Understanding Mechanical Advantage in Pulley Systems

[Gyro]

Let's say I have a 2-pulley system, with the effort force F being pulled upwards. The load is attached to the lower movable pully P1 where the rope is fixed. The rope extends up to the fixed pulley on the ceiling P2, wraps around, then travels back down to the side of the lower pulley, wraps around, and meets the effort force. As shown below:

---------
____|_______F
____|______/
___/P2\ ___/
___|__|___/
___|__/__/
___|_/__/
___|_/_/
___|P1/
____|
____|
L

I'm told the TMA is 3 since there are 3 supporting strands of rope. So since W = Fd is constant, does that mean for every unit of distance I pull at F, the load will rise 1/3 units? If so, how, since there are only 2 other supporting segments of the rope? I mean, how do 2 supporting strands move to make up the 1 unit pulled at F? Wouldn't having 2 supporting strands imply the load is raised 1/2 unit? But if W is constant, ...?
I'm stumped.

 

[Gyro]

this one has a better diagram than my first post

 

[tiny-tim]

Welcome to PF!

Hi Gyro ! Welcome to PF!

(i assume all sections of the rope are vertical)

If P1 and P2 are at heights h₁ and h₂, and if the free end of the rope is at height f,

then the total length of the rope is 2(h₂ - h₁) + (f - h₁),

= f - 3h₁ + constant.

Since the total length is constant, that means ∆f = 3∆h₁

 

[Doc Al]

I'm told the TMA is 3 since there are 3 supporting strands of rope. So since W = Fd is constant, does that mean for every unit of distance I pull at F, the load will rise 1/3 units?

Right.

If so, how, since there are only 2 other supporting segments of the rope? I mean, how do 2 supporting strands move to make up the 1 unit pulled at F? Wouldn't having 2 supporting strands imply the load is raised 1/2 unit?

What do you mean by 2 supporting strands? There are three rope segments pulling up on the load. (Consider P1 as part of the load.)

 

[Gyro]

Thanks for the warm welcome guys!

tiny-tim: I get the total length bit, but what's confusing me is the fact that one of the supporting strands is also the strand I pull on. So I can't pull at F 1 unit, and then divide this distance 3 ways since I'm pulling on one of those strands a fixed amount.
Doc Al: I meant 2 "other" supporting strands. This ties into what I wrote above to Tim. How can the strand I pull also be considered a supporting strand in terms of compensating for the distance I pull at F? If I pull at F 1 unit, won't that specific (supporting) strand move 1 unit? If so, how do the other 2 supporting strands distribute the pulled distance to equal a 1/3 distance move of the load? Since MA reduces the effort force by 3, and sicne W is constant, then my d is divided by 3, yet the strand I pull on is moved by a whole unit, not by 1/3...

 

[Gyro]

correction, "then my distance is multiplied by 3"

 

[Gyro]

Let me think this one over more, and I'll get back to you...

Thanks for your explanations!

 

[tiny-tim]

Hi Gyro!

… I'm pulling on one of those strands a fixed amount.

No, it's not a fixed amount …

because you're moving when you pull!

 

[Gyro]

O I C! You mean because P2 moves up with me, when I pull at F 1 unit, the rope comes up from the bottom of P2 1/3 unit, the rope between P2 and P1 moves 1/3 and then the rope from P1 back down to P2 moves 1/3... is that it?

 

[Gyro]

correction: "... 1 unit, the rope comes up from the bottom of P1..."

 

[Gyro]

I've confused readers by switching P1 and P2 repeatedly... I apologize...

I mean to say "You mean because P1 moves up with me, whe I pull at F 1 unit, the rope comes up from under P1 1/3 unit, the rope between P1 and P2 moves 1/3 unit, and finally the rope between P2 back down to P1 moves 1/3 unit... is that it?"

Sorry for the confusion, but I think I get it now! Thanks guys!

 

[Gyro]

Yep, it makes much more sense now that I've properly visualized how P1 moves when I pull at F. Thanks for your help PF!

 

[tiny-tim]

… Thanks for your help PF!

You're very welcome!

Hint: try using the edit button!