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I've been learning about pressure and fluid dynamics, and I've stumbled onto a problem. Say you have a straw in a cup. The surface area of the water in the cup is much greater than that of the straw.

So, I would expect Atm. Pressure × Large Area (of the cup) gives a much greater force than Atm. Pressure × Small Area (of the straw). In fact, the force is so much larger I would expect a spectacular jet of water, which obviously does not happen.

What am I missing?

Thanks

EDIT: So I chose a really bad example. Consider any amount of fluid with two openings of different size (facing upwards). Shouldn't the force due to the atmosphere shoot the water out the smaller opening, since the force on the larger opening due to the atmosphere is greater?

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  • $\begingroup$ Why do you even need a straw for that to happen? If it did happen that way, the vertical walls of the straw would play no part, so even if you just consider a small cross section of the surface, it should spurt out into a jet! $\endgroup$ – mikhailcazi Apr 27 '14 at 14:03
  • $\begingroup$ I know! But this problem shows up in different ways too... imagine a horizontal tube of water with two (upward facing) vertical tubes leading out of it, with different areas. If the whole system is filled with water, I would expect a jet of water out of the vertical tube with smaller area. I know this is wrong, but I don't understand why by looking at the math. (See this image aplusphysics.com/courses/honors/fluids/images/…) $\endgroup$ – Mahkoe Apr 27 '14 at 14:06
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    $\begingroup$ Haha I like your question though. Funnily, I know why it won't happen, but I can't seem to explain it with the mathematics of it all. Hang on, it'll come to me. $\endgroup$ – mikhailcazi Apr 27 '14 at 14:36
  • $\begingroup$ I think I have the answer. I'm posting an answer. $\endgroup$ – mikhailcazi Apr 27 '14 at 16:18
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You must have learnt about the principle that:

Pressure exerted by a fluid is equal for all points at a certain depth.

That's where you've gone a bit wrong. If you consider two points in the fluid just below the surface - one inside the straw, and one outside, it is the pressure at these two points which will be equal, not the force!

Secondly, you've considered the total force acting on the outside area, and then said that this total force will act upwards on the tiny portion inside the straw. Is this true? No! This is where your mistake lies.

That whole force will not act on the straw! Even though that is the total force acting on the outside-area, it isn't going to be translated throughout the liquid. It is the pressure which is going to be equal at those points. So you can't simply subtract the forces and say that since one is substantially greater than the other, there won't be equilibrium.

If you equate the pressures, you'll see that it is completely in sync. Hope it's clear! :D

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  • $\begingroup$ Ahhhhh so only pressure is "transmitted" through a substance, not force! So that's where I went wrong. Thanks for sticking with me! $\endgroup$ – Mahkoe Apr 27 '14 at 17:17
  • $\begingroup$ @Mahkoe No problem!! :) Force and pressure are very closely related. Pressure is just simply force by area, so it's gonna be wrong to say that force also will not be transmitted. Yes, if you put more force on the water surface, it will put force on the rest of the liquid, but we can't really calculate the value of this force at a certain point. We can, however, calculate the pressure at any depth, by $P_h = P_{atm} + \rho gh$, and then consider an area and calculate the force on that area. :) $\endgroup$ – mikhailcazi Apr 28 '14 at 8:24
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It is easier to think of this in terms of energy. It's a bit like a lever, where a large force on one end balances a small force on the other. Large force * small distance = small force * large distance.

Imagine putting a piston in the straw and pushing a small amount of water down a large distance. This raises the level of a large amount of water outside the straw a small distance.

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The straw plays actually no role in this thought experiment. You can consider the liquid in the cup as consisting of many of such sub-divisions (equal in size to the straw). And each of them exerts pressure in exactly the same way, and of exactly the same value.

In other words, you cannot consider the water within the straw as one (separate) body, and water outside of it as another. This is fluid, which means its particles act separately of each other.

EDIT: You can also consider it in terms of variables and equations. See Wikipedia, it says that:

"Pressure is the amount of force acting per unit area."

$$p=\frac{F}{A}$$

So the force per unit is exactly the same inside and outside the straw.

But then, if you add more water to the cup with the straw already in it, the force per unit will be greater outside the straw, because the depth of water will be greater. Therefor the water inside the straw will be pushed up until it levels off.

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  • $\begingroup$ Maybe I chose a poor example, but see the second comment on the original post. Basically, if you have one fluid with two different areas of opening, the force due to the atmosphere throughout the liquid should shoot it out the smaller opening. I know this is false, but the math won't show it for me. Thanks for your answer! $\endgroup$ – Mahkoe Apr 27 '14 at 14:12
  • $\begingroup$ I understand. However, see the second paragraph in my answer. This method of dividing water does not make the water in the two openings two different bodies. It's still one container exactly equivalent to a simple bucket. $\endgroup$ – bright magus Apr 27 '14 at 14:19
  • $\begingroup$ Ohhhh okay! This answers my question then, and the math works now. Although to be honest I don't really get it... Thanks! $\endgroup$ – Mahkoe Apr 27 '14 at 14:23
  • $\begingroup$ OK. See my edit. $\endgroup$ – bright magus Apr 27 '14 at 14:48

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