Difficult question on pressure in liquids

[jontyjashan]

Homework Statement

There is a container of height 2 m
it is filled with water upto the brim
there are 2 holes made in it
one hole of area 5cm^2 is made at a height of 1.5 m
another hole of area 10cm^2 is made at a height of 0.5m
through which hole will the water rush out with more pressure

Homework Equations

P=hdg

The Attempt at a Solution

I cannot make out through which hole the water will rush out with more pressure.
i don't know whether pressure in liquids depend upon area

 

[minger]

This question is worded quite poorly if you ask me. When people say that a liquid is flowing with a lot of pressure, they often mean that it exerts a lot of force upon a stationary object; or more plainly, it has a high velocity.

If we are to look at pressure alone, then there are two pressures to consider: the pressure just before the fluid leaves the container, and the pressure afterwards. The pressure just before it leaves the container is simply the hydrostatic pressure that you have listed $$p = \rho g h$$. As the fluid then leaves the container, this pressure or energy is converted into kinetic energy, which can be illustrated in Bernoulli's equation which basically says that along a steamline the pressure energy + the velocity energy + the height equals a constant. Now, the final pressure is simply the pressure of the air, ambient.

So, whatever case has the higher initial pressure has the higher velocity. However, this is not pressure, as both fluids will leave at atmospheric pressure.

 

[tiny-tim]

Hi jontyjashan!

I cannot make out through which hole the water will rush out with more pressure.
i don't know whether pressure in liquids depend upon area

Well, pretend there's only one hole, and that the density of water is 1 …

what speed will the water leave the hole?

 

[jontyjashan]

So, whatever case has the higher initial pressure has the higher velocity. However, this is not pressure, as both fluids will leave at atmospheric pressure.

First of all there is only 1 container and 1 fluid not 2 fluids.
how can water from both holes leave with the same pressure.It is simple logic.What wordings inthe question you did not understand by the way.I just want to know from which hole water will come out with more velocity

 

[minger]

I understood the wordings [sic] better than you explained them. Pressure != velocity. Remember that. Any jet discharging into the atmosphere will have an internal pressure of 0 gauge. However, I was trying to explain to you the difference between the two.

Now, onto velocities. Before anyone simply throws you the answer, why don't you give it a try. What do you know, what do you need to find? What are you looking for?

 

[jontyjashan]

sorry ,If i said something wrong but
if we do this experiment practically,velocity of water from the upper hole is much high
but if we look at it mathematically,velocity of water from the lower hole should be high

 

[RoyalCat]

sorry ,If i said something wrong but
if we do this experiment practically,velocity of water from the upper hole is much high
but if we look at it mathematically,velocity of water from the lower hole should be high

Yes, you are incorrect. Take the limit of the 'upper hole', the very top of the container of liquid. The velocity at which it exits the container is 0, it does not leak out at all!

Interestingly enough, the maximum horizontal distance the water will reach isn't achieved from a hole at either extreme, but from a hole at the very middle. :)

 

[tiny-tim]

if we do this experiment practically,velocity of water from the upper hole is much high
but if we look at it mathematically,velocity of water from the lower hole should be high

Hi jontyjashan! Thanks for the PM.

Yes, mathematically the speed depends only on the pressure, and therefore on the depth, not on the size of the hole.

Did you do the experiment yourself?

How did you measure the water coming out of the two holes?

 

[RoyalCat]

Hi jontyjashan! Thanks for the PM.

Yes, mathematically the speed depends only on the pressure, and therefore on the depth, not on the size of the hole.

Did you do the experiment yourself?

How did you measure the water coming out of the two holes?

Wouldn't the fact that $$P\propto{Area^{-1}}$$ have a say in that?
So that means that the acceleration the pressure provides to the liquid is inversely proportional to the area of the hole it's bursting through, doesn't it? (Sorry for the poor wording, by the way. Haven't dealt with pressure properly yet)

And the acceleration it provides to the liquid, is what determines its exit velocity, and this would be determined by making an equation something like the following, right?

$$a=\frac{P}{mA}$$
$$a=Acceleration, A=Area, P=Pressure$$
$$P=\rho gh$$
$$a(A,h)=\frac{\rho gh}{mA}$$

So the acceleration, and in turn, exit velocity, is directly proportional to the height, and inversely to the area, isn't it?
Sorry if I'm way off on this, this is just my intuition talking here, other than knowing the units of pressure, I'm lost, haha!

 

[minger]

The velocity is inversely proportional compared to the area in the case of pipe flow or when the fluid is forced through the opening. In this case of a free jet, where the inlet can be assume to be an infinite resevoir, this is not the case.

The fact of any disparities you may see in a small experiment would be in any difficulties keeping the level of the fluid high enough such that constant pressure is maintained in both holes.

 

[tiny-tim]

… Sorry if I'm way off on this, this is just my intuition talking here, other than knowing the units of pressure, I'm lost, haha!

Sorry, RoyalCat, but you're just burbling.

 

[RoyalCat]

Sorry, RoyalCat, but you're just burbling.

:shy:
Guess I should give this a try once I actually study hydrodynamics, heh.

 

[jontyjashan]

i have tried this experiment practically,and i found that from the upper hole water just trickles out but from the second all the water comes out very fast

 

[tiny-tim]

i have tried this experiment practically,and i found that from the upper hole water just trickles out but from the second all the water comes out very fast

But that is mathematically correct!

(I'd thought you were saying it was the other way round )

At the lower hole, there is more pressure , so the water should come out faster.

