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Difficulty: HardMeasurement of Length and Measuring Instruments

A Vernier caliper has 1010 divisions on its Vernier scale that coincide with 99 main scale divisions of 1 mm1\text{ mm} each. When the measuring jaws are fully closed without any object between them, the zero mark of the Vernier scale lies to the left of the main scale zero mark, and the 7th7\text{th} Vernier division coincides precisely with a main scale mark. When used to measure the internal diameter of a hollow brass ring, the main scale reads 3.5 cm3.5\text{ cm} and the 4th4\text{th} Vernier division coincides with a main scale mark. What is the actual corrected internal diameter of the ring in cm?

Answer: 3.57 cm

Answer

The corrected internal diameter of the hollow brass ring is 3.57 cm3.57\text{ cm}.
The least count of the Vernier caliper is 0.01 cm0.01\text{ cm}. The negative zero error is −(10−7)×0.01 cm=−0.03 cm-(10 - 7) \times 0.01\text{ cm} = -0.03\text{ cm}. The observed reading is 3.5 cm+(4×0.01 cm)=3.54 cm3.5\text{ cm} + (4 \times 0.01\text{ cm}) = 3.54\text{ cm}. Correcting for zero error gives 3.54 cm−(−0.03 cm)=3.57 cm3.54\text{ cm} - (-0.03\text{ cm}) = 3.57\text{ cm}.

Step-by-Step Solution

1
Calculate the least count of the Vernier caliper.
Least count = 0.01 cm0.01\text{ cm} (0.1 mm0.1\text{ mm}).
One main scale division is 1 mm=0.1 cm1\text{ mm} = 0.1\text{ cm}. Ten Vernier divisions equal nine main scale divisions (0.9 mm0.9\text{ mm}), so one Vernier division = 0.09 cm0.09\text{ cm}. Least count = 0.1 cm−0.09 cm=0.01 cm0.1\text{ cm} - 0.09\text{ cm} = 0.01\text{ cm}.
2
Determine the zero error of the instrument.
Zero Error = −0.03 cm-0.03\text{ cm}.
For a negative zero error where the Vernier zero lies to the left of the main scale zero, Zero Error = −(N−n)×least count-(N - n) \times \text{least count}, where N=10N = 10 and n=7n = 7. Thus, Zero Error = −(10−7)×0.01 cm=−0.03 cm-(10 - 7) \times 0.01\text{ cm} = -0.03\text{ cm}.
3
Calculate the uncorrected observed reading.
Observed Reading = 3.54 cm3.54\text{ cm}.
Observed Reading = Main scale reading + (Vernier coincided division \times Least count) = 3.5 cm+(4×0.01 cm)=3.54 cm3.5\text{ cm} + (4 \times 0.01\text{ cm}) = 3.54\text{ cm}.
4
Apply zero error correction to obtain the actual reading.
Corrected Reading = 3.57 cm3.57\text{ cm}.
Corrected Reading = Observed Reading - Zero Error = 3.54 cm−(−0.03 cm)=3.57 cm3.54\text{ cm} - (-0.03\text{ cm}) = 3.57\text{ cm}.

Key Concept

Measurement of length using a Vernier caliper with negative zero error correction
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Topics

Measurement of Length and Measuring Instruments

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