Two uniform strings X and Y made of same material made to vibrate under same tension. If the second overtone of string X is equal to the third overtone of string Y and the radius of string X is thrice that of string Y. The ratio of length of string X to that of Y is

Step-by-step solution

The frequency of the p -th harmonic of a stretched string is given by f = p 2L T The linear mass density is = r^2 , where r is the radius and is the density of the material. Substituting , we get f = p 2Lr T For string X, the second overtone corresponds to the third harmonic, so p_X = 3 . For string Y, the third overtone corresponds to the fourth harmonic, so p_Y = 4 . Since both strings are made of the same material and are under the same tension, and T are constant. Given that the frequencies are equal ( f_X = f_