LEVELJEE Main
Visualized Solution
The Sigma Insight: Vernier Calipers and Screw Gauge
Analyzing the Setup
Imagine you are holding a precision instrument, the screw gauge, designed to measure the microscopic thickness of a wire. Before we even place the wire between its jaws, we need to understand the instrument's unique characteristics.
The problem gives us a crucial piece of information: two full turns of the circular scale advance the screw by on the main scale. We are also told that the circular scale is divided into equal divisions.
Furthermore, the instrument isn't perfect. When the jaws are completely closed, it doesn't read zero. Instead, it has a negative zero error of . This means the instrument naturally reads less than the actual value, and we will need to compensate for this later.
Finally, when measuring the wire, the main scale shows , and the division of the circular scale aligns perfectly with the reference line.
The Master Equation
Pitch and Least Count
To make sense of these readings, we first need to determine the Pitch of the screw gauge. The pitch is the linear distance the screw travels in one complete rotation.
Since two turns cover , we can easily find the pitch by dividing the total distance by the number of turns:
This tells us that every single full rotation of the circular scale moves the jaws by exactly .
Next, we calculate the Least Count (LC), which is the absolute smallest measurement the instrument can reliably make. It is the distance moved when the circular scale is rotated by just one single division.
We find the least count by dividing the pitch by the total number of divisions on the circular scale:
This means every tiny tick mark on the circular scale represents a microscopic !
Final Calculation
The True Reading
Now we have all the tools to calculate the actual diameter of the wire. The master formula for the true reading of a screw gauge is:
Let's break down our specific values. The Main Scale Reading (MSR) is . The Circular Scale Reading (CSR) is , which we must multiply by our least count of . And crucially, we must subtract our negative zero error of .
Substituting these values into our equation gives:
Let's perform the arithmetic carefully. Multiplying the circular scale reading gives . Subtracting a negative number is mathematically identical to adding a positive number, so the zero error correction becomes .
Adding these components together yields our final, highly precise diameter:
And there we have it! By systematically breaking down the instrument's properties and carefully handling the sign of the zero error, we've arrived at the exact thickness of the wire.
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