Question

Two vibrating strings made of the same material but of lengths $L$ and $2L$ have the radii $2r$ and $r$ respectively. They are stretched under the same tension. Both the strings vibrate in their fundamental modes-the one of length $L$ with frequency $v_1$ and the other with frequency $v_2.$ The ratio $\frac{v_1}{v_2}$ is equal to

• A. $4$
• B. $2$
• C. $8$
• D. $1$

Answer 1

The correct option is D $1$

We know that,
$v=\frac{1}{2l}\sqrt{\frac{T}{m}}=\frac{1}{2l}\sqrt{\frac{T}{A\rho }}=\frac{1}{2l}\sqrt{\frac{T}{\pi R^2\rho }}=\frac{1}{2lR}\sqrt{\frac{T}{\pi \rho }}$

As $T$ and $\rho$ are the same for both the wires, $v\propto \frac{1}{lR}$.

Here, $lR=L\times 2r=2L\times r.$

$\therefore v_1=v_2$.