Question

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

Options

• 2 : 1

• 1 : 2

• 1 : 3

• 3 : 1

Answer 1

1 : 3

Explanation:

The first wire's fundamental frequency is

n = `1/(2l_1) sqrt(T/m) = 1/(2l_1) sqrt(T/(pir_1^2p)) = 1/(2l_1r_1) sqrt(T/(pip)`

The first overtone n₁ = 2n = `1/(l_1r_1) sqrt(T/(pip)`

Similarly, the second wire's second overtone will be,

n₂ = `3/(2l_2r_2) sqrt(T/(pip)`

Given that n₁ = n₂

∴ `1/(l_1r_1) sqrt(T/(pip)) = 3/(2l_2r_2) sqrt(T/(pip)`

∴ `3l_1r_1 = 2l_2r_2`

`l_1/l_2 = (2r_2)/(3r_1)`

= `(2r_2)/(3(2r_2))`  .....(∵ r₁ = 2r₂)

= `1/3`