The main scale of a vernier callipers is calibrated in mm and 19 divisions of main scale are equal in length to 20 divisions of vernier scale. In measuring the diameter of a cylinder by this instrument, the main scale reads 35 divisions and 4th division of vernier scale coincides with a main scale division.

Find —

(i) least count and

(ii) radius of cylinder.

(i) As we know,

$\text{Least count (L.C.)} = \dfrac{\text{ Value of 1 main scale div (x)}}{\text{total no. of div on vernier (n)}}$

Given,

Value of one main scale division(x) = 1 mm

Total number of divisions on vernier callipers (n) = 20

Substituting the values in the formula above we get,

$\text{L.C.} = \dfrac{1}{\text{20}} \\[0.5em] \text{L.C.} = 0.05 \text{mm} \\[0.5em] \text{L.C.} = 0.005 \text{cm} \\[0.5em]$

Hence, least count of the vernier callipers is 0.005 cm.

(ii) As we know,

circular scale reading = L.C. x Coinciding division

Given,

Coinciding division = 4

L.C. = 0.005 cm

Substituting the values in the formula above we get,

$\text {circular scale reading} = \text {0.005 cm x 4} \\[0.5em] \Rightarrow \text {circular scale reading} = \text {0.02 cm} \\[0.5em]$

Hence, the circular scale reading = 0.02 cm. …… (1)

Main scale reading = 3.5 cm (as it reads 35th division on mm scale)

Hence, the main scale reading = 3.5 cm. …….. (2)

Substituting the values 1 and 2 in the formula for total reading we get,

$\text{Total reading} = \text{main scale reading } + \text{ vernier scale reading} \\[0.5em] \text{Diameter = Total reading} = \text{3.5 cm + 0.02 cm} \\[0.5em] \Rightarrow \text{Diameter} = \text{3.52 cm} \\[0.5em] \Rightarrow \text{Radius} = \dfrac{3.52}{2}\text{cm} \\[0.5em] \Rightarrow \text{Radius} = 1.76\text{cm} \\[0.5em]$

Hence, the radius of the cylinder = 1.76 cm.