Why is the coefficient two in the work done by friction equation?

[Leo Liu]

Homework Statement

below

Relevant Equations

na

Solution in 2ed manual:

Solution in 1ed manual:

Could someone explain why the work done by friction is multiplied by 2? I get that the distance traveled by the block in one cycle is greater than $x_f+x_i$, but why is the coefficient two?

 

[haruspex]

Homework Statement:: below
Relevant Equations:: na

View attachment 276734
Solution in 2ed manual:
View attachment 276735
Solution in 1ed manual:
View attachment 276736
Could someone explain why the work done by friction is multiplied by 2? I get that the distance traveled by the block in one cycle is greater than $x_f+x_i$, but why is the coefficient two?

It is confusing that the two solutions use x_i and x_f to mean different things.
In the second one (1ed), they are both offsets from the relaxed position instead of from the wall, and x_f means the new rightmost extension after one cycle.
That cycle involves traveling x_i to the left to reach x_o, then another (x_i+x_f)/2 to the left for its first leftmost position, then back through x_o to its new rightmost position.
Total distance, 2(x_i+x_f).