---
title: "Spirit level sensitivity and tilt"
subject: "Metrology"
chapter: 5
key: "semester_3/metrology:spirit-level-sensitivity"
source: https://www.bctengineer.com/q/semester_3/metrology/spirit-level-sensitivity
site: BCT Engineering
---

# Spirit level sensitivity and tilt

Metrology · Chapter 5: Angular and Taper Measurement

- Asked in: year not given, 6 marks
- Video: [Spirit Level in Metrology | GATE Mechanical | SSC JE | RRB JE Complete Explanation](https://www.youtube.com/watch?v=U8Deb-dGZHs) — KN Prasad Academy (NITian)
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/metrology/spirit-level-sensitivity
- Whole chapter: https://www.bctengineer.com/md/semester_3/metrology/chapter-5

## Question

(a) What is a spirit level? Explain how its sensitivity is expressed. (b) A level has a vial whose divisions are 2 mm apart, and a division corresponds to 10 seconds of arc. Calculate the radius of curvature of the vial. (c) This level is placed on a 500 mm long base on a machine slide and the bubble moves 3 divisions. Find the difference in height between the ends of the base.

## Short answer

Spirit level is a curved vial with a bubble; sensitivity θ = p/R per division. For p=2 mm, θ=10", radius R = 41.3 m. On a 500 mm base, 3 division movement gives tilt 30", height difference 0.073 mm.

## Answer

#### (a) Spirit level
A spirit (bubble) level is a sealed glass vial, slightly curved, partly filled with alcohol or ether so that a bubble remains. The bubble always rises to the highest point. If the base is horizontal the bubble is at the centre; if the base tilts by a small angle the bubble moves along the graduated scale. It is used to check the levelness of machine tools, surface plates and slideways.

**Sensitivity** is the tilt of the base per division of bubble movement. It is expressed as an angle (seconds of arc per division) or as slope (mm per metre per division). For a vial of radius $R$ and division length $p$:

$$
\theta = \frac{p}{R} \;\Rightarrow\; R = \frac{p}{\theta}
$$
A larger $R$ gives greater sensitivity (small tilt moves the bubble far) but a longer settling time.

#### (b) Radius
$\theta = 10'' = 10 \times 4.848\times10^{-6} = 4.848\times10^{-5}$ rad

$$
R = \frac{2\ \text{mm}}{4.848\times10^{-5}} = 41\,253\ \text{mm} \approx 41.3\ \text{m}
$$

#### (c) Height difference
Total tilt $= 3 \times 10'' = 30'' = 1.454\times10^{-4}$ rad.

$$
\Delta h = 500 \times 1.454\times10^{-4} = 0.0727\ \text{mm}
$$

(a slope of 0.145 mm per metre).

**Answer: $R = 41.3$ m; end-to-end height difference = 0.073 mm over 500 mm.**

## Check yourself

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

1. In a spirit level, the bubble always:
   A. Moves to the lowest point of the vial
   B. Stays at the centre regardless of tilt
   C. Rises to the highest point of the curved vial
   D. Moves only when the vial is heated
   Answer: C. The bubble rises to the highest point, so tilt moves it along the scale. (Key point: Spirit level principle)

2. For a vial of radius R and division length p, the sensitivity (tilt per division) is theta = p/R. A larger R gives:
   A. Lower sensitivity and faster settling
   B. Greater sensitivity but longer settling time
   C. No change in sensitivity
   D. Greater sensitivity and faster settling
   Answer: B. A larger radius makes the bubble move farther for a small tilt, but it takes longer to settle. (Key point: Sensitivity and radius)

3. For a 2 mm division equal to 10 seconds of arc (4.848 x 10^-5 rad), the radius is about:
   A. 41.3 m
   B. 4.13 m
   C. 413 m
   D. 0.41 m
   Answer: A. R = 2 mm / 4.848 x 10^-5 = 41 253 mm, about 41.3 m. (Key point: Radius calculation)

4. The bubble moves 3 divisions (30 seconds) on a 500 mm base. The height difference between ends is about:
   A. 0.0242 mm
   B. 0.145 mm
   C. 0.0727 mm
   D. 0.727 mm
   Answer: C. Delta h = 500 x 1.454 x 10^-4 = 0.0727 mm. (Key point: Height difference)

## Related in Metrology

- [Straightness of a bed using a spirit level](https://www.bctengineer.com/question.md?key=semester_3%2Fmetrology%3Astraightness-by-spirit-level-numerical) — Metrology, chapter 13

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