Linear and angular measurement

Why it matters

Every inspection report, gauge calibration and machine setting traces back to a few basic length and angle instruments: vernier callipers, micrometers, slip gauges, sine bars and bevel protractors. Knowing how each one reads, what its least count is and where its errors come from tells you which instrument can actually verify a given tolerance. A common rule of thumb is that the instrument's resolution should be about one-tenth of the tolerance being checked.

Key ideas

Accuracy, precision, resolution. Accuracy is closeness of the mean reading to the true value; precision (repeatability) is the closeness of repeated readings to each other; resolution or least count is the smallest change the instrument can display. An instrument can be precise but inaccurate if it has a systematic error such as a zero error.

Errors. Systematic errors (zero error, scale error, thermal expansion, cosine error from misalignment) have a fixed sign and can be corrected. Random errors scatter both ways and are reduced by averaging. Standard reference temperature for length measurement is 20 °C.

Line and end standards. A line standard defines length between two engraved lines (steel rule, international prototype metre of the past); an end standard defines it between two flat end faces (slip gauges, end bars, micrometer anvils). End standards are more accurate in workshop use but are subject to wear and wringing errors. Today the metre is defined from the speed of light and realised with laser interferometry.

Vernier calliper. A main scale and a sliding vernier scale. When n vernier divisions span (n − 1) main-scale divisions, each vernier division is shorter than a main-scale division by the least count. Reading = main-scale reading + (coinciding vernier division × LC), corrected for zero error. Typical LC is 0.02 mm.

Micrometer. A precision screw: one rotation of the thimble advances the spindle by the pitch. LC = pitch / number of thimble divisions, typically 0.5 mm / 50 = 0.01 mm. A ratchet stop gives constant measuring force. Abbe's principle (the measuring axis in line with the scale axis) is satisfied by the micrometer but not by the vernier calliper, which is why the micrometer is more accurate.

Zero error. If the reading is not zero when the faces touch, the instrument has a zero error. Positive zero error (reads above zero) is subtracted from every reading; negative zero error is added. Correct reading = observed reading − zero error (with sign).

Slip gauges (gauge blocks). Hardened, lapped steel or ceramic blocks with two flat parallel faces, wrung together by sliding with a light film so they adhere. A set such as M112 lets almost any size be built in 0.0005 mm steps. Build a stack with the fewest blocks, eliminating the last decimal place first. Grades (calibration, 00, 0, 1, 2) differ in accuracy; take values from the standard (IS 2984 / ISO 3650).

Angular instruments.

• Universal bevel protractor: vernier on a circular scale, least count 5 minutes (12 vernier divisions span 23° of the main scale).
• Sine bar: a hardened bar with two rollers of equal diameter at a precise centre distance L (100, 200 or 250 mm). Placing slip gauges of height h under one roller sets sin θ = h/L. Accuracy falls as θ increases because dθ/dh = 1/(L cos θ); sine bars are not recommended beyond about 45°. Sine centres and sine tables work the same way.
• Angle gauges: a small set of hardened blocks (degrees, minutes and seconds) that are added or subtracted to build any angle.
• Autocollimator: an optical instrument that measures small angular tilts of a reflector very precisely; used for flatness and straightness checks.

Formulas

LC (vernier) = 1 MSD − 1 VSD = s / n (when n VSD = (n − 1) MSD)

• s = value of one main-scale division (mm); n = number of vernier divisions.

Reading = MSR + (VC × LC) − zero error

• MSR = main-scale reading (mm); VC = coinciding vernier or thimble division (number); zero error in mm with its sign.

LC (micrometer) = p / N

• p = screw pitch (mm); N = number of thimble divisions.

sin θ = h / L

• h = slip gauge stack height (mm); L = centre distance of rollers (mm).

dθ = dh / (L·cos θ) (radians)

• Angle error caused by a small height error dh; shows why errors grow as θ approaches 90°.

Worked examples

Example 1 (standard): build a slip gauge stack of 41.3825 mm from an M112 set.

Given: the M112 set contains 1.0005 mm; 1.001–1.009 mm in 0.001 mm steps; 1.01–1.49 mm in 0.01 mm steps; 0.5–24.5 mm in 0.5 mm steps; 25, 50, 75, 100 mm.

1. Remove the fourth decimal first: use 1.0005 → remainder 41.3825 − 1.0005 = 40.382 mm.
2. Remove the third decimal: use 1.002 → remainder 40.382 − 1.002 = 39.380 mm.
3. Remove the second decimal: use 1.38 → remainder 39.380 − 1.38 = 38.000 mm.
4. 38.000 = 25 + 13.0: use 13.0 and 25.

Stack: 1.0005 + 1.002 + 1.38 + 13.0 + 25 = 41.3825 mm (five blocks).

Example 2 (GATE level): sine bar setting and its sensitivity.

Given: 200 mm sine bar; required angle θ = 30°20′; the slip gauge stack has an error of 2 µm.

1. h = L·sin θ = 200 × sin(30.333°) = 200 × 0.50503 = 101.006 mm (built as 100 + 1.006).
2. dθ = dh / (L·cos θ) = 0.002 / (200 × 0.86310) = 1.159 × 10⁻⁵ rad.
3. Convert: 1.159 × 10⁻⁵ × (180/π) × 3600 = 2.39″.

Slip gauge height 101.006 mm; a 2 µm stack error tilts the angle by about 2.4 seconds of arc.

Common mistakes

• Adding a positive zero error instead of subtracting it.
• Taking the vernier least count as 1/n of a main-scale division without checking how many MSD the vernier spans.
• Building slip gauge stacks from the largest block downward; it uses more blocks and accumulates wringing errors. Eliminate the smallest decimal first.
• Using a sine bar at large angles (above about 45°), where small height errors cause large angle errors; use angle gauges or measure the complement.
• Forgetting cosine error: a dial or probe tilted by angle α to the measuring direction reads L/cos α (or L·cos α, depending on the set-up).
• Ignoring temperature: steel grows about 11.5 µm per metre per °C, comparable to fine tolerances.

For GATE PI

Expect least-count and reading questions on vernier callipers and micrometers (with zero error), slip-gauge stack building, sine bar height or angle calculations, and conceptual questions on accuracy versus precision, Abbe's principle, and line versus end standards. Practise converting degrees-minutes-seconds and keeping enough decimals in sine values.

Quick check

1. A vernier has 50 divisions spanning 49 mm of a 1 mm main scale. What is its least count?
2. A micrometer of pitch 0.5 mm has 50 thimble divisions. What is its least count?
3. A 100 mm sine bar is set with a 50 mm stack. What is the angle?
4. Does the vernier calliper obey Abbe's principle?

Answers: 1. 0.02 mm. 2. 0.01 mm. 3. 30°. 4. No; its measuring jaws are offset from the scale axis.