I 1:1 versus 2:1 Mechanical Advantage debate

[cmars]

TL;DR Summary

If a load (a person - 200 lbs.) is on a rope tied in with a knot and the rope goes up to a fixed pulley and back down to the persons hands and he hauls. What is the mechanical advantage in that system?

The rope is tied into the person (the load of 200 pounds) and the rope goes up from the person to a fixed pulley and back down to his hands. He hauls the rope to suspend himself in the air. What is the mechanical advantage of the system? The person will indeed only have to lift half of his body weight (roughly 100 pounds) because he now lessened the load by that same amount. This APPEARS to be a 2:1 because he can hold himself with half the force, but my question is: is that mechanical advantage or is he just equalizing tension on both sides and it’s still a 1:1 MA.

This has been a highly debated topic in my fire department that does technical rescue. I need a real physicist to examine the numbers and system and explain why it is a 2:1 or 1:1.

 

[jedishrfu]

Your self-haul with a single fixed pulley has a 2:1 MA in the ideal case: half the force, twice the rope.

The confusion clears the moment you draw the free-body diagram and remember that the rope lifts you at two contact points (harness and hands).

However, for someone pulling you up from the bottom, it's a 1:1 MA, as they must pull your full 200 pounds.

 

[Filip Larsen]

To put it short, with a 2:1 ratio you do half the load, but then have to do it twice as long to perform the same amount work. Or put another way, in order to lift a weight to a certain height you will need to provide the same amount of work.

In mechanical terms you would say that (reclaimable) work energy (e.g. Joule) equals distance (e.g. meter) times load force (e.g. Newton). From that you can also see that you can do same work by reducing the load force by some factor if you also increase the distance with the same factor. On top of that you would then also in practical systems need to consider frictional losses, so a more accurate rule would say reclaimable work energy equals distance times the sum of all the load and frictional forces, where frictional forces here are considered negative since they oppose the work you are trying to do.

 

[A.T.]

... or is he just equalizing tension on both sides and it’s still a 1:1 MA.

The tension is equal on both sides, and the two tensions have to add up to his weight, so it's only half the weight at his hands, and thus 2:1 MA.

Another way to look at it: To pull himself up by X relative to the ground, he would have to pull down 2X of rope relative to himself.

 

[sophiecentaur]

This has been a highly debated topic

A "debate" implies there's a matter of opinion here. There is only one answer (see above post) which is the same, whichever way you look at it. The same energy / work is needed to raise the CM of the person by x, however you do the job.

However if you use more pulleys then the masses of some of them will also need to be lifted and the actual MA gets less. Which is why there are two different terms to describe the situation. The MA describes the actual work needed and the velocity ratio (VR) describes the idealised geometry (lengths of levers, number of pulleys etc etc.) which ignores dead weight and friction.

I seem to be the only contributing member of PF to draw this distinction but, at school, we were all taught about this so what did they teach all you other guys?

 

[Filip Larsen]

I seem to be the only contributing member of PF to draw this distinction [...]

You mean the distinction where pulleys and ropes are not massless and pulleys has friction? In that case you must have skipped my post.

It is also not my impression that regular contributing members on mechanics issues should have a problem with pointing this out when relevant, but it is probably right that there is an overweight of problems regarding introductory mechanics that tend to only address idealized loss-less mechanical mechanisms and machines. For mechanical engineers going past introductory parts and into, say, machines and power links (mechanical or otherwise) then I would think friction, limits and non-linear effects quickly become a relevant part of the curriculum. At least it was in my university days when dipping my feet in mechanical engineering courses as part of my specialization into computational physics (like numerical study of non-linear effects in power systems).

 

[sophiecentaur]

It is also not my impression that regular contributing members on mechanics issues should have a problem with pointing this out when

We are, of course, basically in agreement about the Physics. However, my experience is that, in mechanics threads the only term that I read is Mechanical Advantage and it is used very often, inappropriately. I'd actually claim never to have read the term "velocity ratio" in threads about extremely practical issues. Moreover, just 'pointing out' something doesn't give a way to calculate / predict the effect of the difference in terms of efficiency - leading to the real need to decide on things like necessary motor power. Engineers need to account for losses as much as basic principles.

Mechanics is a topic that is often discussed by members who make a point of saying that they do not have advanced Phyisics knowledge. These are the guys who use the term MA in a way that implies it's always the right (or only) one. It's then a surprise when their conclusions are not what they expect. Progressing from the ideal to the practical needs some practical method to work with.

If you really have read explicit use of 'velocity ratio' (or equivalent) instead of MA, I'd appreciate examples and quotes.

