A wire of length L , diameter ' d ' density of material ' Q ' is under tension ' T ', having fundamental frequency of vibration n _ A . Another wire of length 2 L , tension 2 T , density 2 Q and diameter 3 d has fundamental frequency of vibration n_B . The ratio n_B: n_A is

Step-by-step solution

The fundamental frequency of vibration for a stretched wire is given by n = 1 2L T , where L is length, T is tension, and is linear mass density. The linear mass density relates to material density and cross-sectional area as = A = d^2 4 , where d is the wire diameter. Substituting gives n = 1 2L T d^2 / 4 = 1 Ld T . For wire A with L_A = L , d_A = d , _A = Q , and T_A = T : n_A = 1 Ld T Q . For wire B with L_B = 2L , d_B = 3d , _B = 2Q , and T_B = 2T : n_B = 1 (2L)(3d) 2T (2Q) = 1 6Ld T Q . The ratio is n_B n_A =