Cylinder Floating: Frequency of Oscillation Explained

[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:
M_cyl = ρ_w*V_displaced= ρ_w*πd²l/4=ρ_cyl*πd²L/4
F_restoring=ρ_w*gπd²x/4
M_cylx"=ρ_w*gπd²x/4
w²=ρ_w*gπd²/4M_cyl
w²=g/l

Why are they taking V_displaced=πd²l/4 Why /4 ?
And why l is equa to 4M_cyl/ρ_w*πd²

Thanks !

 

[TSny]

Why are they taking V_displaced=πd²l/4 Why /4 ?

What's the formula for the area of a circle expressed in terms of its diameter (not radius)?

And why l is equa to 4M_cyl/ρ_w*πd²

This relationship comes from the first equation you wrote.

 

[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 !

 

[TSny]

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.

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