Reading Measurement from Vernier Caliper Scale Configuration

To find the measured value of the diameter $D$, let us determine the least count ($\text{LC}$) of the Vernier caliper, check for any zero error, and read the final measurement:

1. Least Count ($\text{LC}$):

   • The main scale has $10$ divisions in $1\text{ cm}$. Therefore, the value of one Main Scale Division ($\text{MSD}$) is: $1\text{ MSD} = \frac{1\text{ cm}}{10} = 0.1\text{ cm} = 1\text{ mm}$
   • From Figure 1, $10$ Vernier Scale Divisions ($\text{VSD}$) are equal in length to $9\text{ MSD}$: $10\text{ VSD} = 9\text{ MSD} \implies 1\text{ VSD} = \frac{9}{10}\text{ MSD} = 0.09\text{ cm}$
   • The least count of the Vernier caliper is: $\text{LC} = 1\text{ MSD} - 1\text{ VSD} = 0.1\text{ cm} - 0.09\text{ cm} = 0.01\text{ cm}$

2. Zero Error:

   • In Figure 1 (before measurement), the zero mark of the Vernier scale is perfectly aligned with the zero mark of the main scale. Thus: $\text{Zero Error} = 0$

3. Reading for Diameter $D$ (Figure 2):

   • The zero mark of the Vernier scale lies between $0.1\text{ cm}$ and $0.2\text{ cm}$. $\text{Main Scale Reading (MSR)} = 0.1\text{ cm}$
   • The $3^{\text{rd}}$ division of the Vernier scale coincides with a line on the main scale. $\text{Vernier Scale Reading (VSR)} = 3 \times \text{LC} = 3 \times 0.01\text{ cm} = 0.03\text{ cm}$

4. Measured Value of $D$: $D = \text{MSR} + \text{VSR} - \text{Zero Error}$ $D = 0.1\text{ cm} + 0.03\text{ cm} - 0 = 0.13\text{ cm}$

Therefore, the correct option is (C).