# Cylinder floating

1. Mar 12, 2013

### Unicorn.

Hi everyone,
I have an exercise and i don't understand something in the solution:
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?

The diameter of the ﬂoating cylinder is d and it has l of its length submerged in water. The volume of water
displaced by the submerged part of the cylinder in equilibrium condition is πd2l/4. Let the density of water be ρ
and the density of cylinder be ρcyl. Hence the mass of the cylinder is:
Mcyl = ρw*Vdisplaced= ρw*πd2l/4=ρcyl*πd2L/4
Frestoringw*gπd2x/4
Mcylx"=ρw*gπd2x/4
w²=ρw*gπd2/4Mcyl
w²=g/l

Why are they taking Vdisplaced=πd2l/4 Why /4 ???
And why l is equa to 4Mcylw*πd2

Thanks !

2. Mar 12, 2013

### TSny

What's the formula for the area of a circle expressed in terms of its diameter (not radius)?
This relationship comes from the first equation you wrote.

3. Mar 12, 2013

### Unicorn.

Oh okey i didn't notice that it was written in terms of the diameter !
Thanks !

I have another question about oscillations, it was an assignment that we did two weeks ago. I didn't understand the exercise so I didn't do it and my teacher doesn't give the details of the solutions. I'm expecting someone to help me and explain me the steps:

Three block of mass m=0.13kg are connected with three springs of constant k1 k2 k3 and

Two of the normal modes of the system, expressed in terms of displacement are :
|1>=( 1 0 -3) |2>= ( 27 -10 9)

Knowing that the lower frequency that isn't necessarly the first mode frequency is 20 rad/s. Find the spring constants.

All i did is that i gave the equations of motion so i got few points for that, and found the |3> as they're perpendicular between them.

Thank you, I have an exam tomorrow and need to understand this !

4. Mar 12, 2013

### TSny

I think you should post this as a separate question with an appropriate title.