How do I combine equations for absolute independent motion with pulleys?

AI Thread Summary
The discussion focuses on combining equations related to pulleys and independent motion. The user has derived several equations but struggles with how to combine them effectively. A suggestion is made to eliminate variables S_c and S_d by manipulating the equations, such as multiplying and adding them. It is emphasized that the lengths of l_1, l_2, and l_3 do not have absolute values, and understanding velocity and force ratios is crucial. The conversation concludes with advice on labeling forces to simplify the problem-solving process.
JustDerek
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Homework Statement


I've been given a problem with pulleys which I have attached to this post. I've derived the equations shown in the post but I'll also write them below. What I'm struggling with is how to combine them.

Homework Equations


##l_1=S_a+2S_c##
##l_2=S_d+(S_d-S_c)##
##l_3=S_e+(S_e-S_c)##

In a similar example I've been given but with less pulleys it shows :
##l_1+l_2=S_a+4S_d##
This is the part I don't get. I don't get how the two equations combine to become that and I'm sure if I can understand that I can finish the rest.

The Attempt at a Solution

 

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You can add the equations in any way you want. Something that gets rid of S_c and S_d is probably useful.

As an example, you have "+2 S_c" in the first equation and "-S_c" in the second one. You can multiply the second equation (both sides!) with two, then add the two equations.
 
I'm not sure why you are trying to find ##l_1, l_2\ and\ l_3##. There is no absolute value for them. Changing the length of any of them does not affect the velocity ratio nor the forces.
What I think you need to know are the velocity ratios and the force ratios, which you should be able to do by inspection in a simple example like this.
I would suggest the way to deal with the forces is to label one of them F or whatever, then write the others as multiples (or fractions.)
 
Managed to get there myself eventually but thanks for the intended help
 
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