Two strings with circular cross section and made of same material, are stretched to have same amount of tension. A transverse wave is then made to pass through both the strings. The velocity of the wave in the first string having the radius of cross section R is $v_1$, and that in the other string having radius of cross section R/2 is $v_2$. Then $\frac{v_2}{v_1}$ =

<p>To find the ratio of the velocities of transverse waves in two strings with different radii but identical materials and tension, consider the following:</p> <p>The wave velocity $ v $ in a string is given by the formula:</p> <p>$ v = \sqrt{\frac{T}{\mu}} $</p> <p>where $ T $ is the tension and $ \mu $ is the linear mass density of the string, defined as:</p> <p>$ \mu = \rho \pi R^2 $</p> <p>Here, $ \rho $ is the density of the material, and $ R $ is the radius of the string. Given that both strings have the same tension $ T $ and material, we compare their wave velocities $ v_1 $ and $ v_2 $ for radii $ R_1 = R $ and $ R_2 = \frac{R}{2} $.</p> <p>The velocity ratio is expressed as:</p> <p>$ \frac{v_2}{v_1} = \frac{\sqrt{\frac{T}{\rho \pi R_2^2}}}{\sqrt{\frac{T}{\rho \pi R_1^2}}} $</p> <p>Simplifying this expression:</p> <p>$ \frac{v_2}{v_1} = \frac{\sqrt{R_1^2}}{\sqrt{R_2^2}} = \frac{R_1}{R_2} = \frac{R}{\frac{R}{2}} = 2 $</p> <p>Thus, the ratio $ \frac{v_2}{v_1} $ is 2.</p>