Question

A spun yarn made of circular polyester fibres of 10 µm diameter is cut normal to the yarn axis. If the helix angle of a fibre is \( 40^\circ \), then the area (µm\(^2\)) of that fibre projected on the cut-section of the yarn (rounded off to 1 decimal place) is:

Hint:

To calculate the projected area of a fibre in a yarn, use the circular cross-sectional area and adjust it with the cosine of the helix angle to account for the projection effect.

Answer 1

Correct Answer: 100

1. The projected shape is an ellipse where:
- Minor axis = fiber diameter = 10 µm
- Major axis = diameter / cos(angle) = 10 / cos(40°) ≈ 13.05 $µm$
2. Area of ellipse: \[ A = \pi \times \frac{13.05}{2} \times \frac{10}{2} = \pi \times 6.525 \times 5 \approx 102.5 { µm}^2 \] Answer: \boxed{102.5}