Question

Transverse waves are generated in two uniform steel wires A and B when a source of frequency 500 Hz is attached to their free-ends. The diameter of A is half that of B and the tension in A is half that in B. The ratio of the velocity of the waves in wires A and B is

• A. $1:2$
• B. $2:1$
• C. $1:\sqrt{2}$
• D. $\sqrt{2}:1$

Answer 1

The correct option is D $\sqrt{2}:1$

$v=\sqrt{\frac{T}{\mu }}=\sqrt{\frac{T}{\frac{\pi d^2}{4}\rho }}\Rightarrow v=2\sqrt{\frac{T}{\pi d^2\rho }}$
$\Rightarrow v\propto \frac{\sqrt{T}}{d}$
$\frac{v_A}{v_B}=\sqrt{\frac{T_A}{T_B}}\times \frac{d_B}{d_A}=\frac{1}{\sqrt{2}}\times \frac{d_B}{\frac{1}{2}{d}_B}$
$\Rightarrow \frac{v_A}{v_B}=\frac{2}{\sqrt{2}}$
$\therefore v_A:v_B=\sqrt{2}:1$