 

[Lnewqban]

What is the mechanical advantage in that system?

Welcome, @cmars !

Perhaps it is not completely correct to call it mechanical advantage.
The fixed pulley, which is the only device in this man-rope-pulley system, offers no mechanical advantage by itself.

Please, see:
https://en.wikipedia.org/wiki/Mechanical_advantage

What is happening in the discussed case has a different nature, I believe.
We could replace the man with a spring and measure its internal tension via its deflection.

 

[Filip Larsen]

I'd actually claim never to have read the term "velocity ratio" in threads about extremely practical issues.

A quick search for "velocity ratio" seems to give a decent amount of hits, but not as many as for "gearing" for example (which is the term I usually use), so you are probably right that "velocity ratio" is a less often used term than other equivalent or similar terms.

I agree that using the concept of velocity ratio, which almost literally says velocity-in and velocity-out can have a ratio, seems a more useful concept for people who would just like to understand the basics of gearing without necessarily having to understand the physical principles behind why this is so. One can hypothesize that at PF there perhaps is a fair degree of anticipation that laypeople tend to come here to ask "why" rather than "how" and we thus tend to prematurely go directly to the "why" answers first even if "how" would have been sufficient.

However, I do not really see how velocity ratio alone can explain the OP question, since here question specifically is about the reduced sense of "weight" when pulling the rope. How can you explain that a purely geometrical determined velocity ratio also "applies" to load forces (and thus sensed weight in this case) without essentially involving energy = force x distance and conservation of energy?

 

[A.T.]

The fixed pulley, which is the only device in this man-rope-pulley system, offers no mechanical advantage by itself.

It's usually the entire system which creates the MA, not a single pulley or gear.

Perhaps it is not completely correct to call it mechanical advantage.

Please, see:
https://en.wikipedia.org/wiki/Mechanical_advantage

What is happening in the discussed case has a different nature, I believe.

From the wiki:

"Mechanical advantage is a measure of the force amplification achieved by using a tool, mechanical device or machine system. The device trades off input forces against movement to obtain a desired amplification in the output force."

That fits this system, because the man applies half his weight over twice the distance that he is lifted.

 

[A.T.]

How can you explain that a purely geometrical determined velocity ratio also "applies" to load forces (and thus sensed weight in this case) without essentially involving energy = force x distance and conservation of energy?

I think this is similar to explaining the concept of torque from pure statics, without invloving displacment and work / conservation of energy. We had some threads on this:

I Post in thread 'A Question About the Physical Explanation Behind Torque'

Jun 8, 2024

I'm looking for an explanation for why increasing the distance between a force and the center of rotation makes the force better at causing rotation (torque)

There is a static way to understand torque based on linear forces only, by considering simplest truss structures (beams connected with torque-free joints), and solving the equilibrium equations for them.

Also note that torque is not necessarily about rotation. You can accelerate two masses in opposite directions on parallel straight paths. Nothing is rotating here, but the masses gain angular momentum, and the rate of that...

• A.T.

Post in thread 'Law of the lever without (infinitesimal) displacements'

Feb 6, 2018

Is it possible to derive the law of the lever without using any displacements, not even infinitesimal or virtual ones?

You can model the lever as a 3 node truss (triangle as the simplest rigid structure) to derive the static equilibrium, without any mention of work or torque.

• A.T.

Post in thread 'Law of the lever: Conservation of energy or angular momentum'

Sep 19, 2015

So I wonder if the derivation using conservation of energy only works coincidentally, because energy and torque share the same unit.

You can derive the static lever law without invoking the concept of torque, using only linear forces on a truss structure. There were several threads on this here.

• A.T.

 

[sophiecentaur]

A quick search for "velocity ratio" seems to give a decent amount of hits, but not as many as for "gearing" for example (which is the term I usually use), so you are probably right that "velocity ratio" is a less often used term than other equivalent or similar terms.

I agree that using the concept of velocity ratio, which almost literally says velocity-in and velocity-out can have a ratio, seems a more useful concept for people who would just like to understand the basics of gearing without necessarily having to understand the physical principles behind why this is so. One can hypothesize that at PF there perhaps is a fair degree of anticipation that laypeople tend to come here to ask "why" rather than "how" and we thus tend to prematurely go directly to the "why" answers first even if "how" would have been sufficient.

However, I do not really see how velocity ratio alone can explain the OP question, since here question specifically is about the reduced sense of "weight" when pulling the rope. How can you explain that a purely geometrical determined velocity ratio also "applies" to load forces (and thus sensed weight in this case) without essentially involving energy = force x distance and conservation of energy?

