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 solve the problem step by step, we will follow the given information and apply the necessary formulas.

### Step 1: Understand the given values
- **Pitch of screw gauge (P)** = 0.55 mm
- **Number of divisions on circular scale (N)** = 100
- **Instrument reading when nothing is put (error)** = +2 divisions
- **Main scale reading (MSR)** = 8 divisions
- **Circular scale reading (CSR)** = 83rd division

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

### Step 3: Calculate the error in the instrument
The error due to the instrument is calculated as:
\[ \text{Error} = \text{Error in divisions} \times \text{Least Count of Circular Scale} \]
Here, the error in divisions is +2, so:
\[ \text{Error} = 2 \times 0.0055 \text{ mm} = 0.011 \text{ mm} \]

### Step 4: Calculate the total reading for the diameter of the wire
The formula for the diameter of the wire is given by:
\[ \text{Diameter} = (\text{MSR} \times \text{Least Count of Main Scale}) + (\text{CSR} \times \text{Least Count of Circular Scale}) - \text{Error} \]

Substituting the values:
- **MSR** = 8 divisions
- **CSR** = 83 divisions

Calculating:
1. Main Scale Contribution:
\[ \text{MSR Contribution} = 8 \times 0.55 \text{ mm} = 4.4 \text{ mm} \]
2. Circular Scale Contribution:
\[ \text{CSR Contribution} = 83 \times 0.0055 \text{ mm} = 0.4565 \text{ mm} \]

Now, substituting these into the diameter formula:
\[ \text{Diameter} = 4.4 \text{ mm} + 0.4565 \text{ mm} - 0.011 \text{ mm} \]
\[ \text{Diameter} = 4.8455 \text{ mm} \]

### Final Answer
The diameter of the wire is **4.8455 mm**.

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