The diameter of a cylinder is measured using a vernier callipers with no zero error . It is found that the zero of the vernier scale lies between `5.10 and 5.15 cm` of the main scale . The `24 th` division of the vernier scale exactly coincides with one of the main scale divisions . The diameter of the cylinder is

Options

A. 5.112 cm
B. 5.124 cm
C. 5.136 cm
D. 5.148 cm

Answer

To find the diameter of the cylinder using the vernier calipers, we will follow these steps:

### Step 1: Determine the Main Scale Reading (MSR)
The zero of the vernier scale lies between 5.10 cm and 5.15 cm. Therefore, we can take the main scale reading as the lower value:
- **Main Scale Reading (MSR) = 5.10 cm**

### Step 2: Determine the Vernier Scale Reading (VSR)
The 24th division of the vernier scale coincides with a division on the main scale. We need to find the value of one Vernier Scale Division (VSD) first.

### Step 3: Calculate the Least Count (LC)
The least count of the vernier calipers is calculated using the formula:
\[ \text{Least Count (LC)} = \text{Value of 1 MSD} - \text{Value of 1 VSD} \]

1. **Value of 1 MSD**: The difference between the two main scale readings is:
\[ \text{Value of 1 MSD} = 5.15 \, \text{cm} - 5.10 \, \text{cm} = 0.05 \, \text{cm} \]

2. **Value of 1 VSD**: Since there are 50 divisions on the vernier scale, we can find the value of one VSD:
\[ \text{Value of 1 VSD} = \frac{0.05 \, \text{cm}}{50} = 0.001 \, \text{cm} \]

3. **Calculate the Least Count**:
\[ \text{LC} = 0.05 \, \text{cm} - 0.001 \, \text{cm} = 0.049 \, \text{cm} \]

### Step 4: Calculate the Total Reading
The total reading (diameter of the cylinder) can be calculated using the formula:
\[ \text{Total Reading} = \text{MSR} + (\text{VSR} \times \text{LC}) \]

Where:
- VSR = 24 (since the 24th division of the vernier scale coincides with the main scale)

Calculating the total reading:
\[ \text{Total Reading} = 5.10 \, \text{cm} + (24 \times 0.001 \, \text{cm}) = 5.10 \, \text{cm} + 0.024 \, \text{cm} = 5.124 \, \text{cm} \]

### Final Answer
The diameter of the cylinder is:
\[ \text{Diameter} = 5.124 \, \text{cm} \]

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