A uniform rope of length 8.0m and linear mass density 0.12kg/m is stretched between two rigid supports with a constant tension of 54N. A single upward pulse is generated at a point 2.0m from the left support.
(a) Calculate the speed of the pulse along the rope.
(b) The pulse travels to the right support, reflects, travels to the left support, reflects again, and finally returns to the point where it was generated. Determine the total time required for the pulse to complete this sequence.
(c) The tension in the rope is suddenly reduced to half its original value while the pulse is midway between the supports. Assuming the linear mass density remains unchanged, calculate the additional time required for the pulse to finish the remainder of the journey described in part (b).
(d) Derive an expression for the total travel time tn required for the pulse to undergo n complete round trips between the supports (beginning at the point of origin and ending at the same point) in terms of the rope length L, the distance d from the origin to the left support, the tension FT, and the linear mass density μ.