Glycerin is poured into an open U-shaped tube until the height in both sides is 20 cm. Ethyl alcohol is then poured into one arm until the height of the alcohol column is 20 cm. The two liquids do not mix. What is the difference in height between the top surface of the glycerin and the top surface of the alcohol?

Video transcript

Hey, everyone. Let's go through this problem. A U shaped tube has two equal water columns, 30 centimeters high on each side, cooking oil with a density of 920 kg per cubic meter is added to one side of the U tube, creating a 30 centimeter oil column. The two liquids form separate layers, determine the height difference of the top surfaces of liquids, water and oil in the U tube. And we have four multiple choice options. Option, a 2.4 centimeters, option B 27.6 centimeters, option C 16.2 centimeters and option D 13.8 centimeters. All right. So first off, before we try solving this, this is the type of problem that's kind of hard to understand what's going on without drawing a good diagram. So I'm gonna try to sketch what's happening here. So here is our U shaped tube. Here is the other side of its walls. OK. So at some point, this tube was filled with water of equal heights. So I'm gonna draw a little dig a little dotted line here representing where the water levels may have been. And then at some point, then we add 30 centimeters height worth of oil to one side of the tube, which is going. So let's say that the oil has been added to the right hand side of the tube, which is going to lower the water level on this side of the tube. So let's say that the water level has been lowered by a height of h and now as a result, this entire right hand side above that lower dotted line is now filled with oil and the rest of the tube is now filled with the pushed down water. So if we lowered the water level on the right hand side by a level of H, then that means that it raised on the other side of the tube by that same height. So the new water level on the left hand side of the tube is at a distance H above where it originally was as I've drawn in this diagram. Now, in order to determine the height differences between the tops of the fluids, we can try to relate this to the hydrostatic pressure formula because remember that the formula for the pressure due to some depth of fluid is equal to the density of the fluid multiplied by the gravitational acceleration. G multiplied by the depth of the fluid above the point we're analyzing. So if you want to find a relationship of between these two heights due to the pressures of the fluids, let's first pick out two points in the fluid where we know the pressures are going to be equal. And since we know that pressures are generally a function of depth, I'm going to pick out two points that are of the same fluid. So they have the same density and also are at the same height. So I am going to pick out as my point A, I'm going to pick out the surface of the water in the right hand side of the tube just underneath where all the oil is its point A. And then I'm going to define point B to be this in the water at the same height on the left hand side of the tube. So that's going to be represented by the dotted line I'm drawing now which is at another distance H below the original water level. So this, that's what I'm going to call point B. So because points A and B are in the same fluid and at the same heights, that means that we can assume that the pressure at point A is gonna be equal to the pressure at point B. So now let's expand out these pressure formulas a bit. So the more full formula for pressure is that pressure is equal to some absolute pressure plus the height difference formula that we mentioned earlier row G D. So instead of so for part, so for P sub A, I'm going to write the at atmospheric pressure, P knot plus the row G D value for the oil since the point is just underneath the pressure due to the oil and that's where the additional pressure contribution is coming from. So I'm going to use the subscript C to refer to the oil uh because it stands for cooking oil. And so I can not create confusion between the O we're using as the subscript for the atmospheric pressure. So we're adding row sub C, the density of the cooking oil multiplied by the gravitational acceleration G multiplied by D which is going to be defined as the depth of the, of the cooking oil. So the depth of what I've labeled red, the depth of the oil column is D sub C which the problem tells us is just going to be 30 centimeters. So off the, off the side, I'm gonna label that, that D sub C is 30 centimeters. OK. This is going to be equal to the pressure at point B. So first that's going to be so put the pressure at point B is the atmospheric pressure P not plus the row G D formula for the depth of the water above point B. So that's going to be the density of the water row sub W G and D sub W the depth of the water. So first notice that because we have the atmospheric pressure terms on both sides of the equation, they can cancel right out. And the remaining terms both have the G in it for the gravitational acceleration. So they can also cancel out. So we're just left with the formula saying that row C DC is equal to row W D W. Now recall that the density of water is equal to about 1000 kilograms per cubic meter. This means that everything in this expression is known except for D sub W the depth of the water above point B. However, if we look at the diagram that we've been building all this time, we can see that the depth of the water above point B is just going to be two times H so two times the depth through which the water sunk. So instead of D sub W, I'm gonna write two H. Now H is the only unknown here and we can use that to find where the water levels are at. So I'm going to algebraically solve this for H by dividing both sides of the equation by two row sub W. So H is equal to row sub CD, sub C divided by two row sub W. So now let's just plug in the things we've been given for row sub C, the density of the cooking oil. The problem tells us that that's 920 kg per cubic meter. It's 9 20 kg per cubic meter multiplied by the density of the cooking oil which the problem tells us or the, the depth of the oil which the problem tells us is 30 centimeters. So converting that into meters, that's 300.30 m and divided by two, multiplied by the density of the water kg per cubic meter. If you put that into a calculator, then we find that H has a value of about 1.38 m or alternatively in centimeters, 13.8 centimeters. Now remember the problem asked us to find the height difference between the top surfaces of the liquids. So the distance between where the top of the oil is and where the top of the water is. Delta Y is what we actually want to find. And just from looking at the diagram, we can see that we can find delta Y by taking D sub C and then subtracting to H D sub C is given to us and H is what we just found. So if delta Y is equal to D sub C minus two H, then we can very easily plug in what we found. So D sub C is given the problem is 30 centimeters and then we just subtract two multiplied by H which we just found to be 13.8 centimeters. We put this into a calculator and we find a value for delta Y of about 2.4 centimeters. And that there is then the answer to our problem. And if you look at our multiple choice options, we can see that option A does indeed say 2.4 centimeters So that is the answer that's all for now. And I hope this video helped you out if you'd like more practice, please check out some of our other videos which will hopefully hopefully give you more experience with these types of problems, but that's all for now. And I hope you all have a lovely day.