Question

A uniform wire of length $L$, diameter $D$ and density $\rho$ is stretched under a tension $T$. The correct relation between its fundamental frequency $f$, the length $L$ and the diameter $D$ is:

• A. $f\propto \frac{1}{LD}$
• B. $f\propto \frac{1}{L\sqrt{D}}$
• C. $f\propto \frac{1}{D^2}$
• D. $f\propto \frac{1}{L{D}^2}$

Answer 1

The correct option is A $f\propto \frac{1}{LD}$

Given that,

Frequency $=f$

Diameter $=D$

Length $=L$

Density $=\rho$

Tension $=T$

Now, we know that

$f=\frac{c}{\lambda }$

$f=\frac{1}{2L}\times \sqrt{\frac{T}{\mu }}$

$\mu$ is mass per unit length

$\mu =\pi \times \frac{D^2}{4\rho }$

Now, the frequency is

$f=\frac{1}{2L}\times \sqrt{\frac{T}{\pi \times \frac{D^2}{4\rho }}}$

$f\propto \frac{1}{LD}$

Hence, the correct relation is $f\propto \frac{1}{LD}$