A wire of length ' L ', diameter ' d ', density of material ' ' is under tension ' T ' has fundamental frequency of vibration ' n_A '. Another wire of length ' 2L ', diameter ' 3d ', density of material ' 2 ' is vibrated under tension ' 2T ', the fundamental frequency of vibration becomes ' n_B '. The ratio n_B : n_A is

Step-by-step solution

The fundamental frequency of a stretched wire is given by n = 1 2L T m , where L is the length, T is the tension, and m is the linear mass density. The linear mass density m can be expressed in terms of diameter d and density as: m = mass length = volume L = ( r^2 L) L = ( d 2 )^2 = d^2 4 Substituting m in the formula for n : n = 1 2L 4T d^2 = 1 Ld T For the first wire: n_A = 1 Ld T For the second wire, given L' = 2L , d' = 3d , ' = 2 , and T' = 2T : n_B = 1 (2L)(3d) 2T (2 ) n_B = 1 6Ld T Taking the ratio of n_B to