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)`5.112 cm`
B. (b)`5.124 cm`
C. (c)`5.136 cm`
D. (d)`5.148 cm`

Explanation

To find the diameter of the cylinder measured using a Vernier caliper, we will follow these steps:

### Step 1: Identify the Main Scale Reading (MSR)
The problem states that the zero of the Vernier scale lies between 5.10 cm and 5.15 cm. Since it is not specified exactly where it lies, we will take the lower limit as the main scale reading (MSR).

**MSR = 5.10 cm**

### Step 2: Determine the Least Count (LC) of the Vernier Caliper
The least count of the Vernier caliper can be calculated using the formula:

\[ \text{Least Count (LC)} = \text{Value of 1 Main Scale Division (MSD)} - \text{Value of 1 Vernier Scale Division (VSD)} \]

1. **Value of 1 MSD**: Since the main scale reading is not specified, we can assume that each division on the main scale is 0.01 cm (common for Vernier calipers).
2. **Value of 50 Vernier Scale Divisions (VSD)**: Given that 50 divisions on the Vernier scale are equivalent to 2.45 cm, we can find the value of 1 VSD:

\[ \text{Value of 1 VSD} = \frac{2.45 \text{ cm}}{50} = 0.049 \text{ cm} \]

Now, substituting these values into the least count formula:

\[ \text{LC} = 0.01 \text{ cm} - 0.049 \text{ cm} = 0.001 \text{ cm} \]

### Step 3: Identify the Vernier Scale Reading (VSR)
The problem states that the 24th division of the Vernier scale coincides with a main scale division. Therefore, the Vernier scale reading (VSR) can be calculated as:

\[ \text{VSR} = 24 \times \text{LC} = 24 \times 0.001 \text{ cm} = 0.024 \text{ cm} \]

### Step 4: Calculate the Diameter of the Cylinder
The diameter of the cylinder can now be calculated using the formula:

\[ \text{Diameter} = \text{MSR} + \text{VSR} \]

Substituting the values we have:

\[ \text{Diameter} = 5.10 \text{ cm} + 0.024 \text{ cm} = 5.124 \text{ cm} \]

### Conclusion
The diameter of the cylinder is:

\[ \text{Diameter} = 5.124 \text{ cm} \]