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Difficulty: HardMeasurement of Length and Measuring Instruments

A micrometer screw gauge has a pitch of 0.5 mm0.5\text{ mm} and 5050 divisions on its circular thimble scale. When the anvil and spindle are brought into contact without an object between them, the 47th47\text{th} division on the thimble scale aligns exactly with the datum line of the main scale. When used to measure the diameter of a uniform metal rod, the main scale reads 1.5 mm1.5\text{ mm} and the 18th18\text{th} division on the thimble scale aligns with the datum line. What is the true diameter of the metal rod?

  1. A
    1.65 mm1.65\text{ mm}
  2. B
    1.68 mm1.68\text{ mm}
  3. 1.71 mm1.71\text{ mm}Answer
  4. D
    2.15 mm2.15\text{ mm}

Answer

The true diameter of the metal rod is 1.71 mm1.71\text{ mm}.
The least count of the micrometer is 0.01 mm0.01\text{ mm}. Because the zero position when closed shows the 47th47\text{th} division aligned, it has a negative zero error of −0.03 mm-0.03\text{ mm}. Subtracting this negative error from the observed reading of 1.68 mm1.68\text{ mm} gives 1.68−(−0.03)=1.71 mm1.68 - (-0.03) = 1.71\text{ mm}.

Step-by-Step Solution

1
Calculate the least count (precision) of the micrometer screw gauge.
Least count=PitchNumber of thimble divisions=0.5 mm50=0.01 mm\text{Least count} = \frac{\text{Pitch}}{\text{Number of thimble divisions}} = \frac{0.5\text{ mm}}{50} = 0.01\text{ mm}.
The least count determines the value of each thimble division.
2
Determine the zero error of the instrument.
Since the 47th47\text{th} division coincides with the datum line when closed, the zero mark lies above the reference line. Zero error=−(50−47)×0.01 mm=−0.03 mm\text{Zero error} = -(50 - 47) \times 0.01\text{ mm} = -0.03\text{ mm}.
When the thimble zero mark has passed the datum line in the reverse direction, the instrument exhibits a negative zero error.
3
Calculate the observed reading from the main scale and thimble scale.
Observed reading=1.5 mm+(18×0.01 mm)=1.5 mm+0.18 mm=1.68 mm\text{Observed reading} = 1.5\text{ mm} + (18 \times 0.01\text{ mm}) = 1.5\text{ mm} + 0.18\text{ mm} = 1.68\text{ mm}.
The observed reading is the sum of the main scale reading and the thimble scale reading.
4
Compute the true reading by correcting for zero error.
True reading=Observed reading−Zero error=1.68 mm−(−0.03 mm)=1.71 mm\text{True reading} = \text{Observed reading} - \text{Zero error} = 1.68\text{ mm} - (-0.03\text{ mm}) = 1.71\text{ mm}.
True value is found by subtracting the zero error (including its sign) from the observed value.

Key Concept

Zero Error Correction in Micrometer Screw Gauge
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Topics

Measurement of Length and Measuring Instruments

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