Fluid Mechanics - U-tube and mercury

[jkb]

A U-shaped tube, open to the air on both ends, contains mercury. Water is poured into the left arm until the water column is 20.0 cm deep.

How far upward from its initial position does the mercury in the right arm rise?

So far, this is my work:

Using pressure = density(water) x g x height(water column) and neglecting atmospheric pressure as both sides would experience it, I equated that equation to p = density(Hg) x g x height as they would equal on the horizontal axis. I found the height to be 14.7 mm but that answer is not correct.

I am confused and have gone at this in a number of ways. I know that the value 14.7 is not the difference between the new position and the initial position and so I've kind of hit a road block. Any input or thought or help would be appreciated!

 

[MathematicalPhysicist]

i think it should be:
$$\rho_{w}h'g+\rho_{Hg}h'g=\rho_{Hg}hg$$
where h'=20 cm and h is the height of the right arm.
in order to find the difference i think you need this equation:
x-h'=h-x
but I am not sure I am right here, I've done this stuff in high school two years ago.

 

[kyle8921]

Thank you for sharing your work so far. It seems like you have a good understanding of the basic principles involved in this problem. However, there are a few things that could be causing your answer to be incorrect.

Firstly, when using the equation p = density x g x height, it is important to make sure that the units are consistent. In your case, you have used millimeters (mm) for the height of the water column, but you have not specified the units for the density of water. Depending on what value you used for the density of water, this could affect your final answer.

Secondly, it is important to consider the pressure at the bottom of the U-tube. Since the tube is open to the air on both ends, the pressure at the bottom of the tube will be equal to atmospheric pressure. This means that the pressure on both sides of the tube will be equal, and the equation p = density x g x height can be used to find the difference in height between the two sides of the tube.

Lastly, it is important to consider the properties of mercury and how they differ from water. Mercury is a much denser liquid than water, so the height of the mercury column will be significantly smaller than the height of the water column. This means that the difference in height between the two sides of the tube will not be equal to the height of the water column.

To solve this problem, you will need to use the equation p = density x g x height for both the water and the mercury, and then subtract the two equations to find the difference in height between the two sides of the tube. This will give you the correct answer for how far upward the mercury rises in the right arm of the U-tube.

I hope this helps you to understand the problem better and find the correct solution. Keep up the good work!