JEE PYQ: Units & Measurements - Question ID b2ecc5d879fa (JEE Main 2019)

A screw gauge is a precision instrument used to measure small lengths (like wire diameters) by converting linear motion into rotational motion. Its working principle relies on two scales:

• Main Scale (Linear Scale): This is a straight scale with a known least count (here, $1 \text{ mm}$). It measures the coarse part of the length.
• Circular Scale (Rotational Scale): This is a rotating scale attached to the screw. It subdivides the main scale divisions into finer parts, allowing measurement of very small lengths.

The least count (LC) of the screw gauge is the smallest length that can be measured accurately. It is given by:

Here, the least count of the main scale is $1 \text{ mm}$, and we need to find the minimum number of divisions on the circular scale ($N$) such that the least count is $5 \mu \text{m}$ (since the wire diameter to be measured is $5 \mu \text{m}$).

We are given:

• Least count of main scale = $1 \text{ mm} = 1000 \mu \text{m}$.
• Desired least count of screw gauge = $5 \mu \text{m}$.
• Let $N$ = number of divisions on the circular scale.

Using the formula for least count:

Substitute the known values:

Solve for $N$:

Thus, the minimum number of divisions required on the circular scale is 200.

Students often make the following mistakes in this question:

1. Unit Confusion: Forgetting to convert the main scale least count ($1 \text{ mm}$) to micrometers ($1000 \mu \text{m}$) before dividing. This leads to incorrect calculations like $N = \frac{1}{5} = 0.2$, which is nonsensical.
2. Misinterpreting the Question: Some students think the question is asking for the number of divisions to measure $5 \mu \text{m}$ directly on the main scale, ignoring the role of the circular scale entirely.
3. Overcomplicating the Problem: Introducing unnecessary concepts like pitch (distance moved per rotation) or zero error, which are irrelevant here. The question is purely about the relationship between least count and circular scale divisions.

Exam Tip: Always ensure units are consistent before performing calculations. Here, converting $1 \text{ mm}$ to $1000 \mu \text{m}$ simplifies the problem. Also, remember that the least count of the screw gauge is determined by the ratio of the main scale least count to the number of circular scale divisions.