Question

Two strings $A$ and $B$, made of same material, are stretched by same tension. The radius of string $A$ is double of the radius of $B$. A transverse wave travels on $A$ with speed $v_A$ and on $B$ with speed $v_B$. The ratio $v_A/v_B$ is

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

Answer 1

Answer: (A)

$1/2$

Velocity of a transverse wave in a stretched string $v=\sqrt{\frac{T}{\mu }}$
where $T$ is the tension of the string and $\mu$ is mass per unit length of the string.
$\mu =\pi r^2\times \rho$
where $r$ is the radius or the string and $\rho$ is the density of the material of the string.
$\therefore v=\frac{1}{r}\sqrt{\frac{T}{\pi \rho }}$
Since $T,\rho$ are constant
$\therefore v\propto \frac{1}{r}$
$\frac{v_A}{v_B}=\frac{r_B}{r_A}=\left(\frac{r_B}{2r_B}\right)=\frac{1}{2}$.