Manometer with two liquids

GCE O-level 5076 Science (Physics) 2015 November Paper 2 Question 4(c)

Question:

How would you find the density of liquid B? I believe most of you would find the pressure at point X and then equate it with the pressure at point Y. Have you ever wonder why this method works? At least I do.

Let us investigate by considering the pressure at points P and Q. Point P is at the junction between air and liquid A. Point Q is at the same horizontal level in liquid B.

Pressure at P = Atmospheric pressure

Pressure at Q = Atmospheric pressure + Pressure due to the column of liquid above Q

In other words, the pressure at A and at B are different. Hence, we cannot solve the question by equating the pressure at P and pressure at Q. This begs the following question: What's so special about points X and Y?

Explanation:

As the liquids are at rest (i.e. in equilibrium), the pressure at the bottom of the manometer due to the left column must be equal to the pressure due to the right column.

Pressure due to left liquid column = Pressure due to right liquid column.

For points X and Y,

Pressure at X + Pressure due to water column below X = Pressure at Y + Pressure due to water column below Y.

Since Pressure due to water column below X = Pressure due to water column below Y, Pressure at X = Pressure at Y.

For point A and B,
Pressure at P + Pressure due liquid column between P and X + Pressure due to water column below X = Pressure at Q + Pressure due liquid column between Q and Y + Pressure due to water column below Y.

Since Pressure due to water column below X = Pressure due to water column below Y,
Pressure at P + Pressure due liquid column between P and X = Pressure at Q + Pressure due liquid column between Q and Y.

Since liquid A and liquid B have different densities, pressure due liquid column between P and X is NOT equal to pressure due liquid column between Q and Y even though both liquid columns have the same vertical height (i.e. PX = QY). Hence, pressure at P is NOT equal to pressure at Q.