Why are these pulleys equivalent?

[yucheng]

I have to apologize if the diagram is awful: I drew it for the sake of asking... I am analyzing the rope that is hanging the pulleys. From the diagram, the pulley at the left (let's call it L), has two objects with mass M at both ends. The pulley at the right (let's call it R), has an object with mass M at one end, while the other end is fixed. By analyzing the tension, we see that the tension add up together as shown in the free body diagram below, as they are in equilibrium, $a=0$ and $T=M g$. Thus the rope that hangs pulley L and R both exert the same tension. Is this correct? It seems counterintuitive to me, that you can use 1 object to give the same tension on the rope as you could with 2 objects. Thanks in advance.

 

[Delta2]

Summary:: The pulleys seem different at first glance, however the tension of the rope that hangs the pulley is the same. It is counterintuitive, that's why I'm asking.

Is this correct? It seems counterintuitive to me, that you can use 1 object to give the same tension on the rope as you could with 2 objects.

well yes the systems seem to be equivalent (equivalent in terms of the tensions of the ropes) but in static equilibrium only. One system can start to move (if we give a small nudge in one of the masses in the left pulley system) but the other system cannot move (if we give a nudge towards down to the mass in the right pulley, then the Tension will increase and it will prevent the mass from moving.)

Btw, I think you use two objects in both cases, in the right pulley the second object is where you fix the end of rope and it is essentially the whole planet earth!

 

[sophiecentaur]

Interestingly, if you lift the pulley by x, the work you need to do (Force times distance) is the same in both cases.
Two masses Work done = 2mgx
One mass Work done = mg (2x)

 

[sophiecentaur]

(if we give a nudge towards down to the mass in the right pulley, then the Tension will increase and it will prevent the mass from moving.

I love the work and energy approach. Here, you can do no work lifting either of the two balanced suspended masses because 'no' force is involved (the other one will just move down). In the tethered case you can do work on lifting the mass but NONE on pulling it down because the Earth stops you.

 

[sophiecentaur]

@yucheng Do you have a problem with that? You will know how work is defined. One case has twice the force for a given distance and the other involves a single weight force and twice the distance. Simples. You can do no work on the Earth because it doesn't move.

 

[yucheng]

if you lift the pulley by x, the work you need to do (Force times distance) is the same in both cases.

Okay, this confused me. I actually interpreted it as something like this: $2mgx=mgx$, where you pull both pullies by the same distance!

I tried manipulating constraints for : let $h$ be the height of the pulley, and $x$ the height of the block. $h+h-x=constant$, taking the second derivative, $2\ddot{h}=\ddot{x}$. I'm still thinking about its implications.

 

[A.T.]

It seems counterintuitive to me, that you can use 1 object to give the same tension on the rope as you could with 2 objects.

When pulling the pulley up:

Pulley 1 has no mechanical advantage. You pull the total weight and the masses move the same distance that you pull.

Pulley 2 has mechanical advantage. You pull 2x the total weight and the mass moves 2x the distance that you pull.

 

[kuruman]

It seems counterintuitive to me, that you can use 1 object to give the same tension on the rope as you could with 2 objects.

If it is counter-intuitive, then you might consider informing your intuition. Look at the figure below. On the left you see a pulley with two equal masses. On the right you see an identical pulley with its right side hidden by a screen. The pulleys are not accelerating.

Question: Can you tell whether behind the screen there is a hand holding the other end of the string or another hanging mass equal to the one on the left? If you cannot tell the difference, then the tensions in the two figures have to be the same.

 

[sophiecentaur]

Pulley 2 has mechanical advantage. You pull 2x the total weight and the mass moves 2x the distance that you pull.

I read this a few times and I'm not so sure. Mechanical Advantage is load/effort and the effort is actually twice the weight (= 2T) so the MA is actually 1/2. However, the Velocity Ratio VR is 2. The product in both cases is the same (as you say).
A point I am constantly needing to make about Machines in general is that Mechanical Advantage should include the Efficiency of the machine and will always be less than 1/VR; VR comes from the geometry alone but MA includes things like (in this case) the weights of rope and pulley and friction. Afaics, the terminology tends to be used freely in a more sloppy way than most Physicists would dare to use in other aspects of Physics.

 

[Merlin3189]

Not an uncommon situation in mechanics.

 

[sophiecentaur]

Not an uncommon situation in mechanics.
View attachment 274304

I remember a friend of mine who was training of VC10s in the late 60's. He told me that commercial craft (707, I think, was all mechanical linkages) used double control cables for control surfaces and used that sort of system to give the pilot the same 'feel' of controls with one or two cables working. Don't ask me how they made the half with the broken cable go rigid but I guess there would have been a warning light and the pilot could jamb a screwdriver in somewhere to fix things.

 

[Merlin3189]

I haven't done a search yet. The only thing I've thought of, is to have, instead of a freely pivoting link, a rigid T-piece constrained to move parallel to the wires. The force will then be roughly the same (maybe a bit extra friction) whether carried by one wire or two.

I wonder what the idea is? Did the two wires take different routes - maybe one port, one stbd? So that a single piece of shrapnel would likely break only one.
I can't really think it would be because a wire might just wear out or fail unexpectedly. For aircraft I'd hope everything was designed with a good safety margin and I think preventive maintenance is standard.

Oh, just realized you said this was the 707 not the VC10, so they shouldn't have got shot at! I think 707 was even before hijacking and bombs on civil planes. Perhaps there were other local failures they worried about? Wasn't it the Comet that had bits fall off due to metal fatgue? I don't suppose two cables to the elevators were much use if the whole tailplane fell off, but it must have been about then when the 707 was being built.

 

[A.T.]

Pulley 2 has mechanical advantage. You pull 2x the total weight and the mass moves 2x the distance that you pull.

I read this a few times and I'm not so sure. Mechanical Advantage is load/effort and the effort is actually twice the weight (= 2T) so the MA is actually 1/2. However, the Velocity Ratio VR is 2.

This is exactly what I said. I just stated the effort and the velocity ratio.

 

[sophiecentaur]

This is exactly what I said. I just stated the effort and the velocity ratio.

I'm not sure about that:

Pulley 2 has mechanical advantage.

That statement implies more than the wording implies. "Advantage", to most people and to all non - physicists, implies it's 'better' in some way. There was a missing value of 1/2 in the post, which would have cleared it up. Not everyone can cope with minimalist statements and this is not a Higher Physics thread.

 

[sophiecentaur]

Oh, just realized you said this was the 707 not the VC10,

707 was mechanical; VC10 was all electric controls. If all four engines failed there was an emergency air turbine that was lowered to provide the power. ELRAT was Emergency Landing Ram Air Turbine and Roger was more than happy with the fact it was there for him, if necessary!

 

[A.T.]

"Advantage", to most people and to all non - physicists, implies it's 'better' in some way.

I don't like the term "mechanical advantage" either, because "better" depends on what your goal is. If you want to increase speed, then pulley 2 is "better".

Not everyone can cope with minimalist statements

You have cut off the explanatory next sentence.

 

[sophiecentaur]

You have cut off the explanatory next sentence.

I should have left it in, you're right. But it doesn't actually say what's necessary. Is the 'advantage' in greater load or greater distance? MA is not defined in terms of distance. Again, it seemed too sparse to be useful (to someone who doesn't already know the answer).
My feeling is that if one wants to get involved with a thread on an elementary topic, one should be as helpful as possible. Sparse Maths is a serious turn off for many people. And I assume their presence is welcome on PF (?).