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.

 

[paranormal]

The volume of water displaced by the submerged part of the cylinder is equal to the volume of the cylinder that is submerged in water. This volume can be calculated using the formula for the volume of a cylinder, which is πr^2h, where r is the radius and h is the height. In this case, the radius is equal to half of the diameter (d/2), and the height is equal to the length submerged (l). Therefore, the volume of water displaced is equal to π(d/2)^2l, which simplifies to πd^2l/4.

The reason why l is equal to 4Mcyl/ρw*πd2 is because this equation is derived from the equation for the mass of the cylinder (Mcyl = ρcyl*πd2L/4) and the equation for the volume of water displaced (Vdisplaced=πd2l/4). By substituting these values into the equation w²=ρw*gπd2/4Mcyl, we can solve for l, which results in l=4Mcyl/ρw*πd2.

I hope this helps clarify the solution for you. Let me know if you have any further questions.