Animated Solution for Physics - Physics and Measurement: The smallest division on the main scale of a Vernier calipers is $0.1\text{ cm}$. Ten divisions of the Vernier scale correspond to nine divisions of the main scale. The figure below on the left shows the reading of this calipers with no gap between its two jaws. The figure on the right shows the reading with a solid sphere held between the jaws. The correct diameter of the sphere is

A Tale of Two Readings

When dealing with precision instruments like the Vernier caliper, the devil is always in the details. This problem is a classic test of your ability to not just read a scale, but to critically analyze the instrument's inherent flaws before trusting its output. We are presented with two distinct scenarios: one to calibrate the instrument (finding the zero error) and one to take the actual measurement. Let's break this down step-by-step.

Calculating the Least Count

Before we can read any value, we must understand the resolution of our instrument. The fundamental constant of any Vernier caliper is its Least Count (LC). We are given that $10$ Vernier Scale Divisions (VSD) correspond to $9$ Main Scale Divisions (MSD).

Since one MSD is $0.1\text{ cm}$, we can easily find the length of a single Vernier division:

The Least Count is the difference between one main scale division and one Vernier scale division:

This tells us that our caliper can measure accurately down to $0.01\text{ cm}$.

Now, let's look at the first figure where the jaws are completely closed. In a perfect world, the zero of the Vernier scale would align perfectly with the zero of the main scale. However, in our case, the Vernier zero is slightly to the left of the main scale zero.

This is a classic negative zero error. It means the instrument has a "head start" in the negative direction; it reads a value less than zero when it should be exactly zero. To quantify this error, we scan the Vernier scale to find the division that perfectly aligns with any main scale mark. Looking closely, we see that the $6^{\text{th}}$ Vernier division coincides perfectly.

For a negative zero error, the formula is:

Keep this value safe; we will need it to correct our final reading.

Moving on to the second figure, a solid sphere is now held between the jaws. We read the main scale by looking at the mark immediately to the left of the Vernier zero. The Vernier zero has just crossed the $3.1\text{ cm}$ mark, so our Main Scale Reading (MSR) is $3.1\text{ cm}$.

Next, we find the coinciding division for this measurement. Scanning the Vernier scale, the very $1^{\text{st}}$ division perfectly aligns with a main scale mark. So, our Vernier Scale Reading (VSR) is $1$.

The measured value is calculated as:

We have our measured value, but remember, our instrument was flawed from the start. It was reading $0.04\text{ cm}$ less than it should have. To find the true diameter, we must subtract the zero error from our measured value.

And there we have it! By systematically calculating the least count, identifying the negative zero error, taking the raw measurement, and applying the correction, we arrive at the precise diameter of $3.15\text{ cm}$. Always remember: never trust an instrument blindly until you've checked its zero!