Question

Two strings of same material are stretched to the same tension. If their radii are in the ratio $1:2$, then respective wave velocities in them will be in ratio

• A. $4:1$
• B. $2:1$
• C. $1:2$
• D. $1:4$

Answer 1

The correct option is B $2:1$

We know that the velocity of wave in a stretched string is given by:

$v=\sqrt{\frac{TL}{m}}$

Where $T=$tension$=$same for both

$L=$length$=$same for both

$m=$mass

hence

$v\propto \frac{1}{\sqrt{m}}$

we know that

mass$=$volume$\times$ density

$=\pi r^{2}L\rho$

Since L and $\rho$ are equal for both the strings, hence $m\propto r^2$

$\Rightarrow v\propto \frac{1}{\sqrt{m}}\propto \frac{1}{r}$

$\Rightarrow \frac{v_1}{v_2}=\frac{r_2}{r_1}=2:1$.