq, as shown on the left in the figure. Pressing down on the cylinder pushes it deeper into the liquid. Find an expression for the force F needed to push the cylinder a distance x deeper into the liquid and hold it there. Your answer may contain some or all of the variables A. x, rho liq. and any necessary constants. Determine an expression for the amount of work that must be done by the force F to push the cylinder a distance x deeper into the water. (Hint: F is not constant, so an integration is required.) If you push the floating cylinder down and release it, it bobs up and down. So it is like a spring in the sense that it oscillates if displaced from equilibrium. Use your result from part (a) to determine the restoring force on the cylinder. (Remember, a restoring force looks like F = -kx.) Determine the "spring constant" for this system, and determine the period of oscillation of the cylinder.Explanation / Answer
a) Actually integration is not required, because the floating body is prismatic, hence the newly displaced liquid's volume is proportional to the position by which we push the cylinder downward. And we can make this argument without even touching calculus.
If we define the origin as when the cylinder is floating under no restraints other than the liquid's buoyancy, then what the value of x*A represents is the ADDITIONAL volume of fluid that is displaced out of the way.
The force needed to push the cylinder down is equal to the extra buoyancy acting on it by submerging that extra portion of it, which in turn equals the WEIGHT of the additional volume of the newly displaced fluid.
Hence:
F = rho*g*A*x
F is the additional downward force applied to the cylinder
rho is the density of the fluid
A is the cross sectional area of the cylinder
x is the downward displacement of the cylinder
g is Earth's gravitational field.
b) by above we also calculate work done = force x displacement
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