Question

Screw gauge A has a pitch of \(1 \mathrm{~mm}\) and 50 divis1on on its circular scale screw gauge \(B\) has a pitch of \(0.5 \mathrm{~mm}\) and 100 divisions on its circular scale. If \((\mathrm{L} \times \mathrm{C})\) is least count, then which posibility is true ? What is the parilrility? (A) $2(\mathrm{~L} \times \mathrm{C})_{\mathrm{A}}=(\mathrm{L} \times \mathrm{C})_{\mathrm{B}}$ (B) $(\mathrm{L} \times \mathrm{C})_{\mathrm{A}}=4(\mathrm{~L} \times \mathrm{C})_{\mathrm{B}}$ (C) $(\mathrm{L} \times \mathrm{C})_{\mathrm{A}}=(\mathrm{L} \times \mathrm{C})_{\mathrm{B}}$ (D) $(\mathrm{L} \times \mathrm{C})_{\mathrm{A}}=2(\mathrm{~L} \times \mathrm{C})_{\mathrm{B}}$

Short answer

The correct answer is (B): \((\mathrm{L} \times \mathrm{C})_{\mathrm{A}} = 4(\mathrm{L} \times \mathrm{C})_{\mathrm{B}}\).

Step-by-step solution

Understanding the least count

In the context of a screw gauge, the least count is the smallest length that can be accurately measured using the instrument. Mathematically, it is the product of the screw's pitch (length moved per revolution) and the divisions on the circular scale.

Calculate the least count for screw gauge A

For screw gauge A, we are given the pitch of \(1 \mathrm{~mm}\) and 50 divisions on its circular scale. The least count of A, denoted as \((\mathrm{L} \times \mathrm{C})_{\mathrm{A}}\), can be calculated as follows: \[(\mathrm{L} \times \mathrm{C})_{\mathrm{A}} = \frac{\text{pitch of A}}{\text{divisions on circular scale of A}} = \frac{1\ \mathrm{mm}}{50} = 0.02\ \mathrm{mm} \]

Calculate the least count for screw gauge B

For screw gauge B, we are given the pitch of \(0.5 \mathrm{~mm}\) and 100 divisions on its circular scale. The least count of B, denoted as \((\mathrm{L} \times \mathrm{C})_{\mathrm{B}}\), can be calculated as follows: \[(\mathrm{L} \times \mathrm{C})_{\mathrm{B}} = \frac{\text{pitch of B}}{\text{divisions on circular scale of B}} = \frac{0.5\ \mathrm{mm}}{100} = 0.005\ \mathrm{mm} \]

Compare the least counts of A and B

Now we will compare least count of A and B to find which option among A, B, C, or D is true: (A) Check if \(2(\mathrm{L} \times \mathrm{C})_{\mathrm{A}} = (\mathrm{L} \times \mathrm{C})_{\mathrm{B}}\): \[2(0.02\ \mathrm{mm}) = 0.04\ \mathrm{mm} \ne 0.005\ \mathrm{mm}\] (B) Check if \((\mathrm{L} \times \mathrm{C})_{\mathrm{A}} = 4(\mathrm{L} \times \mathrm{C})_{\mathrm{B}}\): \[0.02\ \mathrm{mm} = 4(0.005\ \mathrm{mm}) = 0.02\ \mathrm{mm}\] Since option (B) is satisfied, the correct answer is (B): \((\mathrm{L} \times \mathrm{C})_{\mathrm{A}} = 4(\mathrm{L} \times \mathrm{C})_{\mathrm{B}}\)