There are two vernier calipers both of which have 1 cm divided into 10 equal divisions on the main s

Q: Why can't I use the standard formula $R = \text{MSR} + n \times \text{LC}$ for $C_2$?

The standard formula assumes that the vernier divisions are smaller than the main scale divisions (a direct vernier). In $C_2$, the vernier divisions are larger (a retrograde vernier), so the standard formula breaks down. The universal geometric method always works.

Q: How do I identify which mark is coinciding from the diagram?

Look for the vernier tick mark that forms a perfectly straight, continuous vertical line with a main scale tick mark above it. In both diagrams, this happens at the 7th vernier division.

Q: What is a retrograde vernier caliper?

It's a vernier caliper where the vernier scale divisions are slightly larger than the main scale divisions. The vernier scale is read in the opposite direction to the main scale, though in this specific problem, it's numbered normally, making the geometric subtraction method the safest approach.

Q: How did we determine the coinciding main scale mark for $C_1$ is 3.5?

By counting the main scale divisions from 2.0. The coinciding line is 5 divisions after the 3.0 mark, which corresponds to $3.0 + 5 \times 0.1 = 3.5 \text{ cm}$.

Q: Is the least count the same for both calipers?

Yes! For $C_1$, $\text{LC} = 1 \text{ MSD} - 1 \text{ VSD} = 0.1 - 0.09 = 0.01 \text{ cm}$. For $C_2$, $\text{LC} = 1 \text{ VSD} - 1 \text{ MSD} = 0.11 - 0.1 = 0.01 \text{ cm}$. However, the way this least count is applied differs due to the direct vs. retrograde nature.

Question

There are two vernier calipers both of which have 1 cm divided into 10 equal divisions on the main scale. The vernier scale of one of the calipers ($C_1$) has 10 equal divisions that correspond to 9 main scale divisions. The vernier scale of the other caliper ($C_2$) has 10 equal divisions that correspond to 11 main scale divisions. The readings of the two calipers are shown in the figure. The measured values (in cm) by calipers $C_1$ and $C_2$ respectively, are

Options:

(A) 2.87 and 2.87

(B) 2.87 and 2.83

(C) 2.85 and 2.82

(D) 2.87 and 2.86

Sigma Percentile · Accepted Answer

### The Vernier Challenge

Imagine you are handed two different vernier calipers. At first glance, they look identical. Both have a main scale where $1 	ext{ cm}$ is divided into $10$ equal parts. This means the smallest division on the main scale, the Main Scale Division (MSD), is exactly $0.1 	ext{ cm}$.

However, the trick lies in their vernier scales. Caliper $C_1$ is a standard vernier, where $10$ vernier divisions fit into the space of $9$ main scale divisions. Caliper $C_2$, on the other hand, is a **retrograde vernier**, where $10$ vernier divisions stretch across $11$ main scale divisions. Our mission is to accurately read both calipers from the given diagrams.

### The Universal Reading Formula

Most students memorize the standard formula: $R = 	ext{MSR} + n 	imes 	ext{LC}$. While this works perfectly for standard calipers like $C_1$, it completely breaks down for retrograde calipers like $C_2$. Why? Because in a retrograde caliper, the vernier divisions are *larger* than the main scale divisions, meaning the scale is read backwards relative to the main scale.

To avoid this trap, we use a **universal geometric formula** that relies purely on the physical distance between marks. The reading $R$ is simply the position of the vernier's zero mark. If we find a vernier mark $n$ that perfectly coincides with a main scale mark, we can trace our way back to the zero mark.

$$R = 	ext{MSR}_{	ext{coinciding}} - (n 	imes 	ext{VSD})$$

This formula is foolproof. It doesn't care if the vernier is standard or retrograde; it only cares about the physical geometry of the lines.

### Analyzing Caliper $C_1$

Let's apply our universal formula to $C_1$. First, we need the length of one Vernier Scale Division (VSD). We are given that $10 	ext{ VSD} = 9 	ext{ MSD}$.

$$1 	ext{ VSD} = \frac{9}{10} 	imes 0.1 	ext{ cm} = 0.09 	ext{ cm}$$

Now, we look at the diagram for $C_1$. We scan the vernier scale to find the mark that perfectly aligns with a main scale mark above it. We can see that the $7^{	ext{th}}$ vernier division aligns exactly with the $3.5 	ext{ cm}$ mark on the main scale.

Using our formula:

$$R_1 = 3.5 - (7 	imes 0.09)$$

$$R_1 = 3.5 - 0.63 = 2.87 	ext{ cm}$$

### Analyzing Caliper $C_2$ (The Retrograde Vernier)

Now for the tricky one, $C_2$. Here, $10 	ext{ VSD} = 11 	ext{ MSD}$.

$$1 	ext{ VSD} = \frac{11}{10} 	imes 0.1 	ext{ cm} = 0.11 	ext{ cm}$$

Notice that $1 	ext{ VSD}$ is greater than $1 	ext{ MSD}$. Scanning the diagram for $C_2$, we again find that the $7^{	ext{th}}$ vernier division is the one that perfectly coincides with a main scale mark. This time, it aligns with the $3.6 	ext{ cm}$ mark.

Applying the exact same universal formula:

$$R_2 = 3.6 - (7 	imes 0.11)$$

$$R_2 = 3.6 - 0.77 = 2.83 	ext{ cm}$$

### The Final Verdict

By relying on the fundamental geometry of the scales rather than memorized formulas, we effortlessly bypassed the retrograde vernier trap. The measured values are $2.87 	ext{ cm}$ for $C_1$ and $2.83 	ext{ cm}$ for $C_2$. This perfectly matches option (b).

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Visualized Solution

$1 MSD$

$Universal Formula$

$VSD for C_{1}$

$Coinciding Mark C_{1}$

$Reading R_{1}$

$VSD for C_{2}$

$Coinciding Mark C_{2}$

$Reading R_{2}$

$Final Answer$

The Sigma Insight: Vernier Calipers and Screw Gauge

The Vernier Challenge

The Universal Reading Formula

Analyzing Caliper $C_{1}$

Analyzing Caliper $C_{2}$ (The Retrograde Vernier)

The Final Verdict

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Figure 1 shows the configuration of main scale and Vernier scale before measurement. Fig. 2 shows the configuration corresponding to the measurement of diameter $D$ of a tube. The measured value of $D$ is:

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Visualized Solution

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 1 MSD

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Universal Formula

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } VSD for C1​

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Coinciding Mark C1​

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Reading R1​

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } VSD for C2​

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Coinciding Mark C2​

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Reading R2​

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Final Answer