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 ______.

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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 underlinebb($\frac{1}{2}$). Explanation: Wave speed is given by \nu = \sqrt{\frac{T}{\mathrm{\mu}}} where T is the tension in the string v is the speed of the wave μ is the mass per unit length of the string \mathrm{\mu} = M/L = \rhoV/L = \rho\frac{\left( AL \right)}{L} where M is the mass of the string, which can be written as ρV. L is the length of the string. = \rho\left( $\pi$ r^2 \right) = \rho\left( $\pi$D^$\frac{2}{4}$ \right) \therefore \nu = $\sqrt$(T/\rho$\pi$D^$\frac{2}{4}$) = 2/D$\sqrt$(T/\rho$\pi$) where D is the diameter of the string. Thus, v ∝ 1/D Since, rA = 2rB v_A \propto $\frac{1}{2}$ r_A \propto $\frac{1}{2}$ $\times$ 2 r_B (1) v_{{}_B} \propto \frac{1}{2 r_{{}_B}} (2) From Equations (1) and (2) we get v_A/v_B = $\frac{1}{2}$.

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