A screw gauge has main scale divisions of 1 mm each and its circular scale has 50 division. While measuring the diameter of a ball, cap end is after `19^("th")` marking of main scale and `21^("st")` division of circular scale coincides with the reference line. What is the measured diameter diameter of the ball?

Options

A. 19.42 mm
B. 21.42 mm
C. 19.420 mm
D. None of the Above

Explanation

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

### Step 1: Identify the Main Scale Reading (MSR)
The main scale reading is the value indicated by the main scale just before the circular scale starts moving. According to the problem, the cap end is after the 19th marking of the main scale. Therefore, the Main Scale Reading (MSR) is:
\[ \text{MSR} = 19 \, \text{mm} \]

### Step 2: Identify the Circular Scale Reading (CSR)
The circular scale reading is determined by the division of the circular scale that coincides with the reference line. In this case, the 21st division of the circular scale coincides with the reference line. Therefore, the Circular Scale Reading (CSR) is:
\[ \text{CSR} = 21 \]

### Step 3: Calculate the Least Count (LC)
The least count of the screw gauge is calculated using the formula:
\[ \text{Least Count (LC)} = \frac{\text{Pitch}}{\text{Number of divisions on circular scale}} \]
Given that the pitch (the distance moved by the screw gauge per complete rotation) is 1 mm and the number of divisions on the circular scale is 50, we can calculate the least count:
\[ \text{LC} = \frac{1 \, \text{mm}}{50} = 0.02 \, \text{mm} \]

### Step 4: Calculate the Total Reading
The total reading of the screw gauge can be calculated using the formula:
\[ \text{Total Reading} = \text{MSR} + (\text{CSR} \times \text{LC}) \]
Substituting the values we have:
\[ \text{Total Reading} = 19 \, \text{mm} + (21 \times 0.02 \, \text{mm}) \]

### Step 5: Perform the Calculation
Now, we calculate the value:
\[ 21 \times 0.02 = 0.42 \, \text{mm} \]
Thus, the total reading becomes:
\[ \text{Total Reading} = 19 \, \text{mm} + 0.42 \, \text{mm} = 19.42 \, \text{mm} \]

### Final Answer
The measured diameter of the ball is:
\[ \text{Diameter of the ball} = 19.42 \, \text{mm} \]

---