---
title: "Rotating U-tube pressure and water lost"
subject: "Fluid Mechanics"
chapter: 2
key: "semester_3/fluid_mechanics:rotating-u-tube-suction-leak"
source: https://www.bctengineer.com/q/semester_3/fluid_mechanics/rotating-u-tube-suction-leak
site: BCT Engineering
---

# Rotating U-tube pressure and water lost

Fluid Mechanics · Chapter 2: Pressure and Head

- Asked in: 2072 Chaitra, 6 marks
- 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-suction-leak
- Whole chapter: https://www.bctengineer.com/md/semester_3/fluid_mechanics/chapter-2

## Question

The U-tube shown in figure below is filled with water. It is sealed at A and open to the atmosphere at D. The tube is rotated about vertical axis AB at 1600 rpm. If the U-tube is now spun at 300 rpm what will be the pressure be at A? If a small leaks appear at A, how much water will be lost at D? [Figure: U-tube with vertical limbs AB and CD joined by horizontal base BC; H = 4 cm, L = 2 cm; rotation axis through AB]

## Short answer

At 300 rpm, gauge pressure at A is -197 Pa. If a leak occurs at A, the water surface in limb AB falls by 20.1 mm and an equivalent volume spills out at D.

## Answer

**Reading of the figure:** axis through limb AB (A sealed at the top, on the axis); base BC has length $L=2$ cm; limb CD is vertical with the open end D at the same level as A, $H=4$ cm above the base. The tube is full of water. The clause "1600 rpm" is not needed; the question is for $N=300$ rpm, so $\omega=\dfrac{2\pi\times300}{60}=31.42$ rad/s.

```text
 A (sealed)          D (open)
 |                    |
 | H = 4 cm           |
 B+------- L = 2 cm ---+C
```

#### Pressure at A
Pressure at D is zero. The pressure varies with radius and height as $p=\tfrac12\rho\omega^{2}r^{2}-\rho g z+\text{const}$. Between D ($r=L$, $z=H$) and A ($r=0$, $z=H$), the heights are equal, so only the rotational term acts:

$$
p_A=p_D-\tfrac{1}{2}\rho\omega^{2}L^{2}
$$

$$
p_A=-\tfrac12(1000)(31.42)^{2}(0.02)^{2}=-197.4\ \text{Pa}
$$

**Pressure at A $\approx-197$ Pa (gauge), i.e. a small suction.**

#### Water lost if A leaks
When A is opened to the atmosphere ($p_A=0$), the water surface at A (on the axis) and the surface at D (at $r=L$) must differ in level by

$$
\Delta z=\frac{\omega^{2}L^{2}}{2g}=\frac{(31.42)^{2}(0.02)^{2}}{2(9.81)}=0.0201\ \text{m}=20.1\ \text{mm}
$$

The surface at D cannot rise above the rim, so the water surface in AB falls by 20.1 mm and the same volume spills out at D.

**Answer: $p_A\approx-197$ Pa; water lost at D equals a length of 20.1 mm of tube, i.e. volume $=\dfrac{\pi}{4}d^{2}\times0.0201\ \text{m}^3$ for tube bore $d$.**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. The question mentions 1600 rpm and 300 rpm. Which is used and what is ω?
   A. 300 rpm, ω = 31.42 rad/s
   B. 1600 rpm, ω = 167.6 rad/s
   C. 300 rpm, ω = 300 rad/s
   D. Both are added
   Answer: A. The question asks about 300 rpm, so ω = 2π x 300/60 = 31.42 rad/s. (Key point: Which speed is used)

2. Between D (r = L) and A (r = 0) why does only the rotational term appear in p_A = p_D - 1/2 ρ ω^2 L^2?
   A. A and D are at the same radius
   B. A and D are at the same height, so the hydrostatic term cancels
   C. Gravity is zero in the rotating tube
   D. A is open to the atmosphere
   Answer: B. With equal z, the -ρ g z terms cancel and only 1/2 ρ ω^2 (r^2) differences remain. (Key point: Why only the rotational term)

3. What is the pressure at A?
   A. About +197 Pa
   B. About -19.7 kPa
   C. About -197 Pa (small suction)
   D. Zero
   Answer: C. p_A = -1/2 (1000)(31.42)^2 (0.02)^2 = -197 Pa. (Key point: Pressure at A)

4. If A leaks (p_A = 0), by how much does the water surface in AB fall and spill out at D?
   A. About 20 mm, from dz = ω^2 L^2 / 2g
   B. About 197 mm
   C. None, it stays full
   D. About 2 mm
   Answer: A. With A at atmosphere, the surfaces at r = 0 and r = L differ by ω^2 L^2/(2g) = 20.1 mm; that length spills at D. (Key point: Water lost on leak)

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