Relating pressure and height in a container

In summary, we discussed a container filled with liquid of uniform density and a thin horizontal layer of liquid at a given height. We determined that in order for the system to be in equilibrium, the sum of vertical forces acting on the layer must be zero. Using this information, we derived an equation that includes the pressure, weight, and area of the layer, and found that it does not depend on the thickness of the layer.
  • #1
Linus Pauling
190
0
1. A container of uniform cross-sectional area A is filled with liquid of uniform density rho. Consider a thin horizontal layer of liquid (thickness dy) at a height y as measured from the bottom of the container. Let the pressure exerted upward on the bottom of the layer be p and the pressure exerted downward on the top be p+dp. Assume throughout the problem that the system is in equilibrium (the container has not been recently shaken or moved, etc.).

Since the liquid is in equilibrium, the net force on the thin layer of liquid is zero. Complete the force equation for the sum of the vertical forces acting on the liquid layer described in the problem introduction.




2. 0 = sum of forces in y direction



3. Ok, I know that F_up = pA
F_down = A(p + dp)
weight of the thin layer = pAg dy

So I did:

pA - A(p+dp) - pAgdy = -Ap(d + dyg) = 0

And it's telling me that the answer does not depend on d.
 
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  • #2
Linus Pauling said:
3. Ok, I know that F_up = pA
F_down = A(p + dp)
OK.
weight of the thin layer = pAg dy
Do you mean: rho*Ag dy ?

When you set up your equation, you won't have a 'd' by itself.
 
  • #3
Doc Al said:
OK.

Do you mean: rho*Ag dy ?

When you set up your equation, you won't have a 'd' by itself.


My bad, yes it's rho*Ag dy

Thus:

F = F_up - F_down - rho*Agdy
= pA - A(p + dp) - g*rho*A dy
= -dp - g*rho*A dy

which is incorrect
 
  • #4
Linus Pauling said:
F = F_up - F_down - rho*Agdy
= pA - A(p + dp) - g*rho*A dy
= -dp - g*rho*A dy
Redo the last step--you dropped an A.
 

1. How does pressure change with height in a container?

The pressure in a container decreases as the height increases. This is because as the height increases, there is less weight of the gas molecules above, resulting in a decrease in pressure.

2. Is there a specific mathematical relationship between pressure and height in a container?

Yes, there is a direct relationship between pressure and height in a container. This is described by the equation P = ρgh, where P is pressure, ρ is the density of the gas, g is the acceleration due to gravity, and h is the height.

3. How does temperature affect the relationship between pressure and height in a container?

Temperature has a direct impact on the relationship between pressure and height in a container. An increase in temperature causes an increase in pressure, which in turn increases the height of the column of gas in the container.

4. Can pressure and height in a container be used to determine the density of the gas?

Yes, pressure and height in a container can be used to calculate the density of the gas. This can be done by rearranging the equation P = ρgh to solve for ρ, the density. By plugging in the values for pressure, height, and acceleration due to gravity, the density of the gas can be calculated.

5. How does the shape of the container affect the pressure and height relationship?

The shape of the container does not affect the pressure and height relationship. As long as the height and pressure are measured at the same point, the relationship between the two will remain the same regardless of the shape of the container.

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