Question 9: In a vernier calipers, ( \mathrm{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 :
(1) \frac{1}{10 \mathrm{~N}}
(2) \frac{1}{100(\mathrm{~N}+1)}
(3) 100 N
(4) 10(\mathrm{~N}+1)
Answer: Option (2)
Explanation:
The vernier constant (least count) of a vernier calipers is defined as the difference between one main scale division and one vernier scale division.
Given that (N+1) vernier scale divisions coincide with N main scale divisions.
So,
1\ \mathrm{VSD}=\frac{N}{N+1}\ \mathrm{MSD}.
The value of one main scale division is
1\ \mathrm{MSD}=0.1\ \mathrm{mm}=0.01\ \mathrm{cm}.
The vernier constant is
\mathrm{VC}=1\ \mathrm{MSD}-1\ \mathrm{VSD}.
Substituting the values,
\mathrm{VC}=0.01-\frac{N}{N+1}\times0.01.
\mathrm{VC}=0.01\left(1-\frac{N}{N+1}\right).
\mathrm{VC}=0.01\left(\frac{1}{N+1}\right).
\mathrm{VC}=\frac{0.01}{N+1}\ \mathrm{cm}.
\mathrm{VC}=\frac{1}{100(N+1)}\ \mathrm{cm}.
Hence, the correct answer is option (2).