Hey everyone, hopefully you'll try this practice problem here. So we've got water and oil in a U-shaped tube. The column of oil on the right side here is going to be 25 centimeters tall. Remember that this is H₂, this is 25 centimeters here because, just by convention, this is 2 and this is going to be 1. Alright, so we have that the distance between the two tops of the columns is 9 centimeters. So remember, that means that the top of the oil is here, the top of the water is over here, then the distance between them, which I'm going to draw like this, this is equal to 9 centimeters. And remember, this is the difference of the two heights, so this is my ΔH. Right? So it's important to remember, this is ΔH, not H₁. So H₁ would actually correspond to wherever the interface is between the oil and the water. Remember, that is the interface here. This is also in the other video where point D is. And if you go all the way across, that's also going to be where point B is on the other side. Remember, this distance here, the height of the water column, is going to be H₁, and that is going to be your H₁. Alright? So that's important to recognize here. H₁ and H₂ are always heights in liquids, whereas the ΔH is always sort of the gap that goes through air. Alright? So that's one way to remember it as well.
We want to calculate the density of the oil in this problem. Normally, we'd sort of take for granted that the density of oil is 800 kg/m³, but we're actually going to go ahead and calculate it here. Alright. So in other words, we need ρ₂. We know that ρ₁ is equal to 1,000 kg/m³ because it's water, and ρ₂ is basically what we're going to be looking for. Okay? So if I'm going to find the density of the oil, I'm going to have to relate this to my U-tube equation. Remember, we have that ρ₁ * H₁ is equal to ρ₂ * H₂. So if you're looking for ρ₂, then you have to have 3 out of the other 4 variables. Right? So we have ρ₁, that's the density of water, and we have H₂, which is the height of the column. What about H₁? Now hopefully, you guys realize here, that if we want H₁, we're going to have to use our equation for ΔH. So you could say that this is ΔH = |H₂ - H₁|. You move some stuff around, but it's actually really easy to see just by eyeballing it here that if the top of this whole column here is 25 centimeters and if the gap is 9 centimeters, then that means that this has to be the difference between 25 and 9, which is just 16 centimeters. And it's okay to keep everything in centimeters for right now, just in case we don't have to convert, alright?
So now that we know the H₁, we can go ahead and solve for ρ₂. So ρ₂ is going to be equal to ρ₁ * (H₁ / H₂). And so, therefore, what happens is you're going to have 1,000, that's the density of water, and this is going to be H₁. Remember, H₁ is the smaller one. This is going to be 16 centimeters divided by 25 centimeters. Now, the reason I don't have to convert them is because centimeters cancel with centimeters. Or another way of thinking about it is 16 over 25 is the same thing as if you just did 0.16 over 0.25. So it actually doesn't matter because you cancel out the units anyway. What you should get here is 640 kg/m³. And that makes sense for the density of oil because we know that it's going to be lighter than water because this whole thing has to float on top of water. So we know it's generally less than 1,000 kg/m³. Alright. That's it for this one. Let me know if you guys have any questions.