---
title: "Vernier Caliper & Micrometer Interactive Lab | Sandbox Physics"
description: "Why does a finer scale not automatically give a more accurate length? A finer scale resolves smaller changes, but a shifted zero moves every reading together."
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language: "en"
page_type: "lab"
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  zh-CN: "https://sandboxphysics.com/zh/labs/length-metrology/"
last_modified: "2026-10-28"
---

IN01 · Reading a small length

# Vernier Caliper & Micrometer

**Why does a finer scale not automatically give a more accurate length?** A finer scale resolves smaller changes, but a shifted zero moves every reading together. Check the closed jaws before measuring, look straight at the scale and repeat the contact. Repeating a biased reading does not remove its bias.

Slide a jaw or turn the thimble, read the matching graduations and check the zero. Measure a cylinder, a thin foil or a bore, then compare repeated readings.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Model assumptions

Rigid specimens and parallel measuring faces are held at twenty degrees Celsius. The caliper uses twenty vernier divisions over nineteen millimetres; the outside micrometer advances half a millimetre per turn, with fifty thimble divisions. Contact deformation, screw backlash, thermal expansion, jaw thickness and alignment errors are omitted. Scale-plane separation gives a geometric parallax shift. Seeded reading scatter is a teaching assumption, not a manufacturer specification. Standard uncertainty includes repeat scatter, a conservative resolution floor and the saved zero calibration; uncorrected parallax and later zero drift remain systematic errors. The blind specimen is synthetic. Apparatus proportions and scale close-ups are schematic.

Saved simulated measurements

## Read, correct, repeat

Indicated length $12.35\,\mathrm{mm}$

After zero correction $12.35\,\mathrm{mm}$

Mean of this series $12.35\,\mathrm{mm}$

Standard uncertainty $12.35\,\mathrm{mm}$

Inspect a saved reading

Raw scale divisions and their frozen zero correction

| No. | Main scale | Division | Zero | Corrected |
| --- | --- | --- | --- | --- |

Calibration, uncertainty & model

Repeated readings estimate scatter. A resolution floor and the saved zero uncertainty are added in quadrature. Viewing bias and a zero changed after calibration can still shift the mean; they are not hidden inside that uncertainty.

$$
u_c=\sqrt{\frac{s^2}{n}+\frac{d^2}{12}+u_0^2}
$$

The rigid specimen and parallel faces are at twenty degrees Celsius. Contact deformation, backlash, thermal expansion, jaw thickness and alignment errors are omitted. The bore uses ideal thin inside faces; the outside micrometer only measures outside dimensions. These synthetic records are not instrument certification.

- [Mitutoyo caliper guide, D-34–35](https://www.mitutoyo.com/webfoo/wp-content/uploads/D_section.pdf)
- [Mitutoyo micrometer guide, B-60, B-76](https://www.mitutoyo.com/webfoo/wp-content/uploads/B_section.pdf)
- [NIST TN 1297 §5](https://www.nist.gov/pml/nist-technical-note-1297/nist-tn-1297-5-combined-standard-uncertainty)

Reveal the synthetic reference length

$12.35\,\mathrm{mm}$

The reference is separate from the recorded readings and is never used by the mean or uncertainty calculation.

Physics tutorial

## A matched line gives the fraction

**Background** A ruler tells you which millimetre you have passed. A sliding auxiliary scale tells you how far you have travelled into the next one.

**Why it matters** A useful length measurement needs a contact, a readable scale and a checked zero. Each can fail in a different way.

Start with the essentials

- **Focus question**: Why does a finer scale not automatically give a more accurate length?
- **One-sentence intuition**: Resolution separates small changes; calibration checks where the scale starts.

### Core mathematical model

#### Two scales with slightly different spacing

$$
20d_v=19d_m,\qquad d_m-d_v=0.05\,\mathrm{mm}
$$

The main scale has one-millimetre divisions. Twenty slightly shorter vernier divisions span nineteen millimetres. Each successive line gains another twentieth of a millimetre.

#### The screw turns angle into travel

$$
L_{\rm indicated}=0.5N+0.01j\quad(\mathrm{mm})
$$

Count complete half-millimetre sleeve intervals, then add the thimble fraction. Fifty thimble divisions make one turn. A half-millimetre sleeve mark matters as much as a whole-millimetre mark.

#### Remove a measured zero offset

$$
L=L_{\rm indicated}-z_0
$$

The closed faces provide a zero-length reference. Its indicated reading is saved separately for each instrument; changing the instrument later does not rewrite it.

#### Uncertainty of a repeated result

$$
u_c=\sqrt{\frac{s^2}{n}+\frac{d^2}{12}+u_0^2}
$$

The sample scatter s and count n give the repeat contribution. The scale division d retains a conservative quantization floor, and the last term describes the saved zero. Uncorrected viewing bias and later zero drift are outside this estimate.

### Common difficulties

#### Five identical readings can still be biased

**Typical misconception** If every reading agrees, the result must be exact.

**Better mental model** All readings share the same zero and viewing direction. The repeat scatter can vanish even when the mean is displaced. The resolution floor therefore remains.

### Run the experiment

1. 01

   #### Find the matching line

   Choose Read a cylinder, then Scale close-up. Open the jaw slightly and bring it back into contact.

   **What to observe:** The main scale gives the whole millimetres. The green vernier line gives the remaining fraction. A reading can only be saved after contact.

2. 02

   #### Check a drifting zero

   Increase Zero offset in Instrument errors & seed. Keep a reading, choose Close & check zero, then select the cylinder again and save another.

   **What to observe:** The earlier reading keeps its old calibration. The later one subtracts the newly checked zero. More repeats alone cannot remove that shift.

3. 03

   #### Measure from the inside

   Choose Open into a bore. Drag the bright jaw outward until it reaches the inner wall.

   **What to observe:** The jaws stop at the bore walls. For an outside measurement, the contact stops a closing motion instead.

4. 04

   #### See why the viewing direction matters

   Choose Look from the side and switch between Scale close-up and Parallax side view. Set Viewing angle back to zero.

   **What to observe:** The physical opening stays fixed while the two separated scale planes shift in apparent alignment. Repeating from the same wrong angle preserves the bias.

Physical conclusion

Read both parts of the scale, subtract the saved zero and keep systematic shifts separate from repeat scatter.

## Related experiments and articles

How physics instruments measure · Reading path

### How physics instruments measure

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **How a vernier reads between millimetres**](https://sandboxphysics.com/stories/instruments/length-metrology/)

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)