In a screw gauge, 5 complete rotations of the screw cause if to move a linear distance of `0.25 cm`. There are `100` circular scale divisions. The thickness of a wire measured by this screw gauge gives a reading of 4 main scale divisions and 30 circular scale divisions. Assuming negligible zero error, the thickness of the wire is :

Options

A. `0.3150cm`
B. `0.4300 cm`
C. `0.2150 cm`
D. `0.0430 cm`

Answer

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

### Step 1: Calculate the Least Count of the Screw Gauge
The least count (LC) of the screw gauge can be calculated using the formula:
\[ \text{Least Count} = \frac{\text{Pitch}}{\text{Number of Circular Scale Divisions}} \]
Given that 5 complete rotations of the screw cause it to move a linear distance of 0.25 cm, we first find the pitch (the distance moved in one complete rotation):
\[ \text{Pitch} = \frac{0.25 \text{ cm}}{5} = 0.05 \text{ cm} \]
Since there are 100 circular scale divisions, we can now calculate the least count:
\[ \text{Least Count} = \frac{0.05 \text{ cm}}{100} = 0.0005 \text{ cm} \]

### Step 2: Read the Main Scale and Circular Scale
From the problem, we have:
- Main Scale Reading (MSR) = 4 main scale divisions
- Circular Scale Reading (CSR) = 30 circular scale divisions

### Step 3: Calculate the Total Reading
The total reading (TR) can be calculated using the formula:
\[ \text{Total Reading} = \text{Main Scale Reading} + \left(\text{Circular Scale Reading} \times \text{Least Count}\right) \]
Substituting the values we have:
\[ \text{Total Reading} = 4 \text{ cm} + (30 \times 0.0005 \text{ cm}) \]
Calculating the circular scale contribution:
\[ 30 \times 0.0005 \text{ cm} = 0.015 \text{ cm} \]
Now, adding this to the main scale reading:
\[ \text{Total Reading} = 4 \text{ cm} + 0.015 \text{ cm} = 4.015 \text{ cm} \]

### Step 4: Convert to Appropriate Units
Since the question is about thickness, we can express the final answer in centimeters:
\[ \text{Thickness of the wire} = 4.015 \text{ cm} \]

### Final Answer
The thickness of the wire measured by the screw gauge is **4.015 cm**.

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