Question

In a Vernier caliper, \(N+1\)  divisions of vernier scale coincide with  \(N\) divisions of main scale. If 1 MSD represents 0.1 mm, the vernier constant (in cm) is:

• A. \(\frac{1}{10N}\)
• B. \(\frac{1}{100(N+1)}\)
• C. \(100N\)
• D. \(10N+1\)

Answer 1

The Correct Option is B

To find the vernier constant of a Vernier caliper, we first consider the definition of the vernier constant: it is the smallest measurement that can be calculated using a Vernier caliper and is equal to the least count. In the given problem, it is stated that \(N+1\) divisions of the Vernier scale coincide with \(N\) divisions of the main scale.

The given information can be expressed as:

1 Main Scale Division (MSD) = 0.1 mm

Therefore, \(N\) divisions on the main scale equal:

\(N \times 0.1 \, \text{mm} = 0.1N \, \text{mm}\)

The Vernier scale is set such that \(N+1\) Vernier scale divisions coincide with \(N\) main scale divisions, giving each Vernier scale division:

\( \frac{0.1N}{N+1} \, \text{mm}\)

The Vernier constant (VC) is the difference between one main scale division and one Vernier scale division, which is:

\[ VC = \text{MSD} - \text{VSD} = 0.1 \, \text{mm} - \frac{0.1N}{N+1} \, \text{mm} \]

\[ VC = \frac{0.1(N+1) - 0.1N}{N+1} \]

\[ VC = \frac{0.1}{N+1} \, \text{mm} \]

Since we need the vernier constant in centimeters, convert mm to cm (1 mm = 0.1 cm):

\[ VC = \frac{0.1 \times 0.1}{N+1} \, \text{cm} \]

\[ VC = \frac{0.01}{N+1} \, \text{cm} \]

Expressed as a fraction, this becomes:

\[ \frac{1}{100(N+1)} \, \text{cm} \]

Thus, the vernier constant in centimeters is \(\frac{1}{100(N+1)}\).

Answer 2

Calculating the Vernier Constant (V.C.)

Step 1: Formula for the Vernier Constant (V.C.) 

The Vernier constant is given by:

V.C. = Value of 1 MSD − Value of 1 VSD

Step 2: Relate MSD and VSD values

If (N + 1) VSDs coincide with N MSDs, then:

Value of 1 VSD = Value of 1 MSD × (N / (N + 1))

Step 3: Substitute Values and Simplify

Now substitute the values:

V.C. = 0.1mm − 0.1mm × (N / (N + 1))

V.C. = 0.1 / (N + 1) mm

Quick Tip

To convert to centimeters, use:

V.C. = 1 / 100(N + 1) cm