Animated Solution for Physics - Physics and Measurement: Using screw gauge of pitch $0.1\text{ cm}$ and $50$ divisions on its circular scale, the thickness of an object is measured. It should correctly be recorded as

Imagine you are holding a precision instrument like a screw gauge. It consists of two primary scales: a stationary main scale and a rotating circular scale. When you rotate the circular scale by exactly one full turn, it advances linearly along the main scale. This linear distance covered in one complete rotation is known as the pitch of the screw gauge.

In our problem, the pitch is given as $0.1\text{ cm}$. This means every time you turn the circular scale $360^{\circ }$, the jaws of the screw gauge open or close by exactly $0.1\text{ cm}$.

The true power of a screw gauge lies in its ability to measure tiny fractions of the pitch. This is achieved by dividing the circular scale into multiple equal parts. The smallest measurement the instrument can accurately record is called its Least Count (LC).

The formula for the least count is beautifully simple:

Let's substitute the values provided in the question. We have a pitch of $0.1\text{ cm}$ and $50$ divisions on the circular scale.

This result, $0.002\text{ cm}$, is the absolute minimum distance the screw gauge can measure.

Here is where many students make a silly mistake. A measuring instrument is like a staircase; you can only stand on a specific step, never floating between two steps. Therefore, any valid reading taken by this screw gauge must be an exact integer multiple of its least count.

Mathematically, a valid reading $R$ can be expressed as: where $n$ is a whole number (integer).

To find the correct measurement among the given options, we simply divide each option by our least count ($0.002\text{ cm}$). The correct option will yield a perfect integer.

Let's test them: - Option (a): $\frac{2.121}{0.002}=1060.5$ (Not an integer) - Option (b): $\frac{2.124}{0.002}=1062$ (Perfect Integer!) - Option (c): $\frac{2.125}{0.002}=1062.5$ (Not an integer) - Option (d): $\frac{2.123}{0.002}=1061.5$ (Not an integer)

Since $2.124\text{ cm}$ is the only value that perfectly divides by the least count without leaving a remainder, it is the only physically possible reading for this specific screw gauge.