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 Homework Statement

A cylindrical container (section ##S= 4m^2##) is filled by water up to a height of ##h_1=3m##, the remaining part of satured vapour applies pressure on the water ##P=0.8 atm##. On the bottom there's a hole (section ##S_1=0.2m^2##), linked to a pipe of section ##S_1## which in the far end is closed by a tap ##R## (see picture (a)). In the pipe there's a vertical cylinder ##M##. The diametre of the pipe is negligible compared to ##h_1##. Determine the height ##h_2## which the water reaches in the cylinder.
Then, at the end of the pipe ##S_1## laid a new pipe (section ##S_2=0.1m^2##) (see picture (b)) and after short time the water is in steady state condition. Determine the new height ##h_3## of the water and the speed through ##S_2##.
 Homework Equations

Stevin's law: ##p=p_0+ ρgh##
Bernoulli's principle: ##p+ρgh+1/2ρv^2=constant##
Flow rate constant: ##v_1*S_1=v_2*S_2##
The first part of the problem I just used Stevin's law:
$$p_{atm}=P+ρg(h_1h_2)=> h_2=(Pp_{atm}+ρgh_1)/(ρg) =>h_2=0.94m$$
Is this right? I considered ##ρ=10^3 {kg/m^3}##
About the second part.... how can I be sure that ##h_1## remains unchanged? If it is unchanged, then can I use Bernoulli's principle in this way?
##P+ρgh_1=p_{atm}+1/2ρv_1^2## (##v_1## is the speed through section ##S_1##)
so I could find ##v_2## from ##S_1*v_1=S_2*v_2##
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