Two uniform wires of same material are vibrating under the same tension. If the first overtone of first wire is equal to the 2^ nd overtone of 2^ nd wire and radius of the first wire is twice the radius of the 2^ nd wire then the ratio of length of first wire to 2^ nd wire is

Step-by-step solution

Fundamental frequency of the first wire is n = 1 2 l₁ T m = 1 2 l₁ T r ₁^2 = 1 2 l₁ r ₁ T The first overtone n ₁=2 n = 1 l₁ r ₁ T Similarly, the second overtone of the second wire will be, n ₂= 3 2 l₂ r ₂ T Given that n ₁= n ₂ aligned & 1 l₁ r ₁ T = 3 2 l₂ r ₂ T & 3 l₁ r ₁=2 l₂ r ₂ & & l₁ l₂ = 2 r ₂ 3 r ₁ aligned = 2 r₂ 3 (2 r₂ ) . ( r₁=2 r₂ )= 1 3