---
title: "Straightness of a bed using a spirit level"
subject: "Metrology"
chapter: 13
key: "semester_3/metrology:straightness-by-spirit-level-numerical"
source: https://www.bctengineer.com/q/semester_3/metrology/straightness-by-spirit-level-numerical
site: BCT Engineering
---

# Straightness of a bed using a spirit level

Metrology · Chapter 13: Machine Tool Metrology

- Asked in: year not given, 6 marks
- Video: [Determination of straightness error using spirit level](https://www.youtube.com/watch?v=gk7hEhBWybU) — Lab Tutor
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/metrology/straightness-by-spirit-level-numerical
- Whole chapter: https://www.bctengineer.com/md/semester_3/metrology/chapter-13

## Question

The straightness of a 1200 mm lathe bed is tested with a spirit level having a 200 mm base, placed on six consecutive positions. The sensitivity is 0.02 mm per metre per division. The bubble readings in divisions are: +2, +3, +1, -1, 0, +2 (the positive readings show the level rising in the direction of travel). Plot the profile of the bed by calculating the heights at the positions and find the straightness error relative to the line joining the end points of the bed.

## Answer

#### Rise per division
Sensitivity $= 0.02$ mm per 1000 mm, so for the 200 mm base:

$$
1\ \text{div} = 0.02 \times \frac{200}{1000} = 0.004\ \text{mm}
$$

#### Heights
Each rise is $\text{reading} \times 0.004$ mm; the height at a position is the sum of the rises up to it.

| Position (mm) | Reading (div) | Rise (mm) | Height (mm) | Line ends (mm) | Deviation (mm) |
|---|---|---|---|---|---|
| 0 | | | 0.000 | 0.0000 | 0.0000 |
| 200 | +2 | 0.008 | 0.008 | 0.0047 | +0.0033 |
| 400 | +3 | 0.012 | 0.020 | 0.0093 | +0.0107 |
| 600 | +1 | 0.004 | 0.024 | 0.0140 | +0.0100 |
| 800 | -1 | -0.004 | 0.020 | 0.0187 | +0.0013 |
| 1000 | 0 | 0.000 | 0.020 | 0.0233 | -0.0033 |
| 1200 | +2 | 0.008 | 0.028 | 0.0280 | 0.0000 |

The line joining the end points rises $0.028$ mm over $1200$ mm, so at the $k$th station its height is $0.028k/6$ mm.

#### Profile
```text
 height (mm)
 0.028 |                       *
 0.024 |           *
 0.020 |       *       *   *
 0.016 |
 0.012 |
 0.008 |   *
 0.004 |
 0.000 *------------------------> mm
       0  200 400 600 800 1000 1200
```

The bed is highest above the end line at 400 mm ($+0.0107$ mm) and lowest below it at 1000 mm ($-0.0033$ mm). The straightness error is the width of the zone between two lines parallel to the end line that enclose the profile:

$$
0.0107 - (-0.0033) = 0.014\ \text{mm over 1200 mm}
$$

**Answer: straightness error = 0.014 mm over 1200 mm (maximum hump of 0.0107 mm at 400 mm).**

## Check yourself

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

1. A level has sensitivity 0.02 mm per metre per division and a 200 mm base. What is the rise per division?
   A. 0.02 mm
   B. 0.004 mm
   C. 0.1 mm
   D. 0.0004 mm
   Answer: B. 0.02 x 200/1000 = 0.004 mm per division. (Key point: Rise per division)

2. How are the heights of successive stations found?
   A. Each reading x 0.004 taken alone
   B. Sum of all readings x 0.004 at every station
   C. Cumulative sum of (reading x 0.004) up to each station
   D. Reading divided by 0.004
   Answer: C. Each station's height is the previous height plus its own rise, e.g. 0.008, 0.020, 0.024, 0.020, 0.020, 0.028 mm. (Key point: Heights)

3. Relative to the line joining the end points, which line is used at the kth station?
   A. A horizontal line at zero
   B. A line at the average height
   C. 0.028 mm at every station
   D. 0.028 k/6 since the end line rises 0.028 mm over 1200 mm
   Answer: D. The end-point line rises linearly, so deviation = height minus 0.028k/6. (Key point: Reference line)

4. What is the straightness error given maximum deviation +0.0107 mm (at 400 mm) and minimum -0.0033 mm (at 1000 mm)?
   A. 0.014 mm over 1200 mm
   B. 0.0107 mm
   C. 0.0074 mm
   D. 0.0033 mm
   Answer: A. The error is the width of the zone between two parallel lines enclosing the profile: 0.0107 + 0.0033 = 0.014 mm. (Key point: Straightness error)

## Related in Metrology

- [Spirit level sensitivity and tilt](https://www.bctengineer.com/question.md?key=semester_3%2Fmetrology%3Aspirit-level-sensitivity) — Metrology, chapter 5

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- [Slab base for ISSC 200, 1200 kN](https://www.bctengineer.com/question.md?key=semester_6%2Fdesign_of_steel_and_timber_structure%3Aslab-base-isc200) — Design of Steel and Timber Structure, chapter 6
- [Internal discharge in 0.5 mm air bubble](https://www.bctengineer.com/question.md?key=semester_8%2Fhigh_voltage_engineering%3Aair-bubble-internal-discharge) — High Voltage Engineering, chapter 6, asked 2×
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