GATE (TF) Textile 2018 Question Paper Solution | GATE/2018/TF/51

Question 51 (Textile Technology & Fibre Science)

In a vibroscope, a cotton fibre of 25 mm length is clamped at one end, led over a knife-edge support, tensioned by a weight of 350 mg, and is inclined to vibrate at a fundamental resonant frequency of 2.6 kHz. The linear density of the fibre (in tex), accurate to two decimal places, is __0.19 to 0.21__.

[Show Answer]

Write Here

Given in the question-

Fibre length(l)=25mm

Fibre weight(W)=350mg

Resonant frequency(f)=2.6kHz

Linear density of fibre(in tex)=?

Formula:

$f=\frac{1}{2l}\times\sqrt\frac{Tension}{Linear density}$

$2.6\times10^3=\frac{1}{2\times25\times10^-3}\times\sqrt\frac{350\times10^-3\times10}{Linear density}$

$2.6\times10^3=\frac{1}{2\times25\times10^-3}\times\sqrt\frac{350\times10^-3\times10}{Linear density}$

$2.6\times10^3=\frac{1}{2\times25\times10^-3}\times\sqrt\frac{3.5}{Linear density}$

$2.6=\frac{1}{50}\times\sqrt\frac{3.5}{Linear density}$

$2.6=\frac{1}{50}\times\sqrt\frac{3.5}{Linear density}$

$130=\sqrt\frac{3.5}{Linear density}$

$130^2=(\sqrt\frac{3.5}{Linear density})^2$

$16900=\frac{3.5}{Linear density}$

$Linear density=3.25/16900$

$Linear density(Mass per unit length)=0.000207$

By unitary rule:

1 meter length have=0.000207 gm

1000 meter length have=0.000207*1000

Tex=0.20 (Answer)