Question

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

Options

• 2 : 1

• 3 : 4

• 3 : 2

• 1 : 3

Answer 1

Two uniform strings ‘A’ and ‘B’ made of steel are made to vibrate under same tension. If the first overtone of ‘A’ is equal to second overtone of ‘B’ and radius of ‘A’ is twice that of ‘B’. Then the ratio of length of string ‘A’ to that of ‘B’ is 1 : 3.

Explanation:

Given: n₂ = n₃

∴ 2n₁ = 3n₁    ...[∵ n₂ = 2n₁, n₃ = 3n₁)

Fundamental frequency (n) = `1/(2 l) sqrt (T/m)`

= `1/(2 l) sqrt (T/(pi r^2 rho)`

∴ `2 (1/(2 l_2) sqrt (T/(pi r_1^2 rho))) = 3 (1/(2 l_2) sqrt (T/(pi r_2^2 rho)))`

∴ `l_1/l_2 = (2 xx r_2)/(3 xx r_1)`

= `(2 xx r_2)/(3 xx r_2)`    ...[Given, r₁ = 2r₂]

= `1/3`