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

Step-by-step solution

The frequency of the p -th overtone of a stretched string is given by f = (p+1)v 2L , where v is the wave speed and L is the length of the wire. For the first wire, the first overtone ( p=1 ) frequency is: f₁ = 2v₁ 2L₁ = v₁ L₁ For the second wire, the second overtone ( p=2 ) frequency is: f₂ = 3v₂ 2L₂ Given that f₁ = f₂ , we have: v₁ L₁ = 3v₂ 2L₂ L₁ L₂ = 2v₁ 3v₂ The velocity of a transverse wave in a wire is v = T = T r^2 , where T is tension, r is radius, and is density. Since T and are the same for both wires, v