While measuring the diameter of a wire using a screw gauge, the following readings were noted: Main scale reading is \( 1 \, \text{mm} \) and circular scale reading is equal to 42 divisions. Pitch of the screw gauge is \( 1 \, \text{mm} \), and it has 100 divisions on the circular scale. The diameter of the wire is \( \frac{x}{50} \, \text{mm} \). The value of \( x \) is:
- 142
- 71
- 42
- 21
The Correct Option is B
Approach Solution - 1
Given:
MSR (Main Scale Reading) = 1 mm, CSD (Circular Scale Reading) = 42
Least Count (LC) is calculated as:
\[ \text{Least Count (LC)} = \frac{\text{Pitch}}{\text{Number of CSD}} = \frac{1}{100} \, \text{mm} = 0.01 \, \text{mm}. \]
The diameter is calculated as:
\[ \text{Diameter} = \text{MSR} + \text{LC} \times \text{CSD}. \]
Substitute the values:
\[ \text{Diameter} = 1 + (0.01) \times 42 \, \text{mm} = 1.42 \, \text{mm}. \]
Since the diameter is given as \( \frac{x}{50} \), equate:
\[ \frac{x}{50} = 1.42 \implies x = 71. \]
The value of \( x \) is: [71]
Approach Solution -2
To calculate the diameter of the wire using a screw gauge, we need to use the formula:
\(D = \text{MSR} + \left(\frac{\text{CSR} \times \text{LC}}{1}\right)\)
where:
- \(D\) is the diameter of the wire.
- \(\text{MSR}\) is the Main Scale Reading.
- \(\text{CSR}\) is the Circular Scale Reading.
- \(\text{LC}\) is the Least Count of the screw gauge.
Given:
- Main scale reading (MSR) = \(1 \, \text{mm}\)
- Circular scale reading (CSR) = 42 divisions
- Pitch of the screw gauge = \(1 \, \text{mm}\)
- No. of divisions on the circular scale = 100
The Least Count (LC) is given by:
\(\text{LC} = \frac{\text{Pitch}}{\text{No. of divisions on circular scale}} = \frac{1 \, \text{mm}}{100} = 0.01 \, \text{mm}\)
Now, substituting the values in the formula:
\(D = \text{MSR} + \left(\frac{\text{CSR} \times \text{LC}}{1}\right) = 1 \, \text{mm} + \left(\frac{42 \times 0.01}{1}\right) \, \text{mm}\)
\(D = 1 \, \text{mm} + 0.42 \, \text{mm} = 1.42 \, \text{mm}\)
The problem states the diameter in a different form \(\frac{x}{50} \, \text{mm}\). Therefore,
\(\frac{x}{50} = 1.42 \Rightarrow x = 1.42 \times 50 = 71\)
Thus, the value of \(x\) is 71.
The correct answer is 71.
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