Question

Screw gauge A has a pitch of \(1 \mathrm{~mm}\) and 50 divis1on on its circular scale screw gauge \(B\) has a pitch of \(0.5 \mathrm{~mm}\) and 100 divisions on its circular scale. If \((\mathrm{L} \times \mathrm{C})\) is least count, then which posibility is true ? What is the parilrility? (A) \(2(\mathrm{~L} \times \mathrm{C})_{\mathrm{A}}=(\mathrm{L} \times \mathrm{C})_{\mathrm{B}}\) (B) \((\mathrm{L} \times \mathrm{C})_{\mathrm{A}}=4(\mathrm{~L} \times \mathrm{C})_{\mathrm{B}}\) (C) \((\mathrm{L} \times \mathrm{C})_{\mathrm{A}}=(\mathrm{L} \times \mathrm{C})_{\mathrm{B}}\) (D) \((\mathrm{L} \times \mathrm{C})_{\mathrm{A}}=2(\mathrm{~L} \times \mathrm{C})_{\mathrm{B}}\)

Short answer

The correct answer is (B): \((\mathrm{L} \times \mathrm{C})_{\mathrm{A}} = 4(\mathrm{L} \times \mathrm{C})_{\mathrm{B}}\).

Step-by-step solution

Understanding the least count

In the context of a screw gauge, the least count is the smallest length that can be accurately measured using the instrument. Mathematically, it is the product of the screw's pitch (length moved per revolution) and the divisions on the circular scale.

Calculate the least count for screw gauge A

For screw gauge A, we are given the pitch of \(1 \mathrm{~mm}\) and 50 divisions on its circular scale. The least count of A, denoted as \((\mathrm{L} \times \mathrm{C})_{\mathrm{A}}\), can be calculated as follows: \[(\mathrm{L} \times \mathrm{C})_{\mathrm{A}} = \frac{\text{pitch of A}}{\text{divisions on circular scale of A}} = \frac{1\ \mathrm{mm}}{50} = 0.02\ \mathrm{mm} \]

Calculate the least count for screw gauge B

For screw gauge B, we are given the pitch of \(0.5 \mathrm{~mm}\) and 100 divisions on its circular scale. The least count of B, denoted as \((\mathrm{L} \times \mathrm{C})_{\mathrm{B}}\), can be calculated as follows: \[(\mathrm{L} \times \mathrm{C})_{\mathrm{B}} = \frac{\text{pitch of B}}{\text{divisions on circular scale of B}} = \frac{0.5\ \mathrm{mm}}{100} = 0.005\ \mathrm{mm} \]

Compare the least counts of A and B

Now we will compare least count of A and B to find which option among A, B, C, or D is true: (A) Check if \(2(\mathrm{L} \times \mathrm{C})_{\mathrm{A}} = (\mathrm{L} \times \mathrm{C})_{\mathrm{B}}\): \[2(0.02\ \mathrm{mm}) = 0.04\ \mathrm{mm} \ne 0.005\ \mathrm{mm}\] (B) Check if \((\mathrm{L} \times \mathrm{C})_{\mathrm{A}} = 4(\mathrm{L} \times \mathrm{C})_{\mathrm{B}}\): \[0.02\ \mathrm{mm} = 4(0.005\ \mathrm{mm}) = 0.02\ \mathrm{mm}\] Since option (B) is satisfied, the correct answer is (B): \((\mathrm{L} \times \mathrm{C})_{\mathrm{A}} = 4(\mathrm{L} \times \mathrm{C})_{\mathrm{B}}\)

Key concepts

Screw Gauge

A screw gauge, also known as a micrometer, is a precision instrument used to measure small dimensions with high accuracy. It is commonly used in mechanical engineering to measure the diameter of wires, thickness of sheets, or any small object that fits in its jaws. The main components of a screw gauge include:

• The spindle, which moves horizontally as the screw rotates.
• A frame that holds the entire setup together.
• The screw, which is the core component, allowing for precise movement.
  
• The thimble, which is the rotating part of the screw gauge.
• The circular scale, marked on the rotating thimble, used alongside the linear scale to take measurements.

When using a screw gauge for measurement, the object to be measured is placed between the spindle and the anvil, and the screw is rotated until the object is gently secured. The position of the thimble on the circular scale allows the user to read the measurement.

Pitch

In the context of a screw gauge, the pitch is a crucial element that determines the functionality and accuracy of measurements. The pitch is defined as the linear distance traveled by the screw per complete turn of the thimble. For example, if the pitch of a screw gauge is given as 1 mm, it means that for each full rotation of the thimble, the spindle moves 1 mm horizontally. This small movement allows for precise adjustments and measurements. Pitch is an important factor in determining the least count, enabling users to calculate how small a measurement can be reliably taken by the screw gauge. By understanding the relationship between the pitch and the divisions on the circular scale, users can use the screw gauge effectively for high-precision measurements.

Circular Scale Divisions

The divisions on the circular scale of a screw gauge are essential for accurately determining the least count, hence affecting the precision of measurements conducted using the instrument. These divisions are the marks present on the rotating thimble, serving as reference points for the user. For instance, if a screw gauge has 50 divisions on its circular scale, this means that each division represents a fraction of the total movement resulting from a one-turn rotation. The more divisions present, the lower the least count, allowing for more precise measurements. To calculate the least count using circular scale divisions, one needs to divide the pitch of the screw by the number of these divisions. Meanwhile, it is important to align the main scale's reading concerning these divisions to obtain the result.

Precision Measurement

Precision measurement is the cornerstone of using a screw gauge effectively. The aim is to obtain highly accurate dimensions, often in millimeters, by utilizing the least count principle. The least count determines the smallest increment in measurement that the screw gauge can read. It is calculated by dividing the pitch by the number of circular divisions, providing a direct indication of the gauge's precision. For example, if a screw gauge has a least count of 0.02 mm, it means the smallest measurement it can reliably take is 0.02 mm. Therefore, the precision is high enough to measure even the tiniest differences in size meticulously. Precision measurement ensures that errors are minimized, enhancing the reliability of results when dealing with components requiring exact specifications and dimensions.