A U-tube has one arm of area of cross-section A and the other of cross-section 4A.

Question 24

A U-tube has one arm of area of cross-section A and the other of cross-section 4A. The tube contains water of density 1000 kg m^-3 and oil of density 850 kg m^-3, as shown.

The column of oil on top of the water in the left-hand arm is of length 30.0 cm.

What is the difference in height x between the levels in the two arms of the tube?

A 4.5 cm                     B 6.2 cm                     C 23.8 cm                   D 25.5 cm

Reference: Past Exam Paper – November 2015 Paper 13 Q19

Solution:

Answer: A.

At any specific level, the pressure in a liquid is the same.

Consider the pressure at the lower-most reference line drawn in the diagram – this is the boundary between the oil and the water.

At this level, the pressure on the left arm should be equal to the pressure on the right arm.

Recall that pressure is due to the height of liquid above that point.

Pressure in liquid = hρg

At the oil-water boundary,

The pressures exerted by the oil and by the water at this boundary must be equal.

Pressure in left arm = Pressure in right arm

30 × 850 × g = (30 – x) × 1000 × g

30 – x = 30 × 850 / 1000

30 – x = 25.5

Difference between the level = x = 30 – 25.5 = 4.5 cm

 

Note that the pressure does not depend on the cross-sectional area of the arm. So, the difference in area does not affect the pressure.