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Question Video: Determining the Ratio of Fluid Densities in a Liquid Column Manometer Physics

The diagram shows a U-shaped tube containing both water and oil, which are immiscible. Atmospheric pressure acts equally on the tops of the columns of water and oil. How does the ratio of the densities of the oil to the water, 𝜌_(oil)/𝜌_(water), compare to the heights β„Žβ‚ and β„Žβ‚‚? [A] 𝜌_(oil)/𝜌_(water) = β„Žβ‚‚/β„Žβ‚ [B] 𝜌_(oil)/𝜌_(water) = β„Žβ‚/(β„Žβ‚ + β„Žβ‚‚) [C] 𝜌_(oil)/𝜌_(water) = (β„Žβ‚ + β„Žβ‚‚)/(β„Žβ‚ Γ— β„Žβ‚‚) [D] 𝜌_(oil)/𝜌_(water) = β„Žβ‚/β„Žβ‚‚ [E] 𝜌_(oil)/𝜌_(water) = β„Žβ‚‚/(β„Žβ‚ + β„Žβ‚‚)

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Video Transcript

The diagram shows a U-shaped tube containing both water and oil, which are immiscible. Atmospheric pressure acts equally on the tops of the columns of water and oil. How does the ratio of the densities of the oil to the water, 𝜌 sub oil to 𝜌 sub water, compare to the heights β„Ž one and β„Ž two? (A) 𝜌 sub oil to 𝜌 sub water equals β„Ž two divided by β„Ž one. (B) 𝜌 sub oil to 𝜌 sub water equals β„Ž one divided by β„Ž one plus β„Ž two. (C) 𝜌 sub oil to 𝜌 sub water equals β„Ž one plus β„Ž two divided by β„Ž one times β„Ž two. (D) 𝜌 sub oil to 𝜌 sub water equals β„Ž one divided by β„Ž two. (E) 𝜌 sub oil to 𝜌 sub water equals β„Ž two divided by β„Ž one plus β„Ž two.

In our diagram, we indeed see this U-shaped tube filled with two fluids, oil and water. The column of oil has a height equal to β„Ž two. And if we follow a horizontal line from the bottom of the column of oil to the water, then the height of the column of water above this line on the opposite side of the tube is β„Ž one. Our question asks us about the ratio of the densities of these fluids. A fluidβ€²s density 𝜌 and its height β„Ž are connected through this equation for the pressure generated by such a fluid. This equation tells us, for example, that the pressure generated by our column of oil, that is, the downward pressure exerted here due to the oil, is equal to the oilβ€²s density times the acceleration due to gravity multiplied by the height of the column of oil.

Because oil and water are immiscible, that is, they donβ€²t mix, thereβ€²s a clear boundary here between the two fluids. If we look on the other side of our U-shaped tube at that same elevation, we can notice something interesting. The amount of water below the pink line on the right side of our tube, that is, this amount of water here, is exactly equal to the amount of water below that pink horizontal line on the other side of our tube. Therefore, the pressure along this dashed pink horizontal line is constant. Specifically, the pressure here on the left-hand side is identical to the pressure here on the right-hand side.

Letβ€²s clear some space on screen to work. And we can notice that above this point on the left side of our tube is a column of oil of height β„Ž two. That column creates a pressure, weβ€²ll call it 𝑃 sub oil, of the density of oil times the acceleration due to gravity times β„Ž two. If we then follow this horizontal line to that same elevation point on the right side of our tube, we see that this point has a column of water above it of height β„Ž one. This also produces a pressure, weβ€²ll call it 𝑃 sub water. And it equals the density of water times 𝑔 times β„Ž one.

As we saw earlier, the pressure at any point inside the U-shaped tube along the pink line is the same. That means that the pressure due to the column of oil of height β„Ž two equals the pressure due to the column of water of height β„Ž one. Therefore, we can say that 𝜌 sub oil times 𝑔 times β„Ž two equals 𝜌 sub water times 𝑔 times β„Ž one. Notice that the acceleration due to gravity is common to both sides of this equation. Therefore, if we divide both sides by 𝑔, that factor will cancel out.

Weβ€²re now getting close to our answer. Recall that we want to solve for the ratio of 𝜌 sub oil to 𝜌 sub water. If we divide both sides of this remaining equation by 𝜌 sub water times β„Ž two, then on the left-hand side, the height β„Ž two cancels, and on the right 𝜌 sub water cancels out. That leaves us with this result that the density of oil divided by the density of water equals β„Ž one divided by β„Ž two. Reviewing our answer options, we see that this agrees with option (D). This then is our answer. And note that the height β„Ž one is smaller than the height β„Ž two. This tells us that the density of our oil is less than the density of water.

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