A screw gauge with a pitch of 1 mm has 100 divisions on its circular scale. What it is used to measure the diameter of a thin wire, the main scale reading is 3 mm and the circular scale reading is 45. If D is the diameter of the wire in mm, then find the value of D.

Answer

To solve the problem of finding the diameter \( D \) of the wire using the screw gauge, we can follow these steps:

### Step 1: Understand the given information
- **Pitch of the screw gauge**: 1 mm
- **Number of divisions on the circular scale**: 100
- **Main scale reading (MSR)**: 3 mm
- **Circular scale reading (CSR)**: 45

### Step 2: Calculate the least count of the screw gauge
The least count (LC) of the screw gauge can be calculated using the formula:

\[ \text{Least Count} = \frac{\text{Pitch}}{\text{Number of divisions}} \]

Substituting the values:

\[ \text{Least Count} = \frac{1 \text{ mm}}{100} = 0.01 \text{ mm} \]

### Step 3: Calculate the contribution of the circular scale reading
The contribution of the circular scale reading to the measurement can be calculated as:

\[ \text{Circular Scale Contribution} = \text{CSR} \times \text{Least Count} \]

Substituting the values:

\[ \text{Circular Scale Contribution} = 45 \times 0.01 \text{ mm} = 0.45 \text{ mm} \]

### Step 4: Calculate the total diameter of the wire
Now, we can find the diameter \( D \) of the wire by adding the main scale reading and the circular scale contribution:

\[ D = \text{MSR} + \text{Circular Scale Contribution} \]

Substituting the values:

\[ D = 3 \text{ mm} + 0.45 \text{ mm} = 3.45 \text{ mm} \]

### Final Answer
Thus, the diameter \( D \) of the wire is:

\[ D = 3.45 \text{ mm} \]

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