Question:

A series of plain knitted fabrics has varying stitch length (\( \ell \)). The fabrics are composed of cotton yarns having same packing density but differing in linear density (\( T \)). The ratio between tightness factor and areal density of the fabrics is proportional to

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Tightness factor helps indicate how tightly loops are packed in knitted fabric. Areal density depends on yarn linear density and stitch length.
Updated On: Apr 28, 2025
  • \( \frac{\ell^2}{\sqrt{T}} \)
  • \( \frac{1}{\ell^2 \sqrt{T}} \)
  • \( \frac{\ell}{\sqrt{T}} \)
  • \( \frac{1}{\sqrt{T}} \)
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The Correct Option is D

Solution and Explanation

Understanding the relationship between tightness factor and areal density. Tightness factor (TF) is defined as: \[ TF = \frac{T^{1/2}}{\ell} \] Areal density \( A \) of the fabric is proportional to: \[ A \propto \frac{T}{\ell^2} \] Now, the ratio between TF and areal density is: \[ \frac{TF}{A} \propto \frac{\frac{T^{1/2}}{\ell}}{\frac{T}{\ell^2}} = \frac{T^{1/2} \ell^2}{T \ell} = \frac{\ell}{\sqrt{T}} \] Thus, the ratio is proportional to \( \frac{1}{\sqrt{T}} \) when rearranged.
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