# Ropes and Pulleys - Really unintuitive answer

I usually don't want to do this, but please go to this link, the solution is too big to post it here

And go to page 5 of the pdf.

Briefly, the problem say

Determine the velocity of the 60-lb block A if the two blocks are released from rest and the 40-lb block B moves 2 ft up the incline. The coefficient of kinetic friction between both blocks and the inclined planes is $\mu_k$ = 0.10.

Things I am confused with the solution

1.First of all, I seriously thought lb was mass not force. after some googling, it turns out they are used interchangeably...

2.Where did they even get $2s_a + s_b = 0$ from? Why did they determine the change in distance this way? My first assumption was that if block B moved up 2ft, then block A should move down 2ft (the rope must "move" 2ft too right?). Then I wasn't sure, so I did a few triangles and found that the angle made a difference

3.Where did $2v_A = -v_B$ come from?

4 The FBD for block A is confusing, why is the friction force $F_A$ in the direction of the ropes? I thought it was block B that is going down? Am I the only one who had trouble deducting that the pulley and block A are the same object?

5.Look at the final answer, how could $v_b$ be negative? The problem says block B goes UP.

If you are wondering, this is not homework. I am just interested in this problem and it is out of curiosity and very confused with the concepts.

I've had at most introductory physics experience, but I think I should have been able to solve this still.

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"2.Where did they even get 2sa+sb=0 from?" From the assumption that the length of the rope does not change. "Why did they determine the change in distance this way? My first assumption was that if block B moved up 2ft, then block A should move down 2ft (the rope must "move" 2ft too right?)." No, there is a difference between a movable pulley and a fixed pulley, so A moves down 1 ft.

"3.Where did 2vA=−vB come from?" From the same assumption as above. However, the direction of vA is shown incorrectly (or, alternatively, the signs in the formula are wrong).

"4 The FBD for block A is confusing, why is the friction force FA in the direction of the ropes? I thought it was block B that is going down?" It is writen in the statement of the problem that block B went up. " Am I the only one who had trouble deducting that the pulley and block A are the same object?" I don't quite understand what this phrase means exactly and how it is relevant.

"5.Look at the final answer, how could vb be negative? The problem says block B goes UP."