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Difficulty: MediumMeasurement of Length and Measuring Instruments

A Vernier caliper has 2020 divisions on its Vernier scale that coincide with 1919 main scale divisions of 1 mm1\text{ mm} each. When the jaws of the instrument are brought together without any object, the zero of the Vernier scale lies to the right of the main scale zero mark, and the 3rd3\text{rd} Vernier division coincides with a main scale mark. When used to measure the thickness of a wooden block, the main scale reads 3.5 cm3.5\text{ cm} and the 12th12\text{th} Vernier division coincides with a main scale line. What is the actual thickness of the wooden block?

  1. A
    3.560 cm3.560\text{ cm}
  2. B
    3.575 cm3.575\text{ cm}
  3. 3.545 cm3.545\text{ cm}Answer
  4. D
    3.530 cm3.530\text{ cm}

Answer

The actual thickness of the wooden block is 3.545 cm3.545\text{ cm}.
The least count of a 20-division Vernier caliper with 1 mm1\text{ mm} main scale divisions is 0.1 cm20=0.005 cm\frac{0.1\text{ cm}}{20} = 0.005\text{ cm}. Since the Vernier zero lies to the right of the main scale zero when closed, it has a positive zero error of +(3×0.005 cm)=+0.015 cm+ (3 \times 0.005\text{ cm}) = +0.015\text{ cm}. The observed measurement is 3.5 cm+(12×0.005 cm)=3.560 cm3.5\text{ cm} + (12 \times 0.005\text{ cm}) = 3.560\text{ cm}. Subtracting the positive zero error gives the actual thickness: 3.560 cm−0.015 cm=3.545 cm3.560\text{ cm} - 0.015\text{ cm} = 3.545\text{ cm}.

Step-by-Step Solution

1
Determine the least count (precision) of the Vernier caliper.
Least Count (LC)=1 Main Scale Division (MSD)Number of Vernier Divisions=1 mm20=0.05 mm=0.005 cm\text{Least Count (LC)} = \frac{1\text{ Main Scale Division (MSD)}}{\text{Number of Vernier Divisions}} = \frac{1\text{ mm}}{20} = 0.05\text{ mm} = 0.005\text{ cm}.
The least count is the smallest value that can be measured directly by the instrument.
2
Calculate the zero error of the instrument.
Zero Error=+(3×0.005 cm)=+0.015 cm\text{Zero Error} = +(3 \times 0.005\text{ cm}) = +0.015\text{ cm}.
Because the Vernier zero lies to the right of the main scale zero mark when closed, the instrument has a positive zero error.
3
Calculate the total observed reading from the scales.
Observed Reading=3.5 cm+(12×0.005 cm)=3.5 cm+0.060 cm=3.560 cm\text{Observed Reading} = 3.5\text{ cm} + (12 \times 0.005\text{ cm}) = 3.5\text{ cm} + 0.060\text{ cm} = 3.560\text{ cm}.
The observed reading combines the main scale reading and the coincidental Vernier scale division multiplied by the least count.
4
Apply zero error correction to obtain the actual reading.
Actual Reading=Observed Reading−Zero Error=3.560 cm−(+0.015 cm)=3.545 cm\text{Actual Reading} = \text{Observed Reading} - \text{Zero Error} = 3.560\text{ cm} - (+0.015\text{ cm}) = 3.545\text{ cm}.
Zero error correction always requires subtracting the zero error (with its sign) from the observed measurement.

Key Concept

Measurement of length using Vernier calipers and zero error correction
Estimated Time:1m 30s
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Topics

Measurement of Length and Measuring Instruments

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