--- title: "Pressures in rotating sealed U-tube" subject: "Fluid Mechanics" chapter: 2 key: "semester_3/fluid_mechanics:rotating-u-tube-ab-cd-pressures" source: https://www.bctengineer.com/q/semester_3/fluid_mechanics/rotating-u-tube-ab-cd-pressures site: BCT Engineering --- # Pressures in rotating sealed U-tube Fluid Mechanics · Chapter 2: Pressure and Head - Asked in: 2073 Shrawan, 7 marks - Video: [Physics - Ch 33A Test Your Knowledge: Fluid Statics (6 of 43) Pressure in a rotating Tube](https://www.youtube.com/watch?v=UavtQSOnxXk) — Michel van Biezen - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/fluid_mechanics/rotating-u-tube-ab-cd-pressures - Whole chapter: https://www.bctengineer.com/md/semester_3/fluid_mechanics/chapter-2 ## Question The U-tube AB and CD shown in figure below filled with water. The tube AB is sealed where as tube CD is open to atmosphere. Find the pressure intensities at the points A, B and C where it is rotating with axis Y-Y with uniform rotation of 60 rpm. [Figure: U-tube with sealed limb AB (height 3 m) at 1 m from the axis Y-Y and open limb CD at 2 m on the other side of the axis] ## Short answer Gauge pressures are p_C = 29.4 kPa, p_B = −29.8 kPa, p_A = −59.2 kPa, obtained by summing hydrostatic and centrifugal terms from the open limb D to sealed limb A. ## Answer **Reading of the figure:** the U-tube is filled with water; limb AB (sealed at the top A, height 3 m) is 1 m from the axis Y-Y; limb CD (open at the top D) is 2 m on the other side. Take B and C at the bottom, joined by a horizontal tube through the axis, and D at the same level as A (3 m above C). $N=60$ rpm so $\omega=2\pi=6.283$ rad/s. Pressures are gauge. ```text A(sealed) D(open) | | | 3 m | 3 m | | B+----------Y----------+C 1 m 2 m ``` At rest the open end D is at atmospheric pressure. During rotation, pressure follows $p=\tfrac12\rho\omega^{2}r^{2}-\rho g z+\text{constant}$. At D ($r=2$, $z=3$ above the base), $p=0$. #### At C (bottom of the open limb, $r=2$ m) Hydrostatic in the vertical limb (no change with $r$ along a vertical line): $$ p_C=\rho g(3)=1000(9.81)(3)=29.43\ \text{kPa} $$ #### At B (bottom of the sealed limb, $r=1$ m, same level as C) $$ p_B=p_C+\tfrac{1}{2}\rho\omega^{2}\left(r_B^{2}-r_C^{2}\right) $$ $$ p_B=29.43+\tfrac12(1000)(6.283)^{2}(1-4)=29.43-59.22=-29.79\ \text{kPa} $$ #### At A (top of the sealed limb) $$ p_A=p_B-\rho g(3)=-29.79-29.43=-59.22\ \text{kPa} $$ **Answer: $p_A\approx-59.2$ kPa, $p_B\approx-29.8$ kPa, $p_C\approx+29.4$ kPa (gauge).** The absolute pressure at A is still about 42 kPa, above the vapour pressure of water, so the column does not break. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Why is p_C = +29.43 kPa? A. C lies 3 m below the open top D (p = 0) in a vertical limb, so it is hydrostatic: ρ g (3) B. Rotation adds 1/2 ρ ω^2 r^2 at r = 2 m C. C is at atmospheric pressure D. C is at the axis so rotation has no effect and p = 0 Answer: A. In a vertical limb pressure changes only with height, so p_C = 1000 x 9.81 x 3. (Key point: Pressure at C) 2. B and C are at the same level, B at r = 1 m and C at r = 2 m. Which statement is right? A. p_B equals p_C B. p_B is lower than p_C by 1/2 ρ ω^2 (rC^2 - rB^2) C. p_B is higher than p_C D. p_B is lower than p_C by ρ g (3) Answer: B. Pressure rises with radius at constant level, so B nearer the axis has lower pressure: 29.43 - 59.22 = -29.79 kPa. (Key point: Pressure at B relative to C) 3. How is p_A found from p_B? A. p_A = p_B + ρ g (3) = -0.4 kPa B. p_A = p_B = -29.8 kPa C. p_A = p_B - ρ g (3) = -59.2 kPa D. p_A = 0 since A is sealed Answer: C. A is 3 m above B in the same vertical limb, so pressure drops by the hydrostatic 29.43 kPa. (Key point: Pressure at A) 4. Why does the water column in the sealed limb not separate even though p_A is -59.2 kPa gauge? A. Absolute pressure at A is about 42 kPa, still above the vapour pressure of water B. Gauge pressure is never negative in water C. Sealed tubes cannot cavitate D. Rotation raises the vapour pressure Answer: A. Absolute pressure is about 101 - 59 = 42 kPa, above vapour pressure, so no cavitation. 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