Question

Keeping the speed of all other components of a carding machine unchanged, the angular velocity of a cylinder with damaged wire points in an area of 2 cm \( \times \) 2 cm is doubled. Wavelength of the periodic fault in the card sliver would be:

• A. Same
• B. Halved
• C. Doubled
• D. Tripled

Hint:

For systems involving angular velocity and wavelength, remember that an increase in angular velocity leads to a decrease in wavelength, as they are inversely proportional.

Answer 1

The Correct Option is B

In this problem, we are dealing with the relationship between angular velocity and wavelength of periodic faults. The key here is to understand how angular velocity affects the wavelength. Step 1: Understanding the relationship.
The angular velocity (\(\omega\)) of a cylinder determines how fast the damaged wire points move around the cylinder. The periodic fault refers to the repetition of these damaged wire points along the sliver produced by the machine. Wavelength refers to the distance between two consecutive faults. Step 2: Doubling angular velocity.
When the angular velocity of the cylinder is doubled, the rate at which the damaged wire points move also doubles. As the angular velocity increases, the time between two consecutive faults decreases, leading to a shorter wavelength. Step 3: Wavelength changes.
Since wavelength is inversely proportional to the angular velocity, doubling the angular velocity will halve the wavelength. This is because the frequency of faults increases with the angular velocity, and thus the distance between consecutive faults (wavelength) decreases. Step 4: Conclusion.
Thus, the wavelength of the periodic fault in the card sliver would be halved.