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 } $
$\frac{ v _{ A }}{ v _{ B }}=\frac{ r _{ B }}{ r _{ A }}$
$=\left(\frac{ r _{ B }}{2 r _{ B }}\right)=\frac{1}{2}$