## Cylinder oscillating in water.

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

$f = ω/2\pi$
$Ma = F_{(bouyancy)}$
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.

$dx^2 (M_{(cylinder)}) + x (\rho_{(water)} g Area_{(cylinder face)})= 0$

we know that $M_{(cylinder)} = V_{(cylinder)}\rho_{(cylinder)}$

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

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

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hi ozone!
 Quote by ozone $g\rho_{(water)} / l \rho_{(cylinder)}$
(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)

 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.

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## Cylinder oscillating in water.

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.

 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
 Recognitions: Gold Member Homework Help But in the expression that you derived: ω2 = gρw/lρc, 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.
 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?

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

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