GATE (TF) Textile 2008 Question Paper Solution | GATE/2008/TF/46

If the specific volume of yarn is increased by 21% the percentage increase in yarn diameter would be

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Let, the initial specific volume of yarn=V1

Then , final specific volume of yarn(V2)=V1+V1 x 21%
V2=V1+0.21 V1
V2=1.21 V1

Let, initial dia of yarn=d1

Final dia of yarn=d2

Now, by formula-

$Dia of yarn=0.990 \times \sqrt \frac {Specific volume}{840 \times Ne}$

$d=0.990 \times \sqrt \frac {V}{840 \times Ne}$

i.e. Diameter of yarn is directly proportional to the sqaure root of the specific volume of yarn.

$d \alpha \sqrt V$

$\frac{d_1}{d_2}=\frac{\sqrt V_1}{\sqrt V_2}$

$\frac{d_1}{d_2}=\frac{\sqrt V_1}{\sqrt 1.21 \times V_1}$

$\frac{d_1}{d_2}=\frac{1}{\sqrt 1.21}$

$\frac{d_1}{d_2}=\frac{1}{1.1}$

Reverse the whole equations-

$\frac{d_2}{d_1}=1.1$

$\frac{d_2}{d_1}-1=1.1-1$

$\frac{d_2}{d_1}-1=0.1$

$(\frac{d_2}{d_1}-1)\times 100=0.1 \times 100$

$(\frac{d_2}{d_1}-1)\times 100=10$

i.e.,

The percentage increase in yarn diameter=10 (Ans)