Question

A string of density $7.5gc{m}^{-3}$ and area of cross-section $0.2m{m}^2$ is stretched under a tension of $20N$. When it is plucked at the mid-point, the speed of the transverse wave on the wire is

• A. $116\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.$
• B. $40\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.$
• C. $200\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.$
• D. $80\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.$

Answer 1

The correct option is A

$116\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.$

Step1: Given data and assumptions.

Density of the string, $\rho =7.5gc{m}^{-3}$

Cross-section area, $A=0.2m{m}^2=2\times {10}^{-3}c{m}^2$

Tension of the string,$T=20N$

Step 2: Formula used:

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

Where, $v$ is speed of transverse wave, $T$ is the tension of the string, $\mu$ is the mass per unit length of the string.

Step3: Finding the speed of the transverse wave on the wire.

The mass per unit length of the string is,

$\mu =\rho A$

$\Rightarrow$ $\mu =7.5\times 2\times {10}^{-3}$

$\Rightarrow$ $\mu =0.015\raisebox{1ex}{$g$}\!\left/ \!\raisebox{-1ex}{$cm$}\right.=0.0015\raisebox{1ex}{$kg$}\!\left/ \!\raisebox{-1ex}{$m$}\right.$ ……$\left(i\right)$

Now,

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

$\Rightarrow$$v=\sqrt{\frac{20}{0.0015}}$

$\Rightarrow$$v=116\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.$

Hence, option A is correct. The speed of the transverse wave on the wire is $116\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.$.