# Why should pressure be same at same height?

Suppose I have an U shaped tube, and I fill the left side with mercury and right side with water. Also I take another U shaped tube, and fill it with water completely, as shown in the below picture:

Why the pressure at $P_1$ should be same as the pressure at $P_2$ ?
Why the pressure at $P_3$ should be same as the pressure at $P_4$ ?
Would the pressure at $P_1$ be same as the pressure at $P_3$ (I guess not) ?

(Orange lines indicate they're at the same height, the orange line is functionary and not a tube)

• The mercury would sink below the water, as it's much denser. If you're maintaining that it's fixed to be above the water by some sort of barrier, then the pressure at $P_1$ is not the same as $P_2$, and there is a net force on the barrier between the mercury and the water as a result. – probably_someone Jul 23 '18 at 20:17
• It is not. Pressure is equal at same heights as long as it is the same fluid. Your first tube doesn't satisfy this. – FGSUZ Jul 23 '18 at 20:22

Pressure is determined by the height of a column and the density of the fluid in a column: $p=\rho gh$.

Why the pressure at $P_1$ should be same as the pressure at $P_2$?

If we use dimensions in your diagram, they would not be the same.

What we can say about the first tube, is that the pressure at the bottom of the two columns will be the same. If that was not the case, there would be a horizontal gradient of pressure, which would cause fluids to move until the pressure equalizes.

If so, we can state, that the weight of the mercury column is equal to the weight of the water column. But the fraction of mercury column above the line is much smaller than the fraction of the water column above the line, so their weights and therefore the pressure at the level of the line would not be the same: $P_2$ would be greater than $P_1$.

Why the pressure at $P_3$ should be same as the pressure at $P_4$?

Because the columns of the water on both sides should have the same height (since the horizontal pressure at the bottom of the columns should be the same) and, therefore, equal fractions of the left and right columns will be above the line and, therefore, they will create equal pressure at the level of the line.

Why the pressure at $P_1$ should be same as the pressure at $P_3$?

It is hard to tell, since, in general, the drawing is not accurate (considering that the density of mercury is $13.6$ times greater than the density of water), but if you understand the answers to the first two questions, you should be able to figure that out for any given dimensions.

• Thanks. Sorry but I couldn't understand that why should the pressure at the first tube (the one containing mercury and water) should be same at all points in the bottom ? Can you elaborate a bit on " If that was not the case, there would be a horizontal gradient of pressure, which would cause fluids to move until the pressure equalizes." ? – cdt Jul 24 '18 at 3:55
• @AlexKChen The pressure is not the same "at all points in the bottom" - it is the same along any horizontal line at the bottom part of the tube. The horizontal line you have drawn partially goes through air, so mercury cannot flow to right and water cannot go to left, so pressure could be different. But if you draw a horizontal line at the bottom part, say, right at the bottom of the two columns, the water is free to move horizontally, so, if pressure was different on left and right sides, the water would flow and equalize the pressure. This is the mechanism behind communicating vessels. – V.F. Jul 24 '18 at 4:08
• Yes, by "pressure is same at all points in the bottom" I mean that they're same at all points in the bottom parallel to the horizon. But I don't understand your last line (which was my original question): if there's pressure difference then why should water travel from the higher pressure region to the lower pressure region ? And what's the guarantee that this process would terminate (it might happen that when water goes from the higher pressure region to the lower pressure region, a higher pressure is created again at the higher pressure region so the process never stops) – cdt Jul 24 '18 at 7:41

Pressure equals force over area. Let us observe tube 2. The area os obviously the same. Since the whole tube it uniform in density, then the water level should be the same such that the volume is the same on both sides so that weight is the same. Since weight is the downward force, it will allow pressure to be the same. If one side has a higher water level than the other, then it will have higher pressure. We know that the water will move from the region of higher temperature to the region of lower temperature such that the pressure is balance at equilibrium position.

For the first tube, since mercury is denser, less of it is needed to Ensure that pressure is equal on both sides of the tube, hence, it has a lower water level than water.

Lastly, since $P_1=P_2$ and the level at $P_3$ is not the same as $P_2$, it would imply pressure at $P_1$ and $P_3$ are not the same. Do note that here, we compare the levels of the same fluid.