Analysis of the Vernier Calipers

1. Main Scale Division (MSD): Both calipers have a main scale where 1 cm is divided into 10 equal divisions. Therefore, the value of one main scale division is: $1 \text{ MSD} = \frac{1 \text{ cm}}{10} = 0.1 \text{ cm}$

Caliper C1

1. Least Count (LC1): For caliper C1, 10 Vernier Scale Divisions (VSD) correspond to 9 Main Scale Divisions (MSD). $10 \text{ VSD} = 9 \text{ MSD}$ $1 \text{ VSD} = \frac{9}{10} \text{ MSD} = 0.9 \text{ MSD}$ The least count (LC) for a standard Vernier caliper is given by: $LC_1 = 1 \text{ MSD} - 1 \text{ VSD} = 1 \text{ MSD} - 0.9 \text{ MSD} = 0.1 \text{ MSD}$ Substituting the value of 1 MSD: $LC_1 = 0.1 \times 0.1 \text{ cm} = 0.01 \text{ cm}$

2. Reading of C1: The reading of a Vernier caliper is given by the formula: $\text{Reading} = \text{Main Scale Reading (MSR)} + (\text{Vernier Scale Coincidence (VSC)} \times LC)$ From the figure for C1:

• The zero of the Vernier scale is just past the 2.8 cm mark on the main scale. So, MSR = 2.8 cm.
• The 7th division of the Vernier scale coincides with a division on the main scale. So, VSC = 7.

Calculating the measured value for C1: $\text{Reading (C1)} = 2.8 \text{ cm} + (7 \times 0.01 \text{ cm})$ $\text{Reading (C1)} = 2.8 \text{ cm} + 0.07 \text{ cm} = 2.87 \text{ cm}$

Caliper C2

1. Least Count (LC2): For caliper C2, 10 VSD correspond to 11 MSD. This is a retrograde Vernier caliper. $10 \text{ VSD} = 11 \text{ MSD}$ $1 \text{ VSD} = \frac{11}{10} \text{ MSD} = 1.1 \text{ MSD}$ The least count (LC) for this type of caliper is given by: $LC_2 = |1 \text{ VSD} - 1 \text{ MSD}| = |1.1 \text{ MSD} - 1 \text{ MSD}| = 0.1 \text{ MSD}$ Substituting the value of 1 MSD: $LC_2 = 0.1 \times 0.1 \text{ cm} = 0.01 \text{ cm}$

2. Reading of C2: From the figure for C2:

• The zero of the Vernier scale is just past the 2.8 cm mark on the main scale. So, MSR = 2.8 cm.
• The 7th division of the Vernier scale coincides with a division on the main scale. So, VSC (n) = 7.

For a retrograde Vernier caliper, the reading can be calculated using the formula: $\text{Reading} = \text{MSR} + (N - n) \times LC$ where N is the total number of divisions on the Vernier scale (N=10) and n is the coinciding division (n=7).

Calculating the measured value for C2: $\text{Reading (C2)} = 2.8 \text{ cm} + (10 - 7) \times 0.01 \text{ cm}$ $\text{Reading (C2)} = 2.8 \text{ cm} + (3 \times 0.01 \text{ cm})$ $\text{Reading (C2)} = 2.8 \text{ cm} + 0.03 \text{ cm} = 2.83 \text{ cm}$

Note: An alternative formula for retrograde calipers is Reading = MSR' - n * LC, where MSR' is the main scale reading just ahead of the vernier zero. Here MSR' = 2.9 cm. This gives Reading = 2.9 - 7 * 0.01 = 2.83 cm, which confirms our result. There is a slight inconsistency in the provided diagram for C2, as a reading of 2.83 cm with n=7 should have the coincidence at the 3.6 cm mark, not the 3.5 cm mark as shown. However, applying the standard formula is the intended method.

Conclusion

The measured value by caliper C1 is 2.87 cm. The measured value by caliper C2 is 2.83 cm.

This corresponds to option B.