---
title: "U-tube capillaries find liquid density"
subject: "Fluid Mechanics"
chapter: 1
key: "semester_3/fluid_mechanics:u-tube-capillaries-density"
source: https://www.bctengineer.com/q/semester_3/fluid_mechanics/u-tube-capillaries-density
site: BCT Engineering
---

# U-tube capillaries find liquid density

Fluid Mechanics · Chapter 1: Fluid and its physical properties

- Asked in: 2072 Chaitra, 3 marks
- Video: [A \( U \)-tube is made up of capillaries of bores \( 1 \mathrm{~mm}...](https://www.youtube.com/watch?v=sbXy2Q5v2vw) — PW Solutions
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/fluid_mechanics/u-tube-capillaries-density
- Whole chapter: https://www.bctengineer.com/md/semester_3/fluid_mechanics/chapter-1

## Question

A U-tube is made up of two capillaries of bores 1.5 mm and 2 mm respectively. The U tube is held vertical and partially filled with liquid whose surface tension $\sigma$ = 0.075 N/m. Find out the mass density of the liquid if the difference in two menisci is 2 mm. Assume angle of contact is zero.

## Short answer

Capillary rise difference Δh = (4σ/(ρg))(1/d₁ – 1/d₂), so ρ = (4σ/(g Δh))(1/d₁ – 1/d₂) = [4×0.075/(9.81×0.002)] × (1/0.0015 – 1/0.002) ≈ 2548 kg/m³.

## Answer

**Data:** $d_1=1.5$ mm, $d_2=2$ mm, $\sigma=0.075$ N/m, $\theta=0^\circ$, difference of menisci $\Delta h=2$ mm $=0.002$ m.

Capillary rise in each limb: $h=\dfrac{4\sigma\cos\theta}{\rho g d}$. The narrower limb rises higher, so

$$
\Delta h=h_1-h_2=\frac{4\sigma}{\rho g}\left(\frac{1}{d_1}-\frac{1}{d_2}\right)
$$

$$
\rho=\frac{4\sigma}{g\,\Delta h}\left(\frac{1}{d_1}-\frac{1}{d_2}\right)=\frac{4(0.075)}{9.81(0.002)}\left(\frac{1}{0.0015}-\frac{1}{0.002}\right)
$$

$$
\rho=\frac{0.3}{0.01962}\times166.67=2548\ \text{kg/m}^3
$$

**Answer: $\rho\approx2548\ \text{kg/m}^3$.**

## Check yourself

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

1. In the U-tube with bores 1.5 mm and 2 mm, which limb has the higher meniscus?
   A. The 2 mm limb
   B. Both equal
   C. The 1.5 mm limb, since h is inversely proportional to d
   D. Depends on the density
   Answer: C. h = 4 sigma cos(theta)/(rho g d), so the narrower limb rises more. (Key point: Which limb rises higher)

2. Which expression gives the difference of menisci?
   A. (4 sigma/rho g)(1/d1 - 1/d2)
   B. (4 sigma/rho g)(d2 - d1)
   C. (4 sigma/rho g)(1/d1 + 1/d2)
   D. (2 sigma/rho g)(d1 - d2)
   Answer: A. Delta h = h1 - h2 with each h = 4 sigma/(rho g d) for theta = 0. (Key point: Difference expression)

3. What density results for Delta h = 2 mm and sigma = 0.075 N/m?
   A. About 2548 kg/m^3
   B. About 1000 kg/m^3
   C. About 255 kg/m^3
   D. About 25 480 kg/m^3
   Answer: A. rho = 4(0.075)/(9.81 x 0.002) x 166.67 = 2548 kg/m^3. (Key point: Resulting density)

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