Fluid Statics/depth in a cylinder

[tennisgirl92]

Homework Statement

Mercury is added to a cylindrical container to a depth d and then the rest of the cylinder is filled with water. If the cylinder is 0.8 m tall and the pressure at the bottom is 1.3 atmospheres, determine the depth of the mercury. (Assume the density of mercury to be 1.36

104 kg/m3.)

Homework Equations

P_total=P_atmosphere+P_water+P_mercury
P=density x g x height
1 atmosphere=1.013e⁵ Pa
density of water=1 x 10³ kg/m³
pressure of atmosphere=1 atm

The Attempt at a Solution

I first subtracted out the pressure of the atmosphere, so
.3 atm=P_water + P_mercury
.8m=H_water + H_mercury
P_mercury=1.36 x 10⁴ x 9.8 x H_mercury
P_water=1 x 10³ x 9.8 x H_water

I am having difficulty putting all 4 of these equations together. I feel like there is substitution needed, but which equations should I use?

 

[kuruman]

Perhaps you should be more systematic, Starting with your equation
P_total=P_atmosphere+P_water+P_mercury
substitute
P_water=ρ_waterg h_water
P_mercury=ρ_mercuryg h_mercury
and solve for h_mercury.

 

[Chestermiller]

Homework Statement

Mercury is added to a cylindrical container to a depth d and then the rest of the cylinder is filled with water. If the cylinder is 0.8 m tall and the pressure at the bottom is 1.3 atmospheres, determine the depth of the mercury. (Assume the density of mercury to be 1.36 View attachment 195815 104 kg/m3.)

Homework Equations

P_total=P_atmosphere+P_water+P_mercury
P=density x g x height
1 atmosphere=1.013e⁵ Pa
density of water=1 x 10³ kg/m³
pressure of atmosphere=1 atm

The Attempt at a Solution

I first subtracted out the pressure of the atmosphere, so
.3 atm=P_water + P_mercury
.8m=H_water + H_mercury
P_mercury=1.36 x 10⁴ x 9.8 x H_mercury
P_water=1 x 10³ x 9.8 x H_water

I am having difficulty putting all 4 of these equations together. I feel like there is substitution needed, but which equations should I use?

You have 4 linear algebraic equations in 4 unknowns. Are you familiar with Gaussian elimination?

 

[tennisgirl92]

Perhaps you should be more systematic, Starting with your equation
P_total=P_atmosphere+P_water+P_mercury
substitute
P_water=ρ_waterg h_water
P_mercury=ρ_mercuryg h_mercury
and solve for h_mercury.

Ok, if I do that
.3atm=(1000 x 9.8 x H_water) + (1.36 x 10⁴ x 9.8 x H_mercury)
I still need to find the height of water, right? Where would that come from?

 

[tennisgirl92]

You have 4 linear algebraic equations in 4 unknowns. Are you familiar with Gaussian elimination?

No, never heard of that. What is Gaussian elimination?

 

[kuruman]

I still need to find the height of water, right? Where would that come from?

Since the cylinder is filled to the top,
Height of water = height of cylinder - height of mercury

 

[tennisgirl92]

Since the cylinder is filled to the top,
Height of water = height of cylinder - height of mercury

thank you! Got it!