# Mechanical advantage and force analysis of complex pulley systems

1. Aug 13, 2012

### horace_focker

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

These are two of the excercises where I'm supposed to calculate the mechanical advantage of the pulley systems. Excuse the crude MS paint drawings, I do not have access to a scanner or a proper camera. The red dots are where two cables connect.

3. The attempt at a solution

Unfortunately I have no idea where and how to even begin. I looked for explanations online but I only found general equations for block and tackle systems, without a proper explanation. It may sound a bit silly, but I just don't understand pulley systems.

2. Aug 13, 2012

### Arghzoo

Effectively, the work/energy done by the person pulling the rope has to be conserved. So if the load weighs a lot, mechanical advantage is represented as a reduction in the load felt by the person pulling the rope. Because energy must be conserved, the distance traveled by the load must necessarily lessen. Energy = Force (or weight) * distance traveled

The simplest way of finding mechanical advantage is to find the NUMBER of ropes contributing to holding the load up. Don't ignore the ones "pulling" down, though, because the tension in the ropes also contribute. Essentially, just counting the major ropes will work. This is because as the load is lifted, all those ropes shorten, maintaining the distance the rope is pulled but lessening the distance the weight rises and the weight felt by the puller. If the end is pulling down, then that won't be counted because it's not contributing to lifting the entire load up.

In the first case, we see three ropes pulling on the load. The rope connecting the very top pully to the one below it contributes to the mechanical advantage because it has an internal tension. Its mechanical advantage is three.

In the second case, we see six effective ropes. That's a mechanical advantage of six, reducing the load by a factor of 1/6.

Please let me know if there's anything I can clarify. :D
Good luck!

3. Aug 14, 2012

### horace_focker

Thank you very much, I really appreciate it, it's a very clear explanation. Sorry for responding a little late, I was asleep.

Is the calculation of the mechanical advantage really that simple though? I expected something more convoluted.

So in these other two exercises, the advantage on the left would be 5 and on the right it'd be 4?

4. Aug 14, 2012

### Arghzoo

Yep you got it!

It sounds complex but all pulleys do is decrease the effective load by redirecting force. It seems like you're making it easier by adding more direction changes and pulleys, but all that free work manifests in the longer distance you have to pull the rope through.

5. Aug 14, 2012

### horace_focker

Alright, thanks, you were really helpful!

Oh and wouldn't the advantage in the very first exercise be 4 instead of 3? Or does the rope connecting to load to the pulley not count because it doesn't shorten?

6. Aug 14, 2012

### Arghzoo

No problem!

You're exactly right; the rope doesn't shorten, excluding it from the count. The mechanical advantage is still 3.

7. Aug 15, 2012

### Staff: Mentor

You should verify your method by determining the ratio of the total motion at the "input" to the total motion at the "output". Bifurcating ropes can be tricksy. Hint: The systems are linear so superposition should apply; For the first example fix one rope at a time and determine the motion of the load versus the motion of the other rope. What then is the total motion due to both?