# Mechanical Advantage of 3 Pulley System

I am a physics teacher, but have found myself teaching engineering. I have come across the following pulley question, and am starting to doubt my own understanding of pulleys !

Until now the rule that mechanical advantage equals 2*no. of moveable pulleys seems to have hold fast, but I am not sure that it works in this case. I am wondering if instead a better rule is that mechanical advantage equals no. of supporting ropes ?

For this particular question, would the no. of ropes supporting the load be three or four ?

I am not sure whether or not to count the far left section where the effort is applied.

Any thoughts on which is the most reliable rule, and how to apply it to this question, would be most gratefully received please. Thank you.

• Why not start by solving the force/distance/etc. for the first pulley, replace it with an equivalent weight applied to the next pulley, repeat until all pulleys accounted for? That is the most rigorous path to a solution as well as a mathematical proof. NMech's answer is certainly more straightforward and simple. Commented Apr 5, 2021 at 14:16

The tension on the rope is everywhere the same and its equal to F.

So if you did a free body diagram on the following system by sectioning along the ropes:

what you get from the equilibrium is $$4F = 48[N]$$.

I hope that is sufficient as an explanation, I tend to find that problems with pulleys can have different configurations and as such it is always better to turn to the basics.

• Thank you NMech for the very clear answer. I'm not sure why I didn't think of that. I was placing too much emphasis on worrying about a rule. Anyway, it all makes sense now. I've ticked your answer. Commented Apr 3, 2021 at 8:47
• I've had my share of deadends and head scratching, so I am glad it benefitted someone. I can also understand the lure of a rule (play of words intended), however, when you teach this to people, its best to be able to adapt. Rules are good, but they invariably have exceptions. Commented Apr 3, 2021 at 9:14
• It's not much about what I perceive as a mistake, I align to what at least NIST physics.nist.gov/cuu/pdf/sp811.pdf and BIPM bipm.org/utils/common/pdf/si_brochure_8_en.pdf give as guidelines. In short [*] is the "units of *" operator. It should be used in something like [L]=m for instance meaning that the quantity L is given in metres. Finally I apologize if you felt me rude in defining plague what you wrote, but again, IMHO your beautiful answers deserve this little final tune up Commented Apr 4, 2021 at 5:23
• No, I didn't take it as rude at all. I've read enough of your posts to know you a bit. My point in response to your polite quesiton was that I'm open to correction. Commented Apr 4, 2021 at 11:32
• I am an old schooler too, but only use enclose the ending units in brackets, or parenthesis, for mixed numerical operation without carrying units in every step. For instance, M = 2 x 6 = 12 (N-m), or 12 [kips-ft].
– r13
Commented Apr 4, 2021 at 21:00