Vernier Calliper Reading for Cylinder Diameter Measurement

To determine the measured diameter of the cylinder, we first find the value of one main scale division ($\text{MSD}$) and one Vernier scale division ($\text{VSD}$).

1. Main Scale Division ($\text{MSD}$): The zero of the Vernier scale lies between $5.10\text{ cm}$ and $5.15\text{ cm}$. Thus, the spacing between consecutive markings on the main scale is: $1\text{ MSD} = 5.15\text{ cm} - 5.10\text{ cm} = 0.05\text{ cm}$

   The Main Scale Reading ($\text{MSR}$) is: $\text{MSR} = 5.10\text{ cm}$

2. Vernier Scale Division ($\text{VSD}$): It is given that $50$ divisions of the Vernier scale correspond to $2.45\text{ cm}$: $50\text{ VSD} = 2.45\text{ cm}$ $1\text{ VSD} = \frac{2.45\text{ cm}}{50} = 0.049\text{ cm}$

3. Least Count ($\text{LC}$): The least count of the Vernier calliper is the difference between one main scale division and one Vernier scale division: $\text{LC} = 1\text{ MSD} - 1\text{ VSD} = 0.05\text{ cm} - 0.049\text{ cm} = 0.001\text{ cm}$

4. Total Reading: The $24^{\text{th}}$ division of the Vernier scale coincides with a main scale mark ($n = 24$). Since there is no zero error, the measured diameter is given by: $\text{Diameter} = \text{MSR} + (n \times \text{LC})$ $\text{Diameter} = 5.10\text{ cm} + (24 \times 0.001\text{ cm})$ $\text{Diameter} = 5.10\text{ cm} + 0.024\text{ cm} = 5.124\text{ cm}$

Therefore, the correct option is (B).