Express the pressure as a gauge or vacuum, depending on what value you get.

Express the pressure as a gauge or vacuum, depending on what value you get.

Let’s move on to the next problem in the tutorial. So you’re given an inclined manometer. So that’s different from a u-tube manometer. And it’s used to measure the pressure of a gas within a reservoir, right there.

Using the data on the figure, determine the gas pressure. So that’s what you need to find out. So use the data in the figure, OK? And determine the gas pressure. So Pgas is what you need to find.

Express the pressure as a gauge or vacuum, depending on what value you get.

And finally, they’re asking you what is the advantage does the inclined manometer possess over a u-tube manometer? So let’s deconstruct this problem.

So you’re given this inclined manometer. And what you’re asked to find is this pressure of the gas. Pgas is what we need to find. And, based on the principle of the manometer– so the data that you provided are all of this stuff.

So you’re given this inclined length. That value is given to you. This L equals 15 inch. That’s what that is. You’re given this angle. That’s given as 40 degrees with respect of the horizontal. And then what we do is go ahead and proceed with our analysis.

So, based on our analysis– and the analysis is going to be pretty straightforward here. The analysis will say pressure at point a equals the pressure at point b. That’s basically Pascal’s law. The equality of pressures in a continuous fluid, OK? So Pa equals Pb.

So, if Pa equals Pb– we know Pa is nothing but Pgas, OK– so Pgas has got to be equal to– the hydrostatic pressure, basically, is going to be– well, the pressure at point b is going to be the sum of the atmospheric pressure plus– remember the hydrostatic pressure or rho gh that we’re looking for is this value, right– the vertical distance. That there is H– plus rho of the fluid times g times H.

So, using simple trigonometry, we know, well, this angle is 40 degrees, right? So, using trigonometry, we know H over L equals sine theta, where theta is– or sine 40. Let’s just do that that way, right? So our H equals L sine 40.

So, therefore, your Pgas simply becomes Patmosphere plus rho g times L sine 40. So let’s see what this is. The density of the oil is given to us here. This is what’s given. See this is what happens when you skip a step, right? So that is given. That is the oil density.

You’re given the Patmosphere. Look at the units there. That’s in pound force per inch square, OK? So we need to be a little careful when you assemble all of this together. So let’s just put this down.

So this is going to be 14.7 pound force per inch squared plus density of the [? oil– ?] is 54.2 pound force per foot [? cube– ?] times [? g– ?] is 32 foot per second square– times the L values in inches. So we’ll convert that into foot also.

15 into 12 inches per foot, right? So into 1 over 12 foot per inch. So this is 15. Let’s say. Multiply that by 15 inches.

So let’s just be sure that we have all of the units captured properly here, OK? So that’s that. Oops. Well, [? now ?] that multiplied by sine of 40. Now we have everything that we need, all right?

So from here you get Pgas equals 14.7 pound force per inch square plus 54.2 pound force per foot cube times 32 foot per second square– rho g– times L– was 15 over 12 inches per foot. So, basically, 15 over 12 foot per inch, right? So that is inch times foot per inch. So all of that cancels off, OK? So many– oh, times sine of 40. So that’s what we get.

So then we have 14.7 lbf inch square plus– [INAUDIBLE] how much is this? So let’s tally up the units. It’s not foot cube. It’s not foot square but foot cube, right there.

Oh, that’s the mistake that I made. I’m so sorry. Let me just go back and retrace my path. There you go. That’s not pound force. That’s pound mass, OK? We need to be careful with these kinds of units. English units can trip you. So that’s pound mass.

So we have 54.2 times 32 times 15 over 12 sine 40. And then, if you look at the units, you have pound mass per feet cube times feet per second square times feet.

So let’s just make sure– so that’s all and that. So you have pound mass per foot second square, OK? That’s what you’ll end up with, which is 14.7 pound force per inch square plus whatever this is. [? It’s– ?] we’re Going to do this. 54.2 times 32 times 15 divided by 12 times sine of 40. So that’s 200,673.15. Now these weird set of units now– pound mass per feet second square. So let’s see how we can get out of this mess.

So what we’re going to have to do is– that’s mass. So you need a force divided by mass. I’m going to have to make sure that I have my units right here. Let me just redo this part– it is all of this– and do everything in inch and see if that’ll work.

I’m sorry. But units– it’s kind of good that I go over this. So you understand the difficulty with dealing with English units if you’re not used to it all the time, right? So let’s see. Right there, we have that much. That’s 14.7 pound force per inch square plus 54.2 pound force per foot cube.

Let’s try to do this in terms of inch cube. So foot cube times 1 feet is 12 inches. So 1 over 12 inch cube per foot cube, right there. So we’ll take care of that part. So let me make sure that I got that right. Oh, pardon. I’ll put this in parentheses– times 32 foot per second square, OK?

So this is pound mass. So you might benefit by writing it as pound mass instead of just pound. So lbm is pound mass. It’s always better to write it that way.

So we have this times 1 pound force because we want everything in pound force because the units of pressure here are pound force per inch square. So 1 pound force per 32 foot per second square. That is nothing but mass over acceleration. So force over acceleration– that’s mass, right? That’s going to give you the units in pound mass.

So we have that, well, times lbm. So that’s fine [? because ?] we have this– times 15 inches. We have that. All of this should work out now.

So we cancel off the pound mass and the pound mass. We cancel off the foot cube and foot cube, the foot and foot there, second square second square.

So everything is now in the correct units. 14.7 pound force per inch square plus– that’s going to be 54.2 multiplied by 32– oh, this is going to be 12 cube. Let’s not forget that– divided by 1728– that’s 12 cubed– times 15– so many– pound force per inch square.

All what you get is Pgas. From here, it’s going to end up being approximately– [? let’s ?] see how much this is. 54.2 times 32 times 15– oh, we forgot the sine 40– times sine 40. How could I forget that– sine 40– times sine 40 16.722 [? multiply ?] 1728. What was that? Oh, no 32.

That’s good [? That’s right. ?] 54.2 times 15 times sine of 40. [INAUDIBLE] times 8 [INAUDIBLE] 7. This is 15.0 pound force per inch square, OK? Since Pgas is greater than Patmosphere, Pgauge equals Pgas minus Patmosphere, which is 0.3 pound force per inch square, OK? That’s part b.

So this is the answer to part a. So the last part– I will leave it as a little homework. I want you to figure out why an inclined plane manometer is better– or an incline manometer– is better than a standard u-tube manometer, OK? We’ll stop right here.