In a screw gauge, 5th division of the circular scale coincides with the reference line when the ratc

Q: Why do we subtract the zero error?

Think of a weighing scale that shows 2 kg when empty. If you step on it and it shows 62 kg, your actual weight is 62 - 2 = 60 kg. We always subtract the initial 'false' reading (zero error) from the final observed reading.

Q: How do I know if the zero error is positive or negative?

If the zero of the circular scale is below the reference line (e.g., 5th division coincides), it's positive. If the zero is above the reference line (e.g., 45th division coincides), it's negative.

Q: What does 'pitch' actually mean physically?

Pitch is the linear distance the screw moves forward or backward in one complete 360-degree rotation. It's the distance between two consecutive threads on the screw.

Q: Why is the least count formula Pitch / Total Divisions?

If one full rotation (50 divisions) moves the screw by 0.5 mm, then rotating it by just 1 division moves it by 0.5 mm divided by 50. This unitary method gives the smallest measurable distance.

Q: Can the main scale reading be something like 5.5 mm?

Yes! If the pitch is 0.5 mm, the main scale will have markings every 0.5 mm. If the thimble uncovers the 5.5 mm mark, your MSR would be 5.5 mm.

Question

In a screw gauge, 5th division of the circular scale coincides with the reference line when the ratchet is closed. There are 50 divisions on the circular scale, and the main scale moves by 0.5 mm on a complete rotation. For a particular observation the reading on the main scale is 5 mm and the 20th division of the circular scale coincides with reference line. Calculate the true reading.

Options:

(A) 5.00 mm

(B) 5.25 mm

(C) 5.15 mm

(D) 5.20 mm

Sigma Percentile · Accepted Answer

The screw gauge is a beautiful piece of engineering that allows us to measure dimensions down to a fraction of a millimeter. But with great precision comes the need for careful calibration. In this problem, we are going to walk through the complete anatomy of a screw gauge measurement, from finding its least count to correcting its inherent zero error.

### Understanding the Instrument's Precision

Before we can measure anything, we need to understand the limits of our instrument. The problem states that the main scale moves by $0.5 	ext{ mm}$ for every complete rotation of the circular scale. This distance is known as the **pitch** of the screw gauge.

The circular scale itself is divided into $50$ equal divisions. The **Least Count (LC)**—the smallest value the instrument can measure—is found by dividing the pitch by the total number of circular divisions.

$$ 	ext{LC} = \frac{	ext{Pitch}}{	ext{Total Divisions}} = \frac{0.5 	ext{ mm}}{50} = 0.01 	ext{ mm} $$

This tells us that every single step on the circular scale corresponds to a forward or backward movement of exactly $0.01 	ext{ mm}$.

### The Trap of the Zero Error

In an ideal world, when the jaws of the screw gauge are fully closed (using the ratchet to ensure uniform pressure), the zero mark of the circular scale should align perfectly with the reference line of the main scale. However, our instrument has a slight imperfection.

When closed, the $5^	ext{th}$ division of the circular scale coincides with the reference line. Because the zero mark has already crossed the reference line, the instrument is reading a value even when nothing is between the jaws! This is a **positive zero error**.

To calculate the exact magnitude of this error, we multiply the coinciding division by the least count:

$$ 	ext{Zero Error} = + (5 	imes 0.01 	ext{ mm}) = +0.05 	ext{ mm} $$

Keep this value safe. Since the instrument is over-reading by $0.05 	ext{ mm}$, we will need to subtract this from our final observation to get the true measurement.

### Making the Final Measurement

Now, we place our object between the jaws and take the reading. The main scale clearly shows $5 	ext{ mm}$. This is our **Main Scale Reading (MSR)**.

Next, we look at the circular scale to get the fractional part of the measurement. The $20^	ext{th}$ division perfectly coincides with the reference line. This gives us our **Circular Scale Reading (CSR)**.

The formula for the true reading combines all these elements:

$$ 	ext{True Reading} = 	ext{MSR} + (	ext{CSR} 	imes 	ext{LC}) - 	ext{Zero Error} $$

Let's carefully substitute our values into the master equation. Watch out for the minus sign!

$$ 	ext{True Reading} = 5 	ext{ mm} + (20 	imes 0.01 	ext{ mm}) - (+0.05 	ext{ mm}) $$

$$ 	ext{True Reading} = 5 	ext{ mm} + 0.20 	ext{ mm} - 0.05 	ext{ mm} $$

$$ 	ext{True Reading} = 5.15 	ext{ mm} $$

And there we have it! By systematically breaking down the instrument's parameters, accounting for its initial calibration error, and carefully combining our readings, we arrive at the precise true reading of $5.15 	ext{ mm}$.

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Visualized Solution

$Screw Gauge Parameters$

$Least Count Formula$

$Calculating Least Count$

$Zero Error Analysis$

$Calculating Zero Error$

$Observation Setup$

$True Reading Formula$

$Final Calculation$

$The Way Forward$

The Sigma Insight: Vernier Calipers and Screw Gauge

Understanding the Instrument's Precision

The Trap of the Zero Error

Making the Final Measurement

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Visualized Solution

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Screw Gauge Parameters

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Least Count Formula

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Calculating Least Count

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Zero Error Analysis

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Calculating Zero Error

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Observation Setup

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } True Reading Formula

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Final Calculation

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The Way Forward