A light string is tied at one end to a fixed support and to a heavy string of equal length `L` at the other end as shown in figure. A block of mass `m` is tied to the free end of heavy string. Mass per unit length of the strings are `mu` and `9mu` and the tension is `T`. Find the possible values of frequencies such that junction of two wire point `A` is a node.

Answer

A, B, C, D

Explanation

When A is a node Suppose `n_1 and n_2` are the
complete loops formed on left and right side of
point A. Then,
`f_1 = f_2`
or ` n_1 (v_1/ (2L)) = n_2 (v_2/(2L))`
or ` n_1/n_2 = (v_2/v_1) = sqrt (mu_1/mu_2) = 1/3, 2/6, 3/9,……...., etc.`
( as `v prop 1/sqrt(mu))`
`:.` Possible frequencies are
`v_1/ (2L), 2(v_1/(2L)), (3v_1)/(2L),............,etc. (v_1 = sqrt (T/mu))`
or ` 1/(2L) sqrt(T/mu), 1/L sqrt(T/mu), 3/(2L) sqrt T/(2mu)`,......., etc.
When A is an antinode Suppose `n_1` and `n_2` are
complete loops on left and right side of point A,
`n_1 (lambda_1/2) + (lambda_1/4) = L `
or `f_1 = v_1/L (n_1/2 + 1/4)`
`n_2 lambda_2/2 + lambda_2/4 = L `
` or `f_2 = v_2/L (n_2/2 + 1/4)`
Substituting `f_1 = f_2`,
we get, ` (2n_1 + 1)/(2n_2 +1) = 1/3`
For `n_1 = 1, n_2 = 4`
`n_1 = 2, n_2 = 7 `
n_1 = 3, n_2 = 10`, etc.
Therefore, the possible frequencies are
` v_1/L (1/2 + 1/4), v_1/L (2/2 + 1/4), v_1/L (3/2 + 1/4)`,.........,etc.
or `3/(4L) sqrt(T/mu), 5/(4L) sqrt(T/mu), 7/(4L) sqrt(T/mu)`, ....... , etc. `