Is the balance of forces equation correct for a tube immersed in alcohol?

[ValeForce46]

Homework Statement

A tube of mass $M= 18 kg$, section $A= 300 cm^2$ and lenght $l= 200 cm$ is sealed to the top and opened to the bottom. Initially the tube contains only air at a pressure $p_0= 1 bar$, occupying the whole volume $A*l$ (see picture a). Then, the tube is leaning on a surface of alcol (density $ρ=0.8 g/cm3$) and, keeping the tube vertically, it is immersed until it reaches a position of balance. In this process the air inside the tube is compressed (see picture b).
In conditions of equilibrium, calculate:
a)the pressure $p$ of the air inside the tube;
b)the difference of level $d$ between the surface of the alcol inside and outside the tube;
c)the length $b$ of the tube which emerges.

Relevant Equations

Stevino's Law $p=p_0+ρ*g*h$
Force caused by pressure $F=p*A$

This is a problem from a past exam.
For point a)
the balance between force should be given by the following equation
$$M*g+p_0*A=ρ*g*(A*d)+p*A$$
But I still have two unknown values...
I know that pressure outside the tube at depth d is
$$p(d)=p_0+ρ*g*d$$
and this has to be equal to the pressure at depth d inside the tube, because that's a free surface.
What do I do now? Is the equation of the balance of the forces correct?

 

[BvU]

Hello VF, !

What do I do now?

You will need something to link the pressure inside the tube to the gas volume. Any suggestions ?

Is the equation of the balance of the forces correct?

Looks good

 

[ValeForce46]

Well... The only two things that jump in my mind are Boyle's law and work. I don't think you report on those because I can't figure out how I should use them. It's not even specified if that is a isothermal process...

 

[kuruman]

Well... The only two things that jump in my mind are Boyle's law and work. I don't think you report on those because I can't figure out how I should use them. It's not even specified if that is a isothermal process...

You may assume that the temperature of the gas and everything else around it remains constant throughout the compression of the gas.

On edit:
Upon closer inspection, why is this equation correct?

$M*g+p_0*A=ρ*g*(A*d)+p*A$

The weight of the tube $Mg$ must be equal to the weight of the displaced fluid $\rho V_{under}g$. What is the volume $V_{under}$ of the tube under the surface of the fluid?

 

[ValeForce46]

so if I use Boyle's law I get
$$p_0*(A*l)=p*A*(b+d)$$
that is
$$p_0*l=p*(b+d)$$
But now what should I do to get the pressure $p$?

 

[kuruman]

What does Stevin's (Stevino's) law say about the variation of pressure at a given point under the surface of a liquid? Does it matter whether that point is under the immersed tube or horizontally off by some distance?

 

[ValeForce46]

Stevin's law says that the variation of pressure under the surface of a liquid at the depth h is $Δp=ρ*g*h$. If I'm not wrong, it doesn't matter if that point is under the immersed tube or horizontally off by some distance, because every parallel plane to the free surface is isobaric, right? But this should be valid only if the depth is $h>d$ (in this situation)... Maybe I'm blind but I can't still see the solution :-(

 

[kuruman]

First please read my edited post #4 about the force balance at equilibrium and fix your equation.

Stevin's law says that the variation of pressure under the surface of a liquid at the depth h is $Δp=ρ*g*h$. If I'm not wrong, it doesn't matter if that point is under the immersed tube or horizontally off by some distance, because every parallel plane to the free surface is isobaric, right?

Right.

But this should be valid only if the depth is $h>d$ (in this situation)... Maybe I'm blind but I can't still see the solution :-(

Consider a point right at the lip of the submerged tube. Go a bit off to the right and a bit off to the left and write the same pressure in two different ways. Do you see it now?

 

[ValeForce46]

I THINK I FINALLY GOT IT. Thank you for your time and patience