 

[minger]

Just so we don't confuse him tiny...

At the lower hole, just before the hole there is more pressure, which is a function of depth. That pressure then becomes atmospheric as the fluid leaves the hole, with the energy being converted into kinetic energy, the difference in pressure being the driving force.

 

[modulus]

As minger mentioned, the experiment is practically impossible, because it is not possible to keep a uniform pressure while the water flows out of the holes.

But, if we take this theoretically and conceptually, the hole from which water comes out with more pressure (considering an infinite reservoir of water), is the one from which water goes a larger horizontal distance.

So, I think we might try to attempt this by figuring out the initial velocity with which the water comes out from each hole, and then, use that knowledge to figure out the horizontal range (considering each water particle is a projectile under horizontal projection) with the formula u*sqrt(2h/g).

But, here, we come at another problem- if we use this formula, we must consider the area around the infinite reservoir to be a vaccum, and, if we consider that, then, the water inside the reservoir would have infinite pressure (being exerted on the walls of the container).

Or, we can use the formulas of projectile motion which also consider drag provided by air... but I don't know those...

 

[jontyjashan]

ya,there is need to find the range of the water from both holes.but is it necessary that water having more range will have more velocity

 

[ideasrule]

ya,there is need to find the range of the water from both holes.but is it necessary that water having more range will have more velocity

You need to find the range? I thought you just needed to find out which hole water will rush out faster from.

Also, no, it's not true that higher velocity = higher range. If the hole is too close to the ground, then the water will get to travel almost no distance at all before hitting the ground. if the hole is too high, the water's exit velocity would be very low. Somewhere between those two extremes is the height that gives the longest range.

 

[jontyjashan]

thats a part of it

 

[jontyjashan]

Also, no, it's not true that higher velocity = higher range. If the hole is too close to the ground, then the water will get to travel almost no distance at all before hitting the ground. if the hole is too high, the water's exit velocity would be very low. Somewhere between those two extremes is the height that gives the longest range.

i think what you said is not true

 

[modulus]

Also, no, it's not true that higher velocity = higher range. If the hole is too close to the ground, then the water will get to travel almost no distance at all before hitting the ground. if the hole is too high, the water's exit velocity would be very low. Somewhere between those two extremes is the height that gives the longest range.

Jontyjashan, he's not wrong, he's completely correct.

And, mabey the initial velocity with which the liquid comes out does not determine its pressure.

Again, I agree with minger, the question is worder wrongly- as soon as the water comes out of the vessel, its pressure becomes equal to that of the external environment, so, mabey you want to ask that 'from which hole will come out with greater force'.

 

[jontyjashan]

first of all i have already clarified that i wanted to ask by which hole will the water come out with more velocity and secondly can you give me a proof to show Higher Velocity is not equal to higher range.come out with things of your own.Just don't keep on agreeing with everyone

 

[modulus]

Well, if you want to find out the hole from which water comes out with a higher velocity, here you go:

Firstly, we know that the water exerts an equal pressure on all sides of the vessel, so we can use the pressure exerted by the liquid, in the horizontal direction at the respective point, and then figure out the force with which the water comes out. Next, figure out the relation between force and velocity and you'll get your answer for each hole.

At least, I agree with everyone, instead of contradicting their every statement when they are trying their level best to help you. You better remember that we are just 9th graders and they've got a lot more knowledge than us. So, I'd advise you to stop using your arrogant and hostile attitude towards everyone, and just try to learn a bit of something from them.

Think twice about what you're going to say before you say it.

 

[modulus]

The water accelerates out of the vessel with an instantaneous acceleration (I think). Now we'll just have wait for the dudes who know about differential equations to get online before we can proceed.

 

[jontyjashan]

I am not here for contradicting everyone.I just try to derive sense from every word they speak and i don't blindly agree with them.This site is for help and not for argument.Sorry if I said something wrong but my attitude is like this.
Secondly i also know my limits . Dont try to show a domineering,dominating and patronising attitude.
Again what you said in this post, I don't agree with it .Can you give detailed explanation

 

[modulus]

The force with which the liquid comes out mass*v-0/time.

Here, the initial velocity is zero, and the time is negligible, making the acceleration instantaneous. Since the acceleration of the particles of water are zero, we'll hae to use differential equations- I don't know how to use them.

Let's not fight, and focus on the question.

Please.

I'm sorry if I said anything that hurt you.

 

[jontyjashan]

lets both of us put our brains together.
First of all initial velocity is not zero.
Secondly are we considering only one jet of water or are we taking the water to be infinite

 

[modulus]

Can't we talk about this later? I'm getting tired

 

[rl.bhat]

Homework Statement

There is a container of height 2 m
it is filled with water upto the brim
there are 2 holes made in it
one hole of area 5cm^2 is made at a height of 1.5 m
another hole of area 10cm^2 is made at a height of 0.5m
through which hole will the water rush out with more pressure

When you say the height of the hole is 1.5 m am 0.5 m, I presume that they are from the bottom of the container to the center of the hole. The velocity of the water molecules on the horizontal diameter of the hole will be the same and is equal to sqrt(2gh). The velocities of the molecules above the diameter will gradually decrease and below the diameter the velocities will gradually increase. But the average velocity will be approximately the velocity of the molecules at the center of the hole. So the velocity of the out flow does not depend on the diameter of the hole. It only depends on the depth of the hole from the water surface. The rate of outflow depends on the diameter of the hole.

 

[queenofbabes]

Do you understand Bernoulli's equation? If you use that the problem is pretty trivial...