Hydrostatic pressure in a tilted tube

In summary, the hydrostatic pressure difference between the liquid surface and the base of the rightmost tube can be proven to be ρgh, where h is the vertical distance of the liquid surface from the base. This is done by considering a small cuboidal fluid element and taking into account the equilibrium and pressure equations. Additionally, if the rightmost tube is a thin capillary tube with a curved meniscus, the pressure at the base would be P0 - 2T/r + ρgh, taking into account surface tension.
  • #1
Jahnavi
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Homework Statement


tilted tube.jpg


Prove that the hydrostatic pressure difference between the liquid surface and the base of rightmost tube is ρgh (h is the vertical distance of the liquid surface from the base ) ?

Homework Equations

The Attempt at a Solution



I would like to prove this result just out of curiosity .

The tube is tilted at an angle θ with the vertical . Tube length is L .

Consider a small cuboidal shape fluid element of base area A and height dx at a distance x from the base of the tube .

Weight of the fluid element = (Adx)ρg

Since the fluid is in equilibrium , the net force along the length of the tube is zero .

If P (x) denotes pressure at a distance x from the base ,

P(x)A - P(x+dx)A - (Adx)ρg(cosθ) = 0

dP = ρg(cosθ)dx

P(0) - P(L) = ∫ ρg(cosθ)dx
= ρgL(cosθ)
= ρgh

Is this derivation correct ?
 

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  • #2
That seems correct to me.
 
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  • #3
Thanks !
 
  • #4
If we consider the rightmost tube to be a thin capillary tube such that the surface has a curved meniscus , will the pressure at the base still be P0+ρgh ?

Actually I want to ask whether surface tension is relevant as far as hydrostatic pressure in the liquid is concerned ?
 
  • #5
Jahnavi said:
If we consider the rightmost tube to be a thin capillary tube such that the surface has a curved meniscus , will the pressure at the base still be P0+ρgh ?

Actually I want to ask whether surface tension is relevant as far as hydrostatic pressure in the liquid is concerned ?
Surface tension causes a pressure difference across the free surface. So, do you think this matters?
 
  • #6
Taking surface tension into account ,pressure at the base would be P0 - 2T/r + ρgh . Right ?
 
  • #7
Jahnavi said:
Taking surface tension into account ,pressure at the base would be P0 - 2T/r + ρgh . Right ?
Yes, for a hemispherical surface contour.
 
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Related to Hydrostatic pressure in a tilted tube

1. What is hydrostatic pressure in a tilted tube?

Hydrostatic pressure in a tilted tube refers to the pressure exerted by a fluid at a specific point in a tilted tube due to the weight of the fluid above it.

2. How is hydrostatic pressure affected by the angle of tilt in a tube?

The hydrostatic pressure in a tilted tube is directly proportional to the angle of tilt. This means that the greater the angle of tilt, the higher the hydrostatic pressure will be.

3. What is the equation for calculating hydrostatic pressure in a tilted tube?

The equation for calculating hydrostatic pressure in a tilted tube is P = ρghsinθ, where P is the pressure, ρ is the density of the fluid, g is the acceleration due to gravity, h is the height of the fluid column above the point, and θ is the angle of tilt.

4. What factors can affect hydrostatic pressure in a tilted tube?

The factors that can affect hydrostatic pressure in a tilted tube include the density and height of the fluid, the acceleration due to gravity, and the angle of tilt.

5. How is hydrostatic pressure in a tilted tube used in scientific experiments?

Hydrostatic pressure in a tilted tube is commonly used in experiments to measure the density of a fluid or to demonstrate the relationship between pressure and depth. It is also used in industries such as oil and gas, where the pressure at different depths in a tilted wellbore is crucial for drilling and extraction processes.

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