Cylinder oscillating in water.

ozone

ω1. The problem statement, all variables and given/known data

A cylinder of diameter d floats with l of its length submerged. The total height is L. Assume no damping. At time t = 0 the cylinder is pushed down a distance B and released.


What is the frequency of oscillation?
2. Relevant equations

[itex] f = ω/2\pi [/itex]
[itex] Ma = F_{(bouyancy)} [/itex]
Writing this in our differential form, making proper substitutions, and noting that bouyancy is affected by the distance that our cylinder is submerged we come to.

[itex] dx^2 (M_{(cylinder)}) + x (\rho_{(water)} g Area_{(cylinder face)})= 0 [/itex]

we know that [itex]M_{(cylinder)} = V_{(cylinder)}\rho_{(cylinder)} [/itex]

hence we should have
[itex] ω^2 = (\rho_{(water)} g Area_{(cylinder face)}) / V_{cylinder}\rho_{(cylinder)} = g\rho_{(water)} / l \rho_{(cylinder)}[/itex]

however the solution in my problem set has ω^2 = g/l. Can anyone shed some light on why the densities may cancel??

tiny-tim

hi ozone!
(that's the same as g/L)

i haven't followed what you've done, but i'd guess you've used the wrong expression for the mass of the cylinder

just use a (vertical) force equation for the cylinder (at depth l + x)

ozone

the mass of the cylinder is the density of the cylinder times the area.. but the force from the water depends only on the density of water.. that is why i don't understand how the densities are cancelled out.

TSny

In the denominator of your final expression for ω², is that a small l or a capital L?

You can find an expression for the ratio of the two densities in terms of the ratio of l and L by considering the condition for equilibrium when length l of the cylinder is submerged.

ozone

that would be the lower case l in the solution set answer. You can view it for yourself at this link on page 4.

http://ocw.mit.edu/courses/physics/8...ents/sol1b.pdf

TSny

But in the expression that you derived: ω² = gρ_w/lρ_c, you should have a capital L rather than a lower case l in the denominator. Then you should be able to show that this expression reduces to the correct answer.

ozone

We never learned about fluid dynamics in my mechanics class, but I am guessing that the water displaced in equilibrium is equal to the mass of the cylinder?

TSny

Yes. Archimedes Principle: Buoyant force = weight of fluid displaced

In equilibrium, Buoyant force equals the weight of the floating object.