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 `5.10cm and 5.15cm` of the main scale. The Vernier scale has `50` divisions equivalent to `2.45 cm`. 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. (a) `5.112cm`
B. (b) `5.124cm`
C. ( c ) `5.136cm`
D. (d ) `5.1148cm`

Explanation

To find the diameter of the cylinder measured using a Vernier caliper, 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. Since we take the lower value for the main scale reading, we have:
- **Main Scale Reading (MSR) = 5.10 cm**

### Step 2: Calculate the Value of One Main Scale Division (MSD)
The difference between the two readings gives us the value of one main scale division:
- **1 MSD = 5.15 cm - 5.10 cm = 0.05 cm**

### Step 3: Determine the Value of One Vernier Scale Division (VSD)
The Vernier scale has 50 divisions that are equivalent to 2.45 cm. Therefore, the value of one Vernier scale division can be calculated as:
- **1 VSD = 2.45 cm / 50 = 0.049 cm**

### Step 4: Calculate the Least Count (LC)
The least count of the Vernier caliper is given by the formula:
- **Least Count (LC) = 1 MSD - 1 VSD**
- **LC = 0.05 cm - 0.049 cm = 0.001 cm**

### Step 5: Identify the Coinciding Division
The 24th division of the Vernier scale coincides with one of the main scale divisions. Therefore, we will add this value to our reading:
- **Value of coinciding division = 24 * 1 VSD = 24 * 0.049 cm = 1.176 cm**

### Step 6: Calculate the Total Reading (TR)
The total reading can be calculated using the formula:
- **Total Reading (TR) = MSR + Coinciding Value**
- **TR = 5.10 cm + 1.176 cm = 6.276 cm**

### Step 7: Final Diameter of the Cylinder
Thus, the diameter of the cylinder is:
- **Diameter = 5.10 cm + (24 * 0.049 cm) = 5.10 cm + 1.176 cm = 6.276 cm**

### Conclusion
The diameter of the cylinder is approximately **6.276 cm**.

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