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Chapter 3: Problem 39

The distance moved by the screw of a screw gauge is \(2 \mathrm{~mm}\) in four rotations and there are 50 divisions on its cap. When nothing is put between its jaws, \(30^{\text {th }}\) division of circular scale coincides with reference line, with zero of circular scale lying above the reference line. When a plate is placed between the jaws, main scale reads 2 division and circular scale reads 20 division. Thickness of plate is : (1) \(1.5 \mathrm{~mm}\) (2) \(1.2 \mathrm{~mm}\) (3) \(1.4 \mathrm{~mm}\) (4) \(1.6 \mathrm{~mm}\)

Short Answer

Expert verified
The thickness of the plate is \(2.5 \text{ mm} \).

Step by step solution

01

- Calculate the Pitch

Pitch is defined as the distance moved by the screw per rotation. Given that the distance moved in 4 rotations is 2 mm, the pitch is calculated as follows: Pitch = \(\frac{2 \text{ mm}}{4 \text{ rotations}} = 0.5 \text{ mm per rotation}\).
02

- Calculate the Least Count

The Least Count (LC) of the screw gauge is the smallest measurement it can accurately measure. It is given by the formula: Least Count = \(\frac{\text{Pitch}}{\text{Number of divisions on the circular scale}}\). Given the pitch is 0.5 mm and there are 50 divisions on the circular scale, we get: Least Count = \(\frac{0.5 \text{ mm}}{50} = 0.01 \text{ mm}\).
03

- Determine the Zero Error

Zero error is noted when no object is placed between the jaws, and the position of the circular scale relative to the reference line is observed. Here, the \(30^{th}\) division coincides with the reference line and the zero of the circular scale is above the reference line. This indicates a negative zero error: Zero Error = \(-30 \times \text{Least Count} = -30 \times 0.01 = -0.3 \text{ mm}\).
04

- Main Scale Reading (MSR)

When the plate is placed between the jaws, the main scale reads 2 divisions. This is the main scale reading (MSR): \(\text{MSR} = 2 \text{ divisions} = 2 \text{ mm} \).
05

- Circular Scale Reading (CSR)

When the plate is placed between the jaws, the circular scale reads 20 divisions. This is the circular scale reading (CSR): \( \text{CSR} = 20 \times \text{Least Count} = 20 \times 0.01 = 0.2 \text{ mm} \).
06

- Calculate the Total Reading

The total reading (TR) is the sum of the Main Scale Reading (MSR) and the Circular Scale Reading (CSR): \(\text{TR} = \text{MSR} + \text{CSR} = 2 \text{ mm} + 0.2 \text{ mm} = 2.2 \text{ mm} \).
07

- Determine the Corrected Reading

The corrected reading is found by subtracting the zero error from the total reading: Corrected Reading = Total Reading - Zero Error \(= 2.2 \text{ mm} - (-0.3 \text{ mm}) = 2.2 \text{ mm} + 0.3 \text{ mm} = 2.5 \text{ mm} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

screw gauge
A screw gauge is a precision instrument used to measure small dimensions, like the thickness of a plate.
It consists of a main scale, a circular scale, and a rotating screw. The screw advances every time it completes a rotation, allowing precise measurements.
Screw gauges are often used in laboratories and workshops for accurate measurements.
pitch calculation
Pitch is the distance moved by the screw with each complete rotation. It is calculated by dividing the distance moved by the number of rotations.
For example, if the screw moves 2 mm in 4 rotations, the pitch is calculated as:
\(\text{Pitch} = \frac{2 \text{ mm}}{4 \text{ rotations}} = 0.5 \text{ mm per rotation}\)
Understanding pitch is crucial for accurate screw gauge measurements.
least count determination
The least count of a screw gauge is the smallest measurement it can accurately make. It is given by the formula:
\(\text{Least Count} = \frac{\text{Pitch}}{\text{Number of divisions on the circular scale}}\)
In our example, if the pitch is 0.5 mm and there are 50 divisions on the circular scale:
\(\text{Least Count} = \frac{0.5 \text{ mm}}{50} = 0.01 \text{ mm}\)
This small unit allows for high precision in measurements.
zero error
Zero error occurs when the scale shows a non-zero reading despite there being no object between the jaws. It can be positive or negative.
In our example, the 30th division coincides with the reference line, and zero is above this line, indicating a negative zero error of:
\(\text{Zero Error} = -30 \times \text{Least Count} = -30 \times 0.01 = -0.3 \text{ mm}\)
Zero error must be corrected in the final measurement.
main scale reading
Main Scale Reading (MSR) is taken from the main scale of the screw gauge. It is the integer part of the measurement.
When a plate is placed between the jaws, the MSR in this example is 2 divisions, equivalent to 2 mm:
\(\text{MSR} = 2 \text{ mm}\)
This value contributes significantly to the total measurement.
circular scale reading
Circular Scale Reading (CSR) represents finer measurements and is read from the circular scale.
Each division represents a fraction of the pitch. In this case, if the circular scale reading is 20 divisions:
\(\text{CSR} = 20 \times \text{Least Count} = 20 \times 0.01 = 0.2 \text{ mm}\)
Combining CSR with MSR gives higher measurement precision.
corrected reading calculation
The corrected reading combines the main scale and circular scale readings, then adjusts for zero error.
First, calculate the total reading:
\(\text{Total Reading} = \text{MSR} + \text{CSR} = 2 \text{ mm} + 0.2 \text{ mm} = 2.2 \text{ mm}\)
Then, adjust for zero error:
\(\text{Corrected Reading} = \text{Total Reading} - \text{Zero Error} = 2.2 \text{ mm} - (-0.3 \text{ mm}) = 2.2 \text{ mm} + 0.3 \text{ mm} = 2.5 \text{ mm}\)
Hence, the accurate thickness of the plate is 2.5 mm.

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