How to Determine Elevator Speed with Winch Mechanics?

[ilya]

Please help. I am starting to think the answer in the back of my book is wrong. Refer to picture below. Question: Find the speed of the elevator if the winches A and B both pull the rope at 5 m/s. Answer is 2.14m/s. How? I have no idea. I am not the kind to just give up.

Thank you very much!

 

[ehild]

The length of both ropes changes in a rate -5 m/s. So the sum of the velocities is not zero.

More: notice that xa=xb, xa=xd+xc.

ehild

 

[ilya]

Thank you for your reply. In class we set these types of equations as:

Rope 1:

Step one: Xb+Xc+Xd= length(which is just Some constant)
Step two: Differentiate both sides to get velocity since previous equation is length or displacement.

there fore we get Vb+Vc+Vd=0

Step three: we know Vb=Va=5m/s and we can substitute. 5+Vc+Vd=0 -> Vc+Vd= -5m/s

Rope 2:
Step one: Xa+Xa+Xc= length(some constant)
Step two: differentiate both sides to get 2Va+Vc=0
Step three: substitute known. 2(5) + Vc = 0 -> Vc= -10m/s

Finaly
Solving for the last unknown Vc+Vd=-5 -> -10+Vd=-5 -> Vd = 5m/s

Somewhere I am not doing something right because the answer is 2.14m/s (sign is disregarded)
What doesn't make sense is that Vd and Vc are different velocities, and I think they should be the same because how can the elevator go up with a constant velocity. MY GUESS IS THAT I LABELED THE PROBLEM WRONG

 

[ilya]

 

[rl.bhat]

When the system moves, total changes in each segments of the string must be zero.
In the blue string
change in the 1st segment = Xb - Xc
change in the 2nd segment= Xd - Xc, but Xd = Xb, so
= Xb - Xc
change in the 3rd segment =- Xd - y
So total change in the blue string = (Xb)-2(Xc) - y = 0...(1)
Now in the purple string
change in the 1st segment = Xa - Xf
change in the 2nd segment = y - Xf
change in the 3rd segment = y - Xg
So total change in the purple string = Xa - 3Xf + 2y = 0...(2)
From eq.1 y = (Xb - 2Xc). Substitute it in equation (2). Note that Xf = Xc and Xa = Xb,
Find Xf in terms of Xa. Put Xa = 5m/s and find Xf.