A micrometer screw gauge has a pitch of 0.5\text{ mm} and 50 divisions on its circular thi… — JAMB UTME

A micrometer screw gauge has a pitch of $0.5\text{ mm}$ and $50$ divisions on its circular thimble scale. When its anvil and spindle are fully brought together without any object, the zero mark of the thimble scale lies $4$ divisions past the main scale index line in the tightening direction (a positive zero error). When the wall thickness of a hollow metal pipe is measured with this micrometer, the main scale reading is $3.5\text{ mm}$ and the $32\text{nd}$ thimble division aligns with the index line. If the outer diameter of the pipe is separately measured to be $25.40\text{ mm}$ , what is the corrected inner diameter of the metal pipe in millimeters?

Answer: 17.84 mm

Answer

17.84 mm

The least count of the micrometer screw gauge is

\frac{0.5\text{ mm}}{50} = 0.01\text{ mm}

. The observed wall thickness is

3.5\text{ mm} + (32 \times 0.01\text{ mm}) = 3.82\text{ mm}

. Accounting for the positive zero error of

+0.04\text{ mm}

gives a true wall thickness of

3.82\text{ mm} - 0.04\text{ mm} = 3.78\text{ mm}

. Subtracting twice the wall thickness from the outer diameter yields the inner diameter:

25.40\text{ mm} - 2(3.78\text{ mm}) = 17.84\text{ mm}

Step-by-Step Solution

Calculate the least count (precision) of the micrometer screw gauge

Least count = 0.01 mm

Least count is defined as pitch divided by total circular scale divisions: 0.5 mm / 50 = 0.01 mm.

Determine the observed wall thickness from the instrument readings

Observed thickness = 3.82 mm

Observed reading = Main scale reading + (Thimble division * Least count) = 3.5 mm + (32 * 0.01 mm) = 3.82 mm.

Correct the observed thickness for positive zero error

Corrected wall thickness = 3.78 mm

Positive zero error (+0.04 mm) must be subtracted from the observed reading: 3.82 mm - 0.04 mm = 3.78 mm.

Calculate the corrected inner diameter of the hollow pipe

Inner diameter = 17.84 mm

The inner diameter equals the outer diameter minus twice the wall thickness: 25.40 mm - 2(3.78 mm) = 25.40 mm - 7.56 mm = 17.84 mm.

Key Concept

Measurement of length, micrometer zero error correction, and geometry of hollow cylinders

Estimated Time:2m 30s

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