Pressure not matching at same height in a U-tube

[Avimanyu Ray]

Can we please look at the following attached file image?
It shows a utube filled with a little water and the rest is oil.
Thus, the height of oil in one arm will be more than the height of water in the other.
When we compare the pressure at the height of the surface of water with that corresponding height of the oil, the pressures don't match. Why? Please refer to the attached image file.

 

[jbriggs444]

Let us examine the first equation in your picture:

$$P_0+(h_1-h_2)\ \rho_2\ g = P_0$$

The left hand side looks like a correct computation for the pressure at the oil-water interface in the right hand tube. You take the surface pressure and add the contribution of the column of water with density $\rho_2$. But what leads you to assert that that pressure is equal to $P_0$?

One could conclude from that equation that either $h_1 = h_2$, that water has zero density or that we are working in zero g. All three are silly, so that equation is obviously incorrect.

Edit: We also have the curious situation that although $\rho_2$ is greater than $\rho_1$ (water is denser than oil), we nonetheless have the denser fluid floating on top of the less dense fluid. Possibly this is a capillary U-tube (without an accounting for surface tension).

 

[sophiecentaur]

Can we please look at the following attached file image?

Where did the image come from? Is it an exercise to spot what's wrong or a 'real situation?

 

[gleem]

What are you looking for, the difference in height of the two columns?

 

[gleem]

Consider this diagram
.

 

[Avimanyu Ray]

Where did the image come from? Is it an exercise to spot what's wrong or a 'real situation?

It's an exercise to spot what's wrong.

 

[Avimanyu Ray]

What are you looking for, the difference in height of the two columns?

Precisely, I want to know what are we observing wrong when we concentrate on the dashed line XY. When we are showing both the pressures on either side on the line are equal, where are we wrong?

I want to know where this assumption is going wrong? Because we know, both the heights won't be equal.

 

[Avimanyu Ray]

Consider this diagram
.View attachment 226373

Wow, thanks a lot.
I got the answer by the diagram you indicated here :)