On this page
A U-tube with water in one arm and an unknown oil in the other is one of the most common ways to set a density question. The physics is one line, yet students lose marks by drawing the level in the wrong place. This post shows where to draw it, then works three cases: two liquids meeting directly, two liquids separated by mercury with unequal levels, and a set of readings turned into a graph.
Where is this in the syllabus?
The JEE (Main) 2026 syllabus and the NEET (UG) 2026 syllabus both list "pressure due to a fluid column" and Pascal's law. Chapter 9 of the NCERT Class 11 Physics textbook (2026-27 reprint) describes the open-tube manometer, a U-tube with one end open to the air, and states that "pressure is same at the same level on both sides of the U-tube containing a fluid". Its Exercise 9.9 sets a U-tube with water and methylated spirit separated by mercury. For layered tanks and the force on a wall, see our hydrostatic pressure problems.
Where do you draw the equal-pressure level?
The rule in NCERT holds within one fluid at rest. Two points at the same height have equal pressure only if you can travel from one to the other through a single, continuous liquid without leaving it.
So draw your horizontal reference line through the lowest interface between two liquids. Below that line, both arms hold the same liquid, so the pressures match. Above it, the arms hold different liquids, and equal heights do not mean equal pressures.
Then, for each arm: pressure at the line = atmospheric pressure + ρgh for every column above the line in that arm. Both arms are open, so atmospheric pressure cancels, and so does g.
Worked example 1: oil poured on water, levels compared
A U-tube contains water. Oil is poured into the left arm until the oil column is 20.0 cm tall, measured from the oil-water interface. The water in the right arm stands 17.0 cm above that interface. Find the density of the oil.
At the line through the interface:
ρ(oil) × g × 0.200 = ρ(water) × g × 0.170
ρ(oil) = 1000 × 0.170 / 0.200 = 850 kg/m³, a relative density of 0.85.
Sense check: the oil column is taller, so the oil must be less dense than water.
Worked example 2: two liquids above mercury, levels unequal
A U-tube holds mercury (relative density 13.6, the value NCERT uses in Exercise 9.10). The left arm has 20.0 cm of liquid X on top of the mercury, and the right arm has 10.0 cm of water on top of the mercury. The mercury surface in the right arm is 0.50 cm higher than the mercury surface in the left arm. Find the density of X.
Draw the line through the lower mercury surface, the X-mercury interface in the left arm. Below it, mercury is continuous.
Left arm, above the line: 20.0 cm of X.
Right arm, above the line: 0.50 cm of mercury, then 10.0 cm of water.
Equate (g cancels):
ρ(X) × 0.200 = 1000 × 0.100 + 13 600 × 0.0050
ρ(X) × 0.200 = 100 + 68 = 168
ρ(X) = 840 kg/m³.
Notice the 0.50 cm of mercury adds 68 to the right-hand side, more than two-thirds of what the 10 cm of water adds. Small mercury differences matter; never drop them.
Worked example 3: sample readings and a graph
These are sample readings for practice. A student pours liquid X onto water in a U-tube in four stages and measures both heights from the interface:
X: 4.0 cm, water: 3.4 cm
X: 8.0 cm, water: 6.7 cm
X: 12.0 cm, water: 10.1 cm
X: 16.0 cm, water: 13.5 cm
Plot water height against X height. Since ρ(X) × h(X) = ρ(water) × h(water), the graph is a straight line through the origin with slope ρ(X)/ρ(water). A least-squares fit gives a slope of 0.8425, so ρ(X) ≈ 842 kg/m³, which agrees with example 2 to within reading error. The graph uses all four readings at once and shows straight away if one is off the line.
Why is pressure in a liquid hard to reason about?
Researchers developing a standard fluids test, Goszewski and colleagues (2013), wrote questions to check whether difficulties with pressure already documented in young children are also common in college students. The point for you is that these are not beginner errors that vanish with age. They need a deliberate method, such as the reference-line rule above.
Common mistakes:
Equating pressures across different liquids at the same height.
Measuring heights from the bottom of the tube instead of from the reference line.
Ignoring a small mercury level difference.
Adding atmospheric pressure to only one side. With both arms open, it cancels.
Using miscible liquids. The method needs liquids that do not mix, so the interface stays sharp.
A practice routine
Solve NCERT Exercise 9.9 using the reference-line method, then Exercise 9.10.
Rework example 2 with the mercury level in the right arm 1.0 cm lower than the left instead of higher. Decide first where the reference line goes.
Make up your own four-reading table for a liquid of density 900 kg/m³ and check the slope you get is 0.90.
If the principle is clear but you keep choosing the wrong level, the Fermi Problem Library has a step-by-step solver that explains the reasoning rather than printing an answer.
Today, sketch example 2 from memory, draw the reference line, and write the two pressure expressions before you calculate anything.
