The pitch of a screw gauge having `50` divisions on its circular scale is `1 mm` When the two jaws of the screw gauge are in contact with each other, the zero of the circular scale lies `6` divisions below the line of gradution. when a wire is placed between the jaws, `3` linear scale divisions are clearly visible while `31` division on the circular scale coincides with the reference line. Find diameter of the wire.

Options

A. `3.62mm`
B. `3.50mm`
C. `3.55mm`
D. `3.74mm`

Answer

To find the diameter of the wire using the screw gauge, we will follow these steps:

### Step 1: Calculate the Least Count (LC)
The least count of the screw gauge can be calculated using the formula:
\[ \text{Least Count (LC)} = \frac{\text{Pitch}}{\text{Number of divisions on the circular scale}} \]
Given:
- Pitch = 1 mm
- Number of divisions on the circular scale = 50

Calculating the least count:
\[ \text{LC} = \frac{1 \text{ mm}}{50} = 0.02 \text{ mm} \]

### Step 2: Determine the Zero Error
The zero error is the reading when the jaws are in contact. It is given that the zero of the circular scale lies 6 divisions below the line of graduation. Therefore, the zero error (E) can be calculated as:
\[ E = \text{Number of divisions below} \times \text{Least Count} \]
Calculating the zero error:
\[ E = 6 \times 0.02 \text{ mm} = 0.12 \text{ mm} \]

### Step 3: Calculate the Linear Scale Reading (LSR)
When the wire is placed between the jaws, it is given that 3 linear scale divisions are visible. The linear scale reading can be calculated as:
\[ \text{LSR} = \text{Number of linear scale divisions} \times 1 \text{ mm} \]
Calculating the linear scale reading:
\[ \text{LSR} = 3 \times 1 \text{ mm} = 3 \text{ mm} \]

### Step 4: Calculate the Circular Scale Reading (CSR)
It is given that 31 divisions on the circular scale coincide with the reference line. The circular scale reading can be calculated as:
\[ \text{CSR} = \text{Number of circular scale divisions} \times \text{Least Count} \]
Calculating the circular scale reading:
\[ \text{CSR} = 31 \times 0.02 \text{ mm} = 0.62 \text{ mm} \]

### Step 5: Calculate the Diameter of the Wire
The diameter of the wire can be calculated using the formula:
\[ \text{Diameter} = \text{LSR} + \text{CSR} - \text{Zero Error} \]
Substituting the values:
\[ \text{Diameter} = 3 \text{ mm} + 0.62 \text{ mm} - 0.12 \text{ mm} \]
Calculating the diameter:
\[ \text{Diameter} = 3 + 0.62 - 0.12 = 3.50 \text{ mm} \]

### Final Answer
The diameter of the wire is **3.50 mm**.

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