PhD., University of Maryland
Jonathan is a published author and recently completed a book on physics and applied mathematics.
According to Pascal's Principle, in a uniform fluid, pressure depends only on height. There can be no pressure gradient except for that caused by liquid pressure. This is seen in hydraulic jacks which are used to lift heavy objects.
Let's talk about Pascal's Principle. Pascal said that if you've got a uniform fluid then the pressure can depend only the height within the fluid. So the pressure over on this part of the fluid got to be the same as the pressure over on this part. Now this is only true when it's a uniform fluid. So it could just like be a bath of water or a big tub of mercury but it can't be a tub of mercury and water. It's got to be the same fluid all uniform throughout, so let's see why this works. Well if we consider what would happen if it wasn't working then we've got a pressure over here that's bigger than a pressure over here. Now it doesn't cost the fluid anything to rearrange itself horizontally because it doesn't have to support anymore water or anymore other type of fluid. If it's doing that it's still at the same height above the surface of the earth. So that means it's just going to rearrange itself and in this situation there'll be a flow of fluid over there until the 2 pressures are the same. So basically what we're saying is, if Pascal's principle isn't satisfied then things will rearrange themselves until Pascal's principle is satisfied. So that allows us to assume that Pascal's principle is satisfied.
Alright let's see how we can use this principle to solve a problem. So I'm going to take a u-tube you may not know what one of those things is but here it is, is just a u-shaped tube of glass and I'm going to pour some water in it. Then into the left hand of this u-tube I'm going to pour oil which has a smaller density 800 kilograms per cubic meter versus water is 1,000 kilograms per cubic meter and I'm going to do that until I've got 5 centimeters of oil in the left hand side. And what I want to know is what's the difference in height between the 2 columns of water that I have in the u-tube remaining? Alright we can't use Pascal's principle directly to answer this problem because it's not a uniform fluid, I've got water and I've got oil.
However, the water part is uniform, that's all water so that means that I can draw a horizontal line over like that and these 2 represent the same horizontal heights in the water, which is all uniform and that means that I can apply Pascal's principle. So the pressure here has got to be the same as the pressure here, well the pressure here comes from the column of oil and we know from our discussions of liquid pressure, fluid pressure that, that pressure is equal to the density of the oil times the acceleration due to gravity times the height of the column of the oil. Now that pressure has got to be the same as the pressure over here inside of the column of water. This pressure comes from the height of water, which I'm going to call h, because that's what we want, that's our answer. So I'm going to say well that's equal to density of water times gravity times this height that we want.
Now happily gravity cancels, so our height that we want will be the density of the oil divided by the density of the water times the height of the oil. And now all we got to do is plug in numbers. So we got 800 over 1000 or 8 over 10 times the height of the oil which is 5 centimeters and then we could cancel and our answer is 4 centimeters. Okay so it's just that easy essentially all the problems involving u-tubes like that with different liquids all work the same way. You just have to figure out where is there a uniform fluid that you can apply Pascal's principle to. And as soon as you find that you can go ahead and apply Pascal's principle and just determine what the pressure is going to be on one side. Alright now the main use of Pascal's principle is not in these u-tube things, you might have guessed that. In fact the main use comes from the operation of hydraulic jacks.
Now what a hydraulic jack is, is a machine that uses Pascal's principle to allow us to multiply the effect of a force that I have going in to a fluid. So what I'm going to do is, I'm going to input a force over here f in and I want to see how much force I get over here f out. Now the beautiful thing about this is that Pascal's principle says that the pressure is the same not the force, the pressure. And pressure of course is force divided by area. So we'll have pressure 1 equals pressure 2 that means that force in divided by area 1 has to equal force out divided by area 2. So that means force out is just the ratio of the 2 areas times the force in. Now when I make that jack I can make that ratio of 2 areas essentially anything I want. So I've got ratio for example is 40 then the force that I get out pushing up my car over here on the big side is 40 times the force that I put in. So maybe I can only lift 50 kilograms on a good day but 40 times that right is 2,000 kilograms now I can lift the car. So, but it seems like I got something for nothing, this actually is very similar to the pulley systems because that also allows you to multiply force.
Essentially what happens is, if I want to raise this a certain amount let's say that I wanted to raise it 1 meter well that means that I need to push the fluid down an amount that would increase here 1 meter. Now the volume is going to be the same, so since this is a much bigger area to increase this by 1 meter, I'm going to have to decrease this height by a lot more than 1 meter. So I'm going to have to stand over here going like this with my little in force a bunch of times just to get 1 little crank over here with the big out force. And so it's that type of conservation that makes it so that while this is useful it doesn't violate any physical principles. Alright that's Pascal's principle.