Fair enough but “gearing” is only a part of it and so would pulleys and levers, which have dead weight and friction built in. VR is an actual variable with a value. MA, likewise has a value and is the one which counts in real machines. Even in the simple example in the OP, friction in the pulley can make a difference in the two scenarios, giving a possible difference in the results.
In Electrical Engineering we use complex Impedance in all but the simplest problems. The idea of Reactance is always there and PF acknowledges it in every relevant thread. Why? Because it is a way to produce reliable answers.
You may say that’s just being over-smart but where is that ‘too clever’ when PF discusses machines?
BTW, was your search for VR on PF or a general google?
Machines deserve a more considered treatment, imo ; not just arm waving to acknowledge that they aren’t perfect due to friction.

PS If the rope just passed over a tree branch instead of a pulley, things would be different. Same VR!

 

[Filip Larsen]

BTW, was your search for VR on PF or a general google?

Just here on PF. I did try to search for the string "velocity ratio" (i.e. search with quotes) which did give fewer hits than if searching without the quotes (word search), but while most hits after a quick scan seemed to have the string highlighted in the summary a few of didn't for some reason so the number of posts containing the exact string might be a bit less than the number of search hits (alternatively, all hits did contain the string and the search summary just has a glitch somehow).

 

[sophiecentaur]

That's a relief @Filip Larsen , I couldn't believe PF never used it but I never bumped into it. I don't really like the term VR much because it assumes motion but I'm sure that many people are unaware of it as a real thing with real values.
I mentioned above the idea of using a tree branch and a rope. I just remembered doing that for climbing trees as a lad. The stiction was really handy, too but you had to be clever not to turn turtle with a foot in a loop and the other rope tail in your hand. Control essential.

 

[Filip Larsen]

To be honest, I wrote my reply to the OP question without properly understand that, as I read it now, the question aims to clarify confusion about holding your own weight up vs someone else holding you up (both via same pulley and rope, the rope end just change hands between the two), and also before before jedishrfu post #2 showed up in my browser.

Reading post #1 and #2 now, I must say that a FBD as jedishrfu mentions do seems to be the easiest/simplest way to resolve this specific question since both situations are static.

 

[Lnewqban]

It's usually the entire system which creates the MA, not a single pulley or gear.

That fits this system, because the man applies half his weight over twice the distance that he is lifted.

I respectfully disagree.
What we have in this case is a balanced mass hanging from two ropes, in successive steps.

The only function of the pulley is to keep the amount of tension of both ropes more or less equal between one balanced state and the next by proportionally reducing the length of one of those ropes as the other's length is reduced by muscular effort.

I see no multiplication of applied force but reduction of load to half or more by each pull.
If the firefighter pulls less that half his weight, nothing moves.

 

[A.T.]

What we have in this case is a balanced mass hanging from two ropes, in successive steps.

The steps are irrelevant to the mechanical advantage. You can replace the human with a motor that continuously pulls itself up by pulling a bike chain with a sprocket wheel.

I see no multiplication of applied force

The total force that pulls the motor upwards is twice the force that the motor applies to the chain. But the distance the motor is raised is half of the chain length that the motor has to pull trough.

Classic force multiplication, by applying a smaller force over a longer distance.

 

[jbriggs444]

Are we arguing about the definition of "mechanical advantage" or are we nattering about a choice of reference frame and where to draw the lines for "effort" and "load".

Isn't it irrelevant in any case since we agree about all of the forces and the net effect? The rest is just words.

[Me, I adopt the frame of the person and see the pulley as a class 2 lever pulling down on the ceiling with double force of the applied effort]

 

[jedishrfu]

But the mass isnt balanced. If a fireman on the ground holds the rope then he will feel 200 hundred pounds of force pulling on the rope at his end.

From the hanging fireman, he only feels 100 pounds pulling on his hands. For every foot he pulls the rope raises him 6 inches. Hence he experiences a 2:1 MA.

 

[sophiecentaur]

The rest is just words.

Words are at the centre of creating any mathematical model. Also, the correct words are important (basis of language).
This whole thread is about a problem that involves the observer's empathy with the participant - 'what would I experience if I was the fireman?' Those sorts of problems can. lead people to inappropriate conclusions. Third person, inanimate stuff is easier.

as a class 2 lever pulling down on the ceiling

Yes, I like that.

 

[sophiecentaur]

he only feels 100 pounds pulling on his hands.

.. . .. .. and 100 pounds on his foot or waist to push / pull them up. Easy to forget.
No paradox if you include all the forces.