A transverse wave travels on a taut steel wire
Question:

A transverse wave travels on a taut steel wire with a velocity of $v$ when tension in it is $2.06 \times 10^{4} \mathrm{~N}$. When the tension is changed to $T$, the velocity changed to $v / 2$. The value of $T$ is close to:

1. $2.50 \times 10^{4} \mathrm{~N}$

2. $5.15 \times 10^{3} \mathrm{~N}$

3. $30.5 \times 10^{4} \mathrm{~N}$

4. $10.2 \times 10^{2} \mathrm{~N}$

Correct Option: , 2

Solution:

(2) The velocity of a transverse wave in a stretched wire is given by

$v=\sqrt{\frac{T}{\mu}}$

Where,

$T=$ Tension in the wire

$\mu=$ linear density of wire

$(\because V \propto T)$

$\therefore \frac{v_{1}}{v_{2}}=\sqrt{\frac{T_{1}}{T_{2}}}$

$\Rightarrow \frac{v}{v} \times 2=\sqrt{\frac{2.06 \times 10^{4}}{T_{2}}}$

$\Rightarrow T_{2}=\frac{2.06 \times 10^{4}}{4}=0.515 \times 10^{4} \mathrm{~N}$

$\Rightarrow T_{2}=5.15 \times 10^{3} \mathrm{~N}$