Hydrostatics: U-Tube Homework Solution

In summary, a U-tube with uniform cross section and vertical limbs of height 20cm contains mercury of relative density 13.6 in its bottom. Each limb has a column of mercury 6.9cm high. When water is poured into one limb until it reaches the top, the height of the water column can be found using the equation P = ρgh. The pressure is equal in both limbs at the reference line, where the height of the water column plus the height of the mercury is 20 cm. The relative density of mercury is given with respect to water, so there is no need to convert the densities. The error in the attempted solution was multiplying everything when calculating the pressure in the right limb instead of adding
  • #1
Darth Frodo
212
1

Homework Statement


A u-tube of uniform cross section with vertical limbs of height 20cm has mercury of relative density 13.6 in the bottom of it. The columns of mercury in each limb are 6.9cm high. Water is poured into one limb until it just reaches the top. Find the height of the water column.


Homework Equations



P = [itex]\rho[/itex]gh

The Attempt at a Solution



Please find attached.

Please excuse the massive x. I got angry.
 

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  • #2
Your picture is good, but I would put the reference at the bottom line.

There is 6.9 cm +x height of mercury in the left limb of the U-tube, and (6.9-x) cm height of mercury +h cm height of water in the right one. The length of both vertical limbs is 20 cm, the right limb is filled with water to the top, the height of the water column +the height of mercury is 20 cm: h+(6.9-x)=20 cm. The pressure is equal in both limbs at the reference line (the red one in my picture)..

It is said that the relative density of mercury is 13.6 - given with respect to the density of water. No need to convert the densities to kg/m3. If you do it, convert also cm to m.

Your error was that you multiplied everything when calculated the pressure in the right limb. The pressure of the water column is added to that of the mercury.


ehild
 

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  • #3
EXCELLENT! Thanks very much, this is extremely helpful!
 

1. What is hydrostatics?

Hydrostatics is the branch of fluid mechanics that deals with the study of fluids at rest and the forces acting on them.

2. What is a U-tube?

A U-tube is a simple device used for measuring fluid pressure and determining the density of a fluid. It consists of a glass tube bent into the shape of a U and partially filled with a liquid.

3. How does a U-tube work?

A U-tube works by utilizing the principle of hydrostatic equilibrium, which states that the pressure at any point in a fluid at rest is the same in all directions. When a fluid is added to one side of the U-tube, it exerts pressure on the liquid in the other side, causing it to rise until the pressures are equal.

4. How is a U-tube used in hydrostatics?

In hydrostatics, a U-tube is used to demonstrate the concept of Pascal's law, which states that pressure applied to a confined fluid is transmitted equally to every point within the fluid. It can also be used to measure the density of a fluid by comparing the heights of two different liquids in the U-tube.

5. What factors affect the reading on a U-tube?

The reading on a U-tube can be affected by the density of the fluid, the length and diameter of the tube, and the gravitational force acting on the fluid. The type of liquid used in the U-tube can also affect the reading, as different liquids have different densities and thus exert different pressures on the fluid in the tube.

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