Question

Two strings $A$ and $B$ of same material are stretched by same tension. The radius of the string $A$ is double the radius of string $B$. Transverse wave travels on string $A$ with speed ‘$V_A$’ and on string $B$ with speed ‘$V_B$’. The ratio $\frac{V_A}{V_B}$ is

• A. $\frac{1}{4}$
• B. $\frac{1}{2}$
• C. $2$
• D. $4$

Answer 1

Answer: (B)

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
$\therefore v\propto \frac{1}{r}$
$\frac{v_A}{v_B}=\frac{r_B}{r_A}$
$=\left(\frac{r_B}{2r_B}\right)=\frac{1}{2}$