Question about a u-tube manometer with different diameters

[MattGeo]

TL;DR Summary

Fluid Height in the evacuated limb of a u-tube manometer when other limb is at atmospheric pressure. Manometer has 2 different diameters for each limb.

Let's suppose that a water filled u-tube is open to the atmosphere at one end and at the other end it is capped and evacuated. Suppose also that the diameter of the capped evacuated end is drastically larger than the open end. (Ignore effects of boiling or vapor pressure). Would the much larger evacuated limb of the manometer see the water raise up to a height of 34 feet? I feel like this must be the case but something about it feels incorrect. I have tried to convince myself with calculations and diagrams but I am having trouble seeing how a limb of very small diameter at atmospheric pressure could raise the much larger water column in the opposite limb.

 

[BvU]

Hello Matt, $\qquad$ $\qquad$ !

Yes, 34 feet or thereabouts (provided enough water is available ). Keyword here is pressure.

You know (right?) that $\Delta p = \rho g \Delta h$ and you know that pressure at same levels in a fluid is the same.

Mercury barometer is smaller, some 76 cm, because of 13.6 times higher $\rho$. Principle is the same.

 

[A.T.]

I am having trouble seeing how a limb of very small diameter at atmospheric pressure could raise the much larger water column in the opposite limb.

Note that to rise the level in the wider closed arm by a certain amount, the atmospheric pressure has to depress the level in the thinner open arm by much more.

It's a bit like a lever: Smaller force applied along a larger distance vs. larger force applied along a smaller distance.

 

[anorlunda]

I am having trouble seeing how a limb of very small diameter at atmospheric pressure could raise the much larger water column in the opposite limb.

I'm sure you have seen hydraulic jacks like the one in the picture. How can a man with little effort lift a 4 ton load? It is the same principle.

 

[MattGeo]

Note that to rise the level in the wider closed arm by a certain amount, the atmospheric pressure has to depress the level in the thinner open arm by much more.

It's a bit like a lever: Smaller force applied along a larger distance vs. larger force applied along a smaller distance.

I was trying to envision it as the atmosphere being the force applied to a problem using Pascal's Principle. It should be the same thing, essentially? There is just something very counter-intuitive about it to me. I guess because we have to accept that the pressure applied to the small area is distributed equally to everywhere else in the fluid.

 

[MattGeo]

I'm sure you have seen hydraulic jacks like the one in the picture. How can a man with little effort lift a 4 ton load? It is the same principle. View attachment 253409

My hunch is that it would just be like Pascal's Law and that we would treat the atmosphere as the applied force. Something about it seemed counter-intuitive to me because I suppose a constraint would be simply having enough water to actually fill the larger column. Also sometimes I forget to consider that a pressure change applied to the fluid is a change distributed throughout the fluid, so it has to work to raise the large column. T

 

[A.T.]

I was trying to envision it as the atmosphere being the force applied to a problem using Pascal's Principle. It should be the same thing, essentially?

Yes, see also Pascal's barrel, where a small amount of liquid in a tall pipe generates a huge pressure.

https://en.wikipedia.org/wiki/Pascal's_law#Pascal's_barrel

 

[MattGeo]

Hello Matt, $\qquad$ $\qquad$ !

Yes, 34 feet or thereabouts (provided enough water is available ). Keyword here is pressure.

You know (right?) that $\Delta p = \rho g \Delta h$ and you know that pressure at same levels in a fluid is the same.

Mercury barometer is smaller, some 76 cm, because of 13.6 times higher $\rho$. Principle is the same.

Hello Matt, $\qquad$ $\qquad$ !

Yes, 34 feet or thereabouts (provided enough water is available ). Keyword here is pressure.

You know (right?) that $\Delta p = \rho g \Delta h$ and you know that pressure at same levels in a fluid is the same.

Mercury barometer is smaller, some 76 cm, because of 13.6 times higher $\rho$. Principle is the same.

Thanks,

So it really is the same as Pascal's Law. The atmosphere provides the pressure to the small diameter side and the pressure change is transmitted to all parts of the fluid so the evacuated larger section will rise in accordance (Pressure x Area = force). So a large force will be experienced to raise the column to 34 feet.

I suppose it helps to consider the fact that if force and area increase in constant linear proportions to equal the same pressure, and that doubling the area of a geometric object at constant density will also double the mass, you'd need double the force to hold it up, but pressure and height are ultimately the same.

 

[MattGeo]

Yes, see also Pascal's barrel, where a small amount of liquid in a tall pipe generates a huge pressure.

https://en.wikipedia.org/wiki/Pascal's_law#Pascal's_barrel

Ahhh yes, I have actually heard of this experiment before. This is actually very insightful to consider. Thanks