Question

A string tied between $x=0$ and $x=2.45\text{m}$ vibrates in fundamental mode. The amplitude of vibration is $1\text{m}$, tension in the string is $9.8\text{N}$ and mass per unit length of the string is $1\text{g}$. Then, total energy of the string is

• A. ${\pi }^2\text{J}$
• B. $2{\pi }^2\text{J}$
• C. $\frac{{\pi }^2}{2}\text{J}$
• D. $\pi \text{J}$

Answer 1

The correct option is A ${\pi }^2\text{J}$

Here,
$l=\frac{\lambda }{2}$ or $\lambda =2l$
Angular wave number $k=\frac{2\pi }{\lambda }=\frac{2\pi }{2l}=\frac{\pi }{l}$
The amplitude of vibration at a distance $x$ from $x=0$ is given by $A=a\sin kx$
Mechnical energy at $x$ of elemental length $dx$ is
$dE=\frac{1}{2}\left(dm\right)A^2{\omega }^2=\frac{1}{2}\left(\mu dx\right)a\sin kx{)}^2(2\pi v{)}^2$
$\Rightarrow dE=2{\pi }^2\mu f^2{a}^2{\sin }^2\text{kx}dx$
Using , $v=f\lambda$ and the data given,

$dE=2{\pi }^2\frac{T}{4l^2}{a}^2{\sin }^2\left\{\left(\frac{\pi x}{l}\right)\right\}dx$
$\therefore$ Total energy of the string.
$E=\int dE={\int }_0^{l}2{\pi }^2\frac{T}{4l^2}{a}^2{\sin }^2\left(\frac{\pi x}{l}\right)dx=\frac{{\pi }^{2}T{a}^2}{4l}$

Substituting the values of $T=9.8\text{N},a=1\text{m}$ and $l=2.45\text{m}$
We have , $E=\frac{{\pi }^{2}T{a}^2}{4l}=\frac{{\pi }^2\times 9.8\times 1^2}{4\times 2.45}={\pi }^2\text{J}$