Two uniform strings A and B made of steel are made to vibrate under the same tension. If first overtone of A is equal to the second overtone of B and if the radius of A is twice that of B , the ratio of the length of string B to that of A is

Step-by-step solution

R ₂=2 R ₁, ~T ₁= T ₂ As the first overtone of A is equal to second overtone of B, we get aligned 2 n_A=3 n_B ( first overtone = second aligned harmonic)...(i) Also, n = 1 2 l ~T ~m . Where, m = mass per unit length = ( R ^2 l ) l . .( ii ) aligned & 2 1 2 l_ A ~T ~m _ A =3 1 2 l_B ~T ~m _ B ...(from(i)) & l_ A l_ B = 2 3 ~m _ B ~m _ A = 2 3 r _ B ^2 r _ A ^2 & l_ A l_ B = 2 3 r _ B ^2 (2 r _ B )^2 = 2 3 1 4 = 1 3 & l_ B l_ A = 3 1 aligned ...(from (ii))