Linear Measurement

(a) Principle and least count

A vernier scale slides along the main scale. Its $n$ divisions equal $(n-1)$ divisions of the main scale, so each vernier division is slightly shorter than a main scale division. The difference of one main division and one vernier division is the least count (LC), the smallest length that can be read:

$$\text{LC} = 1\ \text{MSD} - 1\ \text{VSD} = \frac{1\ \text{MSD}}{n}$$

The reading is: main scale reading (just left of the vernier zero) + (coinciding vernier division $\times$ LC).

(b) Vernier caliper

1 VSD $= 49/50 = 0.98$ mm, so

$$\text{LC} = 1 - 0.98 = 0.02\ \text{mm}$$

Observed reading $= 34 + 27 \times 0.02 = 34.54$ mm.

Zero error is positive: $+3 \times 0.02 = +0.06$ mm (it reads 0.06 when closed). Corrected reading $=$ observed $-$ zero error:

$$34.54 - 0.06 = 34.48\ \text{mm}$$

Answer: LC = 0.02 mm; diameter = 34.48 mm.

(c) Micrometer

Least count $= \dfrac{\text{pitch}}{\text{thimble divisions}} = \dfrac{0.5}{50} = 0.01$ mm.

Observed reading $= 12.5 + 27 \times 0.01 = 12.77$ mm.

The zero error is $-0.02$ mm, so the correction is $+0.02$ mm:

$$12.77 - (-0.02) = 12.79\ \text{mm}$$

Answer: corrected micrometer reading = 12.79 mm.

Slip gauges (gauge blocks) are rectangular blocks of hardened, stabilised steel (or ceramic, tungsten carbide) with two opposite faces flat and parallel to a very high accuracy and a precise distance apart. They are the working end standards of length used to calibrate instruments, set comparators and sine bars, and for direct precision measurement.

Wringing

When two clean gauge faces are slid together with slight pressure and a twisting motion, they stick together. The adhesion is due to a thin film of oil or moisture (molecular attraction and atmospheric pressure). Wringing removes the air gap, so the length of the stack is the sum of the individual sizes with no measurable error. Gauges must be wrung just before use and separated soon after.

Grades (IS 2984 / BS 4311)

• Grade 00 (reference/calibration) for calibration laboratories.
• Grade 0 for inspection and master use.
• Grade 1 for workshop inspection and setting tools.
• Grade 2 for general workshop use and tool room.

Precautions

Keep at standard $20^\circ$C and let the gauges soak; hold them with minimum contact so that body heat is not conducted; clean with a soft lint-free cloth and a solvent; wring correctly; coat with petroleum jelly or rust inhibitor after use; avoid dropping; use the minimum number of gauges to reduce cumulative error; use protector gauges at the end of stacks that are used for measuring.

Building 58.275 mm

Always eliminate the last decimal place first.

Check: $1.005 + 1.27 + 6.0 + 50 = 58.275$ mm.

Answer: 58.275 mm = 1.005 + 1.27 + 6.0 + 50 mm (4 gauges).

External micrometer

It works on the principle of a screw and nut. A precision screw (spindle) of pitch 0.5 mm turns inside a fixed nut in the barrel (sleeve). A thimble attached to the spindle is divided into 50 parts, so one division moves the spindle $0.5/50 = 0.01$ mm.

   anvil  spindle   sleeve  thimble  ratchet
  |==|   |=========|-----|=|||||||||=[R]
  |  |___|         |
  |__________frame__________|

Parts: U-shaped frame, fixed anvil, spindle, locking nut, sleeve with datum line and the main scale (0.5 mm marks), thimble, and a ratchet stop that gives constant measuring force (about 5 to 10 N) to avoid error due to excessive pressure.

Reading: sleeve reading + thimble division on datum line $\times$ 0.01 mm. Used for outside diameters and thicknesses to 0.01 mm (vernier micrometers read 0.001 mm).

Other instruments

Common errors of a micrometer

1. Zero error (anvil and spindle in contact do not read zero).
2. Wear of anvil and spindle faces and of the screw thread; backlash.
3. Non-parallelism or flatness error of measuring faces.
4. Error in the pitch of the screw and thimble graduation.
5. Variable measuring force if the ratchet is not used; deflection of the frame.
6. Temperature effects, e.g. holding the frame in the hand, and parallax in reading.