Why Do the Densities Cancel in the Oscillation Frequency of a Floating Cylinder?

[ozone]

ω

Homework Statement

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?

Homework 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??

 

[tiny-tim]

hi 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)

 

[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 canceled 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/...ons-and-waves-fall-2004/assignments/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]

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.