The distance advanced by the screw of a screw gauge is 2 mm in 4 rotations. Its cap is divided into 50 divisions. There is no zero error. If the screw reads 3 divisions on the main scale and 32 divisions of the cap, then the diameter (in mm ) of the wire is

Answer

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

### Step 1: Calculate the Pitch of the Screw Gauge
The distance advanced by the screw of the screw gauge is given as 2 mm in 4 rotations.

\[ \text{Pitch} = \frac{\text{Total distance}}{\text{Number of rotations}} = \frac{2 \text{ mm}}{4} = 0.5 \text{ mm} \]

### Step 2: Determine the Least Count of the Screw Gauge
The screw gauge is divided into 50 divisions on the circular scale. The least count (LC) can be calculated using the formula:

\[ \text{Least Count} = \frac{\text{Pitch}}{\text{Number of divisions}} = \frac{0.5 \text{ mm}}{50} = 0.01 \text{ mm} \]

### Step 3: Read the Main Scale and Circular Scale
The screw gauge reads 3 divisions on the main scale and 32 divisions on the circular scale.

- **Main Scale Reading (MSR)**:
\[ \text{MSR} = \text{Pitch} \times \text{Main Scale divisions} = 0.5 \text{ mm} \times 3 = 1.5 \text{ mm} \]

- **Circular Scale Reading (CSR)**:
\[ \text{CSR} = \text{Least Count} \times \text{Circular Scale divisions} = 0.01 \text{ mm} \times 32 = 0.32 \text{ mm} \]

### Step 4: Calculate the Total Reading
The total reading (TR) is the sum of the main scale reading and the circular scale reading.

\[ \text{Total Reading} = \text{MSR} + \text{CSR} = 1.5 \text{ mm} + 0.32 \text{ mm} = 1.82 \text{ mm} \]

### Conclusion
The diameter of the wire is therefore:

\[ \text{Diameter of the wire} = 1.82 \text{ mm} \]