---
title: "Optical lever comparator magnification"
subject: "Metrology"
chapter: 7
key: "semester_3/metrology:optical-lever-comparator-numerical"
source: https://www.bctengineer.com/q/semester_3/metrology/optical-lever-comparator-numerical
site: BCT Engineering
---

# Optical lever comparator magnification

Metrology · Chapter 7: Comparators

- Asked in: year not given, 6 marks
- Video: [Optical Lever Comparator | detailed explanation](https://www.youtube.com/watch?v=z4Gua9vIXBM) — Rohit's Online Classes
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/metrology/optical-lever-comparator-numerical
- Whole chapter: https://www.bctengineer.com/md/semester_3/metrology/chapter-7

## Question

A mechanical-optical comparator uses a mirror pivoted at one end and pushed by the plunger at a distance of 5 mm from the pivot. The reflected beam falls on a scale 250 mm from the mirror. Derive the magnification of this single optical lever and find it. If the scale divisions are 0.5 mm apart, what is the least count? The instrument is set to zero with a 25.000 mm slip gauge. A shaft is then put in and the reading is +3.2 divisions. Find the size of the shaft, and say whether it is within the limits $25 \pm 0.015$ mm.

## Short answer

Magnification M = 2L/x = (2×250)/5 = 100. Least count = 0.5/100 = 0.005 mm. Deviation = 3.2×0.005 = 0.016 mm, so shaft size = 25.016 mm; it exceeds the upper limit 25.015 mm and is rejected (oversize by 0.001 mm).

## Answer

#### Derivation
Let the plunger move by $\delta$ and the mirror rotate by $\theta$ about the pivot, with the plunger acting at distance $x$ from the pivot:

$$
\theta \approx \frac{\delta}{x}
$$

A reflected ray turns through $2\theta$ when the mirror turns by $\theta$. On a scale at distance $L$ the spot moves by

$$
y = 2\theta L = \frac{2L}{x}\,\delta
$$

$$
\text{Magnification } M = \frac{y}{\delta} = \frac{2L}{x} = \frac{2(250)}{5} = 100
$$

#### Least count
$$
\text{LC} = \frac{\text{scale division}}{M} = \frac{0.5}{100} = 0.005\ \text{mm}
$$

#### Shaft size
The deviation from the master is $+3.2 \times 0.005 = +0.016$ mm, so

$$
\text{size} = 25.000 + 0.016 = 25.016\ \text{mm}
$$

Limits are $24.985$ to $25.015$ mm. The shaft is larger than the upper limit by $0.001$ mm, therefore it is **not acceptable** (oversize, can be reworked).

**Answer: $M = 100$, LC = 0.005 mm, size = 25.016 mm, rejected (oversize by 0.001 mm).**

## Check yourself

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

1. For a mirror pivoted at one end, plunger acting at distance x from the pivot, scale at distance L, the magnification is:
   A. L/x
   B. 2L/x
   C. x/2L
   D. L/2x
   Answer: B. Mirror turn theta = delta/x, the beam turns 2 theta, so the spot moves 2 theta L and M = 2L/x. (Key point: Optical lever magnification)

2. With x = 5 mm and L = 250 mm, the magnification is:
   A. 50
   B. 200
   C. 100
   D. 25
   Answer: C. M = 2 x 250 / 5 = 100. (Key point: Magnification value)

3. With scale divisions of 0.5 mm and M = 100, the least count is:
   A. 0.05 mm
   B. 0.01 mm
   C. 0.0005 mm
   D. 0.005 mm
   Answer: D. LC = scale division / M = 0.5/100 = 0.005 mm. (Key point: Least count)

4. A reading of +3.2 divisions from a 25.000 mm master gives size 25.016 mm. For limits 25 ± 0.015 mm, the shaft is:
   A. Accepted, being within limits
   B. Rejected, oversize by 0.001 mm
   C. Rejected, undersize
   D. Accepted after rework
   Answer: B. Deviation = 3.2 x 0.005 = 0.016 mm, exceeding the upper limit of 25.015 mm by 0.001 mm. (Key point: Acceptance check)

## Related in Metrology

- [Classification and merits of comparators](https://www.bctengineer.com/question.md?key=semester_3%2Fmetrology%3Aclassification-of-comparators) — Metrology, chapter 7
- [Vernier caliper and micrometer readings](https://www.bctengineer.com/question.md?key=semester_3%2Fmetrology%3Avernier-and-micrometer-readings) — Metrology, chapter 4
- [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
- [Fringe measurement of flatness error](https://www.bctengineer.com/question.md?key=semester_3%2Fmetrology%3Ainterference-fringes-flatness-numerical) — Metrology, chapter 8
- [Spirit level sensitivity and tilt](https://www.bctengineer.com/question.md?key=semester_3%2Fmetrology%3Aspirit-level-sensitivity) — Metrology, chapter 5
- [Sine bar principle, setting and error](https://www.bctengineer.com/question.md?key=semester_3%2Fmetrology%3Asine-bar-setting-and-error) — Metrology, chapter 5

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