SHM and Fluid

A plastic cylinder of height 2h and cross-sectional area A floats in equilibrium in a fluid of density p in such a way that the bottom of the cylinder is a distance h below surface of the fluid. This situation is shown in Figure (a), where the x-axis has been chosen to point vertically downwards.A plastic cylinder of height 2h and cross-sectional area A floats in equilibrium in a fluid of density p in such a way that the bottom of the cylinder is a distance h below surface of the fluid. This situation is shown in Figure (a), where the x-axis has been chosen to point vertically downwards.


(a) Derive an expression in terms of known quantities and g for the force that must be applied to the top of the cylinder to depress it by a distance x, as shown in Figure (b).


(b) State why, when the cylinder is released, its vibrations will be simple harmonic, and write an expression for the force constant in terms of known quantities and g.


(c) What is the maximum theoretical value of x up to which the vibration will remain simple harmonic? Briefly explain your reasoning.


(You will need to use Archimedes's principle, which states that the upward force on an object in a fluid is equal in magnitude to the weight of the fluid displaced. When the cylinder is floating, the weight of the displaced fluid equals the weight of the cylinder. The force required to depress the cylinder below its equilibrium position will therefore be equal to the weight of the additional fluid displaced when the cylinder is depressed.)


The link is the figure reference
http://s675.photobucket.com/albums/vv116/mebo8866/?action=view&current=Image3.jpg