In a complete rotation, spindle of a screw gauge advances by `(1)/(2)mm`. There are `50` divisions on circular scale. The main scale has `(1)/(2)mm` marks to (is graduated to `(1)/(2)mm`)
If a wire is put between the jaws, `3` main scale divisions are clearly visible, and `20th` division of circular scale coincides with reference line. Find diameter of wire in correct significant figures.

Options

A. `1.7mm`
B. `1.70mm`
C. `3.40mm`
D. `3.20mm`

Explanation

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

### Step 1: Determine the Least Count of the Screw Gauge
The least count (LC) of the screw gauge can be calculated using the formula:
\[ \text{Least Count} = \frac{\text{Advance in one complete rotation}}{\text{Number of divisions on circular scale}} \]
Given that the spindle advances by \(0.5 \, \text{mm}\) in one complete rotation and there are \(50\) divisions on the circular scale, we can calculate:
\[ \text{Least Count} = \frac{0.5 \, \text{mm}}{50} = 0.01 \, \text{mm} \]

### Step 2: Identify the Main Scale Reading (MSR)
The main scale reading is determined by counting the number of main scale divisions that are clearly visible. According to the problem, \(3\) main scale divisions are visible. Since each main scale division is \(0.5 \, \text{mm}\):
\[ \text{Main Scale Reading (MSR)} = 3 \times 0.5 \, \text{mm} = 1.5 \, \text{mm} \]

### Step 3: Identify the Circular Scale Reading (CSR)
The circular scale reading is determined by the division on the circular scale that coincides with the reference line. The problem states that the \(20^{th}\) division of the circular scale coincides with the reference line. Since each circular scale division is \(0.01 \, \text{mm}\):
\[ \text{Circular Scale Reading (CSR)} = 20 \times 0.01 \, \text{mm} = 0.20 \, \text{mm} \]

### Step 4: Calculate the Total Diameter of the Wire
The diameter of the wire can be calculated by adding the main scale reading and the circular scale reading:
\[ \text{Diameter of Wire} = \text{MSR} + \text{CSR} = 1.5 \, \text{mm} + 0.20 \, \text{mm} = 1.7 \, \text{mm} \]

### Step 5: Determine the Correct Significant Figures
Since the least count is \(0.01 \, \text{mm}\), the final answer should be reported with two decimal places:
\[ \text{Diameter of Wire} = 1.70 \, \text{mm} \]

### Final Answer
The diameter of the wire is \(1.70 \, \text{mm}\).

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