Question

The pitch of a spherometer is 1 mm and there are 100 divisions on its disc. It reads 3 divisions on the circular scale above zero when it is placed on a plane glass plate. When it rests on a convex surface, it reads 2 mm and 63 divisions on a circular scale. If the distance between its outer legs is 4 cm, the radius R of curvature of the convex surface is:
A.) 4 cm
B.) 2.6 cm
C.) 10.4 cm
D.) 6.3 cm

Answer 1

Hint: To solve the question, we should be aware of a few formulas. The formulas include the concepts of least count, zero count and then finally the spherometer formula. At first, we should be placing the given quantities as per the order. Then we need to find the least count and zero count. Finally, the answer is obtained by using the spherometer formula.

Step by step answer:
At first we are required to put the value of the quantities that are mentioned as per the question.

The pitch of the spherometer is represented as p. So,

$\text{p = 1 mm}$
$\text{Number of division N = 100}$

We know that Least Count is the ratio of the smallest reading of the main scale and the total number of divisions of the spherometer. So,

$\text{Least Count C =}\dfrac{\text{P}}{\text{N}}\text{= 0}\text{.01 mm}$

Now we have to calculate the zero count. Zero count is represented by e. It is the multiplication between the number of divisions (which is 3 in case of spherometer) and the least count.

$\text{Zero count = e = +nC = +3}\left( \text{0}\text{.01} \right)\text{ = +0}\text{.03 mm}$

Let us consider that H is the height of the central screw above the plane. So,

$\text{H = 2 +nC = 2 + 63}\left( \text{0}\text{.01} \right)\text{ = 2}\text{.63 mm}$

We know that, the height of the central leg above the feet of the outer legs = h
Now,
 $\begin{array}{*{35}{l}}
   \text{h = H }\text{ e = 2}\text{.63 }\text{ 0}\text{.03 = 2}\text{.60 mm = 0}\text{.26 cm} \\
   \text{a = 4 cm} \\
\end{array}$

Using spherometer formula
 $\text{R =}\dfrac{{{\text{a}}^{\text{2}}}}{\text{6h}}\text{+}\dfrac{\text{h}}{\text{2}}$

Here, R represents the spherometer constant and a is the distance between the outer legs.

The value of h is obtained as 0.26 and value of a is given in the question as 4.
Substituting in spherometer formula

$\text{= }\!\!~\!\!\text{ }\dfrac{{{\text{(4)}}^{\text{2}}}}{\text{6(0}\text{.26)}}\text{+ }\!\!~\!\!\text{ }\dfrac{\text{0}\text{.26}}{\text{2}}\text{=10}\text{.4 cm}$

Therefore, the correct answer is 10.4 cm. So, the correct option is Option C.

Note: By least count, we mean the smallest value that can be measured by using the measurement that we are talking about.
The instrument that is mentioned in the question is a spherometer. The main purpose of the spherometer, is to measure the radius of curvature of a sphere or of a curved surface, precisely.