---
title: "Sine bar principle, setting and error"
subject: "Metrology"
chapter: 5
key: "semester_3/metrology:sine-bar-setting-and-error"
source: https://www.bctengineer.com/q/semester_3/metrology/sine-bar-setting-and-error
site: BCT Engineering
---

# Sine bar principle, setting and error

Metrology · Chapter 5: Angular and Taper Measurement

- Asked in: year not given, 8 marks
- Video: [sine bar | why Sine bar is not preferred for angles greater than 45 degree? | How to use sine bar](https://www.youtube.com/watch?v=nmHJglu9v48) — Modi Mechanical Engineering Tutorials
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/metrology/sine-bar-setting-and-error
- Whole chapter: https://www.bctengineer.com/md/semester_3/metrology/chapter-5

## Question

(a) Explain the principle of the sine bar and state its limitations. (b) Calculate the height of slip gauge stack required to set a 200 mm centre-distance sine bar to an angle of $28^\circ 30'$. (c) If the stack has an error of 0.002 mm, find the resulting angular error at $28^\circ 30'$ and at $60^\circ$, and comment.

## Short answer

Sine bar sets an angle with stack height h = L sinθ. Limitations: inaccurate above 45°. For L=200 mm, θ=28°30', h=95.432 mm. With 0.002 mm height error, angular error is 2.35 s at 28.5°, 4.13 s at 60°; larger error at steeper angles.

## Answer

#### (a) Principle
A sine bar is a hardened steel bar with two equal precision rollers fixed at a known centre distance $L$ (100, 200 or 300 mm). When one roller rests on slip gauges of height $h$ and the other on the surface plate, the bar is inclined at angle $\theta$ to the plate:

```text
        ________  <- sine bar
      /   L    /(roller)
    /  theta  /|
   /_________/_| h  (slip gauges)
   surface plate
```
$$
\sin\theta = \frac{h}{L}
$$

Used to set a component at a known angle, or to measure an unknown angle by adjusting $h$ until the work surface is level (checked by a dial indicator).

**Limitations**
- Accuracy falls at large angles (above $45^\circ$) because $\cos\theta$ becomes small (see (c)); not suitable above $60^\circ$.
- Needs a surface plate and a good set of slip gauges; cannot measure steep tapers directly.
- Errors in centre distance, roller diameters and parallelism of rollers directly give angle error.
- Only conical or flat surfaces that can be set against the bar can be checked; tedious for large work.

#### (b) Stack height
$$
h = L\sin\theta = 200\sin(28.5^\circ) = 200 \times 0.47716 = 95.432\ \text{mm}
$$

**Answer: slip gauge height = 95.432 mm.**

#### (c) Error
Differentiating $h = L\sin\theta$ gives $\delta h = L\cos\theta\,\delta\theta$, so

$$
\delta\theta = \frac{\delta h}{L\cos\theta}
$$

| Angle | $\cos\theta$ | $\delta\theta$ (rad) | $\delta\theta$ |
|---|---|---|---|
| $28.5^\circ$ | 0.8788 | $\dfrac{0.002}{200(0.8788)} = 1.138\times10^{-5}$ | 2.35 s |
| $60^\circ$ | 0.5 | $\dfrac{0.002}{200(0.5)} = 2.0\times10^{-5}$ | 4.13 s |

The same stack error causes a larger angular error at steeper angles. Hence the sine bar is best used below $45^\circ$, and for large angles the complement is measured.

## Check yourself

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

1. A sine bar of centre distance L is raised on a slip gauge stack of height h. The angle theta satisfies:
   A. cos(theta) = h/L
   B. sin(theta) = h/L
   C. tan(theta) = h/L
   D. sin(theta) = L/h
   Answer: B. The bar's rollers at distance L and height difference h give sin(theta) = h/L. (Key point: Sine bar principle)

2. The stack height for a 200 mm sine bar at 28°30' is:
   A. 95.432 mm
   B. 87.88 mm
   C. 100.0 mm
   D. 47.716 mm
   Answer: A. h = 200 sin(28.5°) = 200 x 0.47716 = 95.432 mm. (Key point: Stack height)

3. The error in angle for a stack height error delta h is:
   A. delta h / (L sin theta)
   B. delta h x L cos theta
   C. delta h / (L cos theta)
   D. delta h / L
   Answer: C. Differentiating h = L sin theta gives delta theta = delta h / (L cos theta). (Key point: Angular error)

4. Why is the sine bar unsuitable for angles above about 45° to 60°?
   A. sin theta becomes zero
   B. cos theta becomes small, so the same stack error produces a larger angular error
   C. The rollers slip off
   D. Slip gauges cannot be used
   Answer: B. At larger angles cos theta falls, so delta theta grows: for 0.002 mm the error is 2.35 s at 28.5° but 4.13 s at 60°. (Key point: Large angle limitation)

## Related in Metrology

- [Slip gauges: wringing and building a stack](https://www.bctengineer.com/question.md?key=semester_3%2Fmetrology%3Aslip-gauge-wringing-and-stack) — Metrology, chapter 4
- [Micrometers, height gauge and depth gauge](https://www.bctengineer.com/question.md?key=semester_3%2Fmetrology%3Atypes-of-micrometers-and-errors) — Metrology, chapter 4
- [Fringe measurement of flatness error](https://www.bctengineer.com/question.md?key=semester_3%2Fmetrology%3Ainterference-fringes-flatness-numerical) — Metrology, chapter 8
- [Error propagation and effect of averaging](https://www.bctengineer.com/question.md?key=semester_3%2Fmetrology%3Aerror-propagation-and-averaging) — Metrology, chapter 2
- [Thread errors and effect of pitch error](https://www.bctengineer.com/question.md?key=semester_3%2Fmetrology%3Ascrew-thread-errors-and-pitch-effect) — Metrology, chapter 12
- [Optical lever comparator magnification](https://www.bctengineer.com/question.md?key=semester_3%2Fmetrology%3Aoptical-lever-comparator-numerical) — Metrology, chapter 7

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