Micrometer Decision on a Copper Wire Coil
A technician must decide whether to accept a 200 m coil of copper wire specified as 0.80 mm diameter with a tolerance of ±0.02 mm. The micrometer has a zero error: when the jaws are closed it reads −0.03 mm. Ten readings taken at different positions along the wire, as displayed, are 0.77, 0.78, 0.78, 0.79, 0.77, 0.78, 0.79, 0.78, 0.77 and 0.79 mm. A vernier caliper used hurriedly gives 0.8 mm. The workshop needs a decision before the coil is used to wind a small transformer, where both diameter and insulation clearance matter. The technician also has to report a value with an uncertainty and explain why the two instruments disagree. The micrometer resolution is 0.01 mm; the caliper resolution is 0.05 mm. No temperature correction data are available, and the wire is at room temperature.
1. Calculate the corrected mean diameter and a simple uncertainty, then decide whether the coil meets the specification. Show each correction explicitly.
Read the worked answer
The displayed mean is
[ \bar d_{\text{disp}}=\frac{3(0.77)+4(0.78)+3(0.79)}{10}=0.780\ \text{mm} ]
A zero reading of −0.03 mm means the instrument reads 0.03 mm too low, so the correction is +0.03 mm:
[ \bar d=0.780+0.030=0.810\ \text{mm} ]
The ten readings span 0.02 mm, so a simple half-range uncertainty is 0.01 mm. A better estimate from the standard deviation of the mean is about 0.003 mm; adding the 0.01 mm resolution contribution gives roughly 0.004 mm. Report (d=0.810\pm0.004\ \text{mm}) (or about ±0.01 mm using the simple method).
The specification limits are 0.78 mm and 0.82 mm. The corrected mean 0.810 mm lies inside the limits, so the coil can be accepted, but it is only 0.010 mm below the upper limit. Even the simple uncertainty does not take it outside 0.82 mm, so rejection is not justified on these data; the batch should be accepted with a note to monitor the upper limit.
2. Explain why the caliper reading of 0.8 mm is not evidence that the micrometer is wrong, and state one improvement to the measurement protocol.
Read the worked answer
The caliper reading is quoted to one decimal place, but its resolution is 0.05 mm. A single reading of 0.8 mm therefore represents a range of about 0.775–0.825 mm, which includes both the micrometer mean and the specification limits. The caliper also has larger jaws and may compress the wire or include insulation, whereas the micrometer measures across the conductor at a defined point. The two instruments are not in serious conflict; the caliper is simply too coarse for a 0.02 mm tolerance.
A suitable improvement is to take readings at more positions around the circumference as well as along the length, and to record the micrometer zero error before and after the measurements. This checks whether the zero error is stable and reduces the chance that a local thick or thin section is missed.