Understanding Vernier Caliper Least Count

A vernier caliper is a precision instrument used for measuring lengths, depths, and external or internal diameters. Its accuracy is determined by its least count, which is the smallest measurement that can be made reliably with the instrument.

The least count of a vernier caliper depends on the main scale least count and the relationship between the divisions on the main scale and the vernier scale.

Calculating the Least Count of the Vernier Caliper

The problem provides the following information:

• The main scale reads in millimetres.
• The least count of the main scale (MSLC) is 0.1 mm. This means the smallest division on the main scale is 0.1 mm.
• Ten divisions on the vernier scale (N) correspond to nine divisions of the main scale. So, \(N = 10\).

Let's denote the value of one main scale division as MSD and the value of one vernier scale division as VSD.

From the given information:

• \(MSD = \text{MSLC} = 0.1 \, \text{mm}\)
• 10 VSD = 9 MSD

We can find the value of one vernier scale division (VSD):

\(10 \times \text{VSD} = 9 \times \text{MSD}\)

\(\text{VSD} = \frac{9}{10} \times \text{MSD}\)

\(\text{VSD} = \frac{9}{10} \times 0.1 \, \text{mm}\)

\(\text{VSD} = 0.9 \times 0.1 \, \text{mm}\)

\(\text{VSD} = 0.09 \, \text{mm}\)

The least count (LC) of a vernier caliper is defined as the difference between the value of one main scale division and the value of one vernier scale division.

The formula for the least count is:

\(\text{LC} = \text{MSLC} - \text{VSD}\)

Alternatively, using the relationship between the scales:

If N vernier scale divisions match (N-1) main scale divisions, then:

\(\text{N} \times \text{VSD} = (N-1) \times \text{MSD}\)

\(\text{VSD} = \frac{N-1}{N} \times \text{MSD}\)

Since MSLC = MSD,

\(\text{LC} = \text{MSLC} - \text{VSD} = \text{MSLC} - \frac{N-1}{N} \times \text{MSLC}\)

\(\text{LC} = \text{MSLC} \times \left(1 - \frac{N-1}{N}\right) = \text{MSLC} \times \left(\frac{N - (N-1)}{N}\right) = \text{MSLC} \times \frac{1}{N}\)

So, the least count can also be calculated as:

\(\text{LC} = \frac{\text{MSLC}}{N}\)

Using the given values:

• MSLC = 0.1 mm
• \(N = 10\)

\(\text{LC} = \frac{0.1 \, \text{mm}}{10}\)

\(\text{LC} = 0.01 \, \text{mm}\)

This result matches the difference method as well:

\(\text{LC} = \text{MSLC} - \text{VSD} = 0.1 \, \text{mm} - 0.09 \, \text{mm} = 0.01 \, \text{mm}\)

Therefore, the least count of the vernier caliper is 0.01 mm.

Revision Table: Vernier Caliper Key Concepts

Additional Information: Using a Vernier Caliper

Using a vernier caliper involves a few steps:

1. Zero Correction: First, check for zero error by closing the jaws completely. If the vernier zero coincides with the main scale zero, there is no zero error. If not, note the zero error. Positive zero error occurs when the vernier zero is ahead of the main scale zero; negative zero error when it is behind.
2. Taking a Reading: Place the object between the jaws. Note the main scale reading just before the zero mark of the vernier scale. This is the main scale reading (MSR).
3. Finding Vernier Coincidence: Look for the vernier scale division that exactly coincides with a main scale division. Note the number of this vernier division (VCD).
4. Calculating Fractional Part: The fractional part of the reading is \(VCD \times LC\).
5. Calculating Observed Reading: The observed reading is \(MSR + (VCD \times LC)\).
6. Applying Zero Correction: The final corrected reading is Observed Reading - Zero Error (with its sign).

Understanding the least count is crucial for correctly interpreting the vernier caliper readings.