The pitch of a screw gauge is `0.55mm` and there are 100 divisions on its circular scale. The instrument reads `+2` divisions when nothing is put in between its jaws. In measuring the diameter of a wire, there are 8 divisions on the main scale and `83^(rd)` division coincides with the reference. Then the diameter of the wire is

Options

A. `4.05mm`
B. `4.8455mm`
C. `3.05mm`
D. `4.8675mm`

Answer

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

### Step 1: Identify the given values
- Pitch of the screw gauge (P) = 0.55 mm
- Number of divisions on the circular scale (N) = 100
- Zero error = +2 divisions (indicating a positive zero error)
- Main scale reading (MSR) = 8 divisions
- Circular scale reading (CSR) = 83 divisions

### Step 2: Calculate the least count (LC) of the screw gauge
The least count of the screw gauge is calculated using the formula:
\[ \text{Least Count (LC)} = \frac{\text{Pitch}}{\text{Number of divisions on circular scale}} = \frac{P}{N} = \frac{0.55 \, \text{mm}}{100} = 0.0055 \, \text{mm} \]

### Step 3: Calculate the total reading from the screw gauge
The total reading (TR) can be calculated using the formula:
\[ \text{Total Reading (TR)} = \text{Main Scale Reading (MSR)} + \left(\text{Circular Scale Reading (CSR)} \times \text{Least Count (LC)}\right) - \text{Zero Error} \]

Substituting the values:
\[ \text{Total Reading (TR)} = 8 \, \text{divisions} + (83 \times 0.0055 \, \text{mm}) - 0.01 \, \text{mm} \]
(Note: Zero error in mm = 2 divisions × 0.0055 mm/division = 0.011 mm, but since it is a positive zero error, we subtract it.)

Calculating the circular scale contribution:
\[ 83 \times 0.0055 = 0.4565 \, \text{mm} \]

Now substituting back into the total reading:
\[ \text{Total Reading (TR)} = 8 + 0.4565 - 0.011 = 8.4455 \, \text{mm} \]

### Step 4: Finalize the diameter of the wire
Since the total reading gives us the diameter of the wire, we conclude that:
\[ \text{Diameter of the wire} = 8.4455 \, \text{mm} \]

### Final Answer:
The diameter of the wire is approximately **8.4455 mm**.

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