Consider a string of total length L, made up of three segments of equal length.
ID: 1483833 • Letter: C
Question
Consider a string of total length L, made up of three segments of equal length. The mass per unit length of the first segment is , that of the second is 2, and that of the third /4. The third segment is tied to a wall, and the string is stretched by a force of magnitude Ts applied to the first segment; Ts is much greater than the total weight of the string.
A.
How long will it take a transverse wave to propagate from one end of the string to the other?
Express the time t in terms of L , , and Ts .
Explanation / Answer
In any case,
t = d / v
t_total = t_1 + t_2 + t_3 [Broke up the segments]
v1 = sqrt(T_s / )
v2 = sqrt(T_s / 2 ) = 1/sqrt(2) sqrt(T_s/ ) = 1/sqrt(2) V1
v3 = sqrt(T_s / 1/4 ) = 2sqrt(T_s / ) = 2 V1
t_total = L/3 [1 / V1 + sqrt(2) / V1 + 1 / (2V1)]
t_total = L sqrt( / T_s) / 3 [1 + sqrt(2) + 1/2]
t_total = L sqrt( / T_s) / 3 * [1/2 (3 + 2sqrt(2))]
t_total = (L / 6) * sqrt( / T_s) * (3 + 2sqrt(2))