# Dynamics -- Velocity of a Block in a system of Pulleys

1. Jan 15, 2017

Hey guys, I have been a long time lurker here but this problem has eluded me for a few hours now and my frustration at not being able to figure out what it is I am missing prompted me to finally pull the trigger on making an account here. I hope the way I wrote everything out is clear to understand but if it is not I would be happy to do my best to elaborate further. Thank you!

1. The problem statement, all variables and given/known data

The system shown starts from rest and each component moves with a constant acceleration. The relative acceleration of block C with respect to collar B is 60mm/s^2 upward and the relative acceleration of block D with respect to block A is 110mm/s^2 downward. Determine the velocity of block C after 6s.

2. Relevant equations
Relative Acceleration: ac = ac/b + ab
v = v0 + a*t

3. The attempt at a solution
My first thought to solve this problem involved deriving an equation for the length of the cable that connects blocks A B and C that I could differentiate twice to get an acceleration equation and an equation for the length of the cable that connects blocks A B and D to do the same with. For the length of cable 1 I got 2SA + 2SB + SC = Constant taking my reference point to be the line that goes through the center of the 3 top pulleys. For the other cable I wasn't 100% certain what it should be though for what the question is asking I don't believe it is necessary to know. I can't seem to find any way to utilize this equation for the length of the cable to give me an acceleration for block B that I can use combined with the relative velocity equation to give me the acceleration for block C that I can use with the kinematic equation to find the velocity. My next thoughts for this problem was to find some way to use the given relative acceleration to simply obtain accelerations for the blocks directly however if this is possible then despite my best efforts I haven't found a way to do so. Any help at solving this would be GREATLY appreciated. I feel like it is easier than I am making it out to be which makes it all the more frustrating that I cannot solve it.

2. Jan 17, 2017

### Staff: Mentor

Let $\delta_A$, $\delta_B$, and $\delta_D$ be the downward displacements of A, B, and D respectively. In order for the bottom string to remain constant in length, $$(\delta_D-\delta_A)+(\delta_D-\delta_B)=0$$