A screw gauge of pitch 0.5 ~mm is used to measure the diameter of uniform wire of length 6.8 ~cm , the main scale reading is 1.5 ~mm and circular scale reading is 7 . The calculated curved surface area of wire to appropriate significant figures is : [Screw gauge has 50 divisions on its circular scale]

Units and Measurements PYQ — JEE Main 2022 · 26 Jul · Shift 1 · Q63 with Solution

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Correct answer: B

Find least count of the screw gauge

Given:

Pitch $= 0.5\,\text{mm}$
Circular scale divisions $= 50$

So,

\text{Least count} = \frac{\text{Pitch}}{\text{Number of divisions}} = \frac{0.5}{50} = 0.01\,\text{mm}

Find diameter of the wire

Given readings:

Main scale reading $= 1.5\,\text{mm}$
Circular scale reading $= 7$

Circular scale contribution:

7 \times 0.01 = 0.07\,\text{mm}

Hence diameter,

d = 1.5 + 0.07 = 1.57\,\text{mm}

Convert units to cm

d = 1.57\,\text{mm} = 0.157\,\text{cm}

Length of wire:

l = 6.8\,\text{cm}

Curved surface area of the wire

The wire is cylindrical, so curved surface area is

\text{CSA} = \pi d l

Substitute values:

\text{CSA} = \pi \times 0.157 \times 6.8

First multiply:

0.157 \times 6.8 = 1.0676

Then,

\text{CSA} = \pi \times 1.0676 \approx 3.35\,\text{cm}^2

Apply significant figures

Length $6.8\,\text{cm}$ has 2 significant figures.
Diameter $1.57\,\text{mm}$ has 3 significant figures.

Therefore final answer should be reported to 2 significant figures.

So,

3.35\,\text{cm}^2 \approx 3.4\,\text{cm}^2

Check options

A: $6.8\,\text{cm}^2$ — incorrect
B: $3.4\,\text{cm}^2$ — correct
C: $3.9\,\text{cm}^2$ — incorrect
D: $2.4\,\text{cm}^2$ — incorrect

Therefore, the correct option is B.

More from Units and Measurements

View written solutionFree

Correct answer: B

Find least count of the screw gauge

Given:

Pitch $= 0.5\,\text{mm}$
Circular scale divisions $= 50$

So,

\text{Least count} = \frac{\text{Pitch}}{\text{Number of divisions}} = \frac{0.5}{50} = 0.01\,\text{mm}

Find diameter of the wire

Given readings:

Main scale reading $= 1.5\,\text{mm}$
Circular scale reading $= 7$

Circular scale contribution:

7 \times 0.01 = 0.07\,\text{mm}

Hence diameter,

d = 1.5 + 0.07 = 1.57\,\text{mm}

Convert units to cm

d = 1.57\,\text{mm} = 0.157\,\text{cm}

Length of wire:

l = 6.8\,\text{cm}

Curved surface area of the wire

The wire is cylindrical, so curved surface area is

\text{CSA} = \pi d l

Substitute values:

\text{CSA} = \pi \times 0.157 \times 6.8

First multiply:

0.157 \times 6.8 = 1.0676

Then,

\text{CSA} = \pi \times 1.0676 \approx 3.35\,\text{cm}^2

Apply significant figures

Length $6.8\,\text{cm}$ has 2 significant figures.
Diameter $1.57\,\text{mm}$ has 3 significant figures.

Therefore final answer should be reported to 2 significant figures.

So,

3.35\,\text{cm}^2 \approx 3.4\,\text{cm}^2

Check options

A: $6.8\,\text{cm}^2$ — incorrect
B: $3.4\,\text{cm}^2$ — correct
C: $3.9\,\text{cm}^2$ — incorrect
D: $2.4\,\text{cm}^2$ — incorrect

Therefore, the correct option is B.

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