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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 100100 equal divisions on its circular thimble scale. When the anvil and spindle are fully closed without any object, the zero mark on the circular scale lies 44 divisions below the datum line. The instrument is then used to measure the total thickness of a tightly bound stack of 1010 identical metal sheets. The main scale reading is 2.5 mm2.5\text{ mm} and the 38th38\text{th} division on the circular scale aligns with the datum line. What is the actual mean thickness of a single metal sheet?

  1. A
    0.269 mm0.269\text{ mm}
  2. B
    0.271 mm0.271\text{ mm}
  3. 0.267 mm0.267\text{ mm}Answer
  4. D
    0.284 mm0.284\text{ mm}

Answer

The actual mean thickness of a single metal sheet is 0.267 mm0.267\text{ mm}.
The least count of the micrometer is 0.5 mm100=0.005 mm\frac{0.5\text{ mm}}{100} = 0.005\text{ mm}. With the zero mark 4 divisions below the datum line, there is a positive zero error of +0.020 mm+0.020\text{ mm}. The observed total reading for 10 sheets is 2.5 mm+(38×0.005 mm)=2.690 mm2.5\text{ mm} + (38 \times 0.005\text{ mm}) = 2.690\text{ mm}. Subtracting the positive zero error yields an actual thickness of 2.690 mm−0.020 mm=2.670 mm2.690\text{ mm} - 0.020\text{ mm} = 2.670\text{ mm} for 10 sheets, which gives 0.267 mm0.267\text{ mm} per sheet.

Step-by-Step Solution

1
Calculate the least count of the micrometer screw gauge
Least count=PitchNumber of circular scale divisions=0.5 mm100=0.005 mm\text{Least count} = \frac{\text{Pitch}}{\text{Number of circular scale divisions}} = \frac{0.5\text{ mm}}{100} = 0.005\text{ mm}
The least count determines the value of each division on the circular scale.
2
Determine the zero error
Zero error=+4×0.005 mm=+0.020 mm\text{Zero error} = +4 \times 0.005\text{ mm} = +0.020\text{ mm}
Since the zero mark on the thimble is below the main scale datum line when closed, the error is positive.
3
Calculate the observed reading for 10 sheets
Observed reading=Main scale reading+(Circular scale division×Least count)=2.5 mm+(38×0.005 mm)=2.5 mm+0.190 mm=2.690 mm\text{Observed reading} = \text{Main scale reading} + (\text{Circular scale division} \times \text{Least count}) = 2.5\text{ mm} + (38 \times 0.005\text{ mm}) = 2.5\text{ mm} + 0.190\text{ mm} = 2.690\text{ mm}
Combines the main scale and circular scale readings to find the raw measured value.
4
Calculate the corrected reading for 10 sheets
Corrected reading=Observed reading−Zero error=2.690 mm−0.020 mm=2.670 mm\text{Corrected reading} = \text{Observed reading} - \text{Zero error} = 2.690\text{ mm} - 0.020\text{ mm} = 2.670\text{ mm}
Subtracting a positive zero error gives the true thickness of the 10 sheets.
5
Find the thickness of a single sheet
Thickness per sheet=2.670 mm10=0.267 mm\text{Thickness per sheet} = \frac{2.670\text{ mm}}{10} = 0.267\text{ mm}
Dividing the total corrected thickness by the total number of sheets gives the average thickness per sheet.

Key Concept

Micrometer Screw Gauge Least Count and Zero Error Correction
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Topics

Measurement of Length and Measuring Instruments

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