Question

Sinusoidal waves $5.00cm$ in amplitude are to be transmitted along a string having a linear mass density equal to $4.00\times {10}^{-2}kg/m$. If the source can deliver an average power of $90W$ and the string is under a tension of $100N$, then find the frequency at which the source can operate. (take ${\pi }^2=10)$.

Answer 1

Given,

$A=5cm=0.05m$

$\mu =4\times {10}^{-2}\frac{kg}{m}$

$P_{avg}=90w$

$T=100N$

${\pi }^2\approx 10$

To find, $f=?$

Power transmitted by string wave $=\frac{1}{2}\mu {\omega }^2{A}^{2}v$

Here, $v=\sqrt{\frac{T}{\mu }}=\sqrt{\frac{100}{4\times {10}^{-2}}}$

$\Rightarrow v=\frac{1}{2}\times {10}^2=0.5\times {10}^2\frac{m}{s}$

$P=\frac{1}{2}\mu {\omega }^2{A}^{2}v$

$\Rightarrow \frac{2P}{\mu A^{2}v}={\omega }^2$

${\omega }^2=\frac{2\times 90}{4\times {10}^{-2}\times (0.05{)}^2\times 0.5\times {10}^2}=\frac{180}{4\times 0.5\times (0.05{)}^2}=3.6\times {10}^4$

${\omega }^2=(2\pi f{)}^2$

$\Rightarrow 4{\pi }^2{f}^2=3.6\times {10}^4$

$\Rightarrow f^2=900$

$\Rightarrow f=30H_z$

The frequency at which the source can operate is $30H_z$