So this isn’t quite the monkey on a rope problem but I figured it’d get your attention. My question is what’s the mechanical advantage in the following scenario:
I’m in a rock climbing harness at the gym with attached to a rope that’s looped over a single fixed pulley on the ceiling. I have the free end in my hands and start to pull it toward me. Then I go up. If it weren’t against the rules I could go all the way up. Assume friction is not negligible, the rope is dynamic but that’s not too big of a deal (I don’t think) because all of the slack comes out pretty quick under load.
The problem is I can’t understand where the advantage is. I know there has to be one because it’s ridiculously easy to do but when I’m stationary and another climber is on the rope I can’t pull them up. I can climb the same rope with just my hands (no feet) but it’s significantly harder than just pulling my harnessed self up. I asked two different physics professors, one said there was definitely an advantage but did a poor job explaining it, and the other said there wasn’t one. Clearly, I think there is one, but what is it! Oh great minds, help me
(standard disclaimers about the rope being light compared to you, friction being low, etc.)
You are exerting a force equal to half of your weight on the rope you’re pulling. Since the pulley at the top has low friction and the rope is light, there’s the same amount of tension everywhere in the rope. The rope is attached to you at two points: At your harness, and at your hands where you’re pulling. At each of those two points, it’s exerting a force on you equal to half of your weight. Half your weight plus half your weight is your full weight, so the force is enough to lift you.
If you had a more complicated block-and-tackle setup, the mechanical advantage would be equal to the number of vertical lengths of rope. For instance, if the rope were anchored at the top, went down to a pulley attached to your harness, then up to a pulley on the ceiling, and then back down to your hands, you’d have a mechanical advantage of 3, and would only need to pull with a force equal to 1/3 your weight.
Firstly, is it actually a pulley at the top, or is it just a cylinder or carabiner (i.e. something that the rope slides over instead of rotating with it). I’ve never seen the former at a climbing gym, and I thing it matters to the analysis.
Assuming that it is not a pulley (or is a pulley with a lot of friction on it), the capstan equation tells you why you can hold yourself (or someone else) up with much less force than your weight.
When you’re climbing the rope just using your hands, and you just want to stay still, you have to hold the rope tight enough that you’re carrying 100% of your weight.
If you’re pulling the other end, looped over the top, then by the capstan equation, you only need to pull exp(-mu * phi) of your weight to hold still, where mu is the coefficient of friction between the rope and the top thing, and phi is the angle of contact. Sometimes the rope is even looped twice around the top thing for an extra factor of exp(-2pimu).
Of course, in order to get to the top (assuming that you are the only non negligible mass in the system) you have to expend the same total energy either way, but I think “perceived difficulty” is probably improved significantly by being able to exert less force while resting. I don’t know – my analysis gets kind of hand-wavey here .
(also and what Chronos said. In fact you’re probably better off listening to him than me.)
Actually, the more I think about this, the less comfortable I am with this analysis. Yes, you are being held by the rope at two points, but those two points both originate with you – the pulley just changes the direction of the force.
My understanding was that in a completely frictionless world, the mechanical advantage of a pulley system is determined by how much the load changes height when you pull the holding rope (so to balance mgh = Fd). In the case of single pulley, pulling the holding rope down by a small amount moves you up by the same amount for a mechanical advantage of 1 (and since d = h, F = mg).
If you had the more complex system with a pulley at the top, passing a rope down through the harness, and then going to to fix to the ceiling again, pulling the holding rope down by a small amount would lift you by only half that amount for a mechanical advantage of 2 (i.e. h = d/2 -> F = mg/2).
I don’t think it matters that the force is being applied by the same object that is the load, but maybe I’m off on that.
At any rate, friction definitely plays a significant role in rock climbing. A belay device is basically just an object take advantage of friction through the capstan equation to reduce the required loading force. So I don’t think a frictionless analysis is likely to be a good approximation of reality. (For that matter, in long belays the weight of the rope contributes noticeably to the force as well.)
For a single pulley at the top, with one end of the rope tied to you, and you pulling on the other end, if you pull the rope down by a foot, you only go up a half of a foot.
Say you had 100 feet of rope, so you’re hanging 100/2 = 50 feet below the pulley. Pull the rope by 1 foot, and you’ve got 99 feet of rope (plus the dangling 1 foot). So now you’re 99/2 = 49.5 feet below the pulley.
I tried this once.
I was about 220lb (diet starts next week) and the skinny guy at the other end could not have been more than 190lb.
When I reached for a handhold, missed, and slipped and dropped, he was yanked about 5 feet into the air. The only reason he did not keep going was a practical one - the rope was looped through a ring at the top of the escarpment, and running on a thick folded blanket up there (to prevent chafing and abrasion). So friction stopped him… and me.
If it’s a perfect, very low friction setup simple pulley, a person should be able to haul up a load lighter than themselves. If you can’t haul up your buddy, he’s the fatty. Most of those gym climbing walls rely on friction to prevent minute differences from allowing you to re-enact the bricklayer’s song -they use a rope through a loop, not a nice free-wheeling pulley.
If you’re bored some day, try the experiment. Both stand on the ground, you pull him up. If you can’t (but your arms are strong enough to lift yourself) walk over to the weight rack, tie a 20-lb weight through your belt, rinse and repeat. The weight difference required before you stay put and he goes up is the resistance of friction. If you two are balanced, then you can see-saw. tighten the rope standing on a chair. Step off while he jumps, and he’ll be hanging in mid-air. you jump up, he lands, repeat. Just like a teeter-totter.
If the pulley is fixed to the ceiling and you pull a rope over it down one foot, the other end goes up one foot. Only the direction has changed, and you’ll have a mechanical advantage of one. So saith every physical science book ever.
If you are hanging from one end, and holding the other end, and you pull the end you are holding down 1 foot relative to you, you have shortened the amount of rope between the end tied to you and the end you’re holding by 1 foot. Since the rope is doubled, you only rise 1/2 foot.
Nope. I just tried it (didn’t need to, I knew how it would come out) with a string over a nail and a ruler. When one end of the string went down one inch, the other went up one inch. To shorten the distance of the output of a simple machine, it must have a mechanical advantage higher than one, and a fixed pulley does not do that. A fixed pulley with a rope over it does not change anything except the direction the rope is pulled. Try it and see!
Ah, but when the load pulls on the string its a different case to something else pulling up the load… When the load pulls 10 “of string down, the load moved up 5”. Thats your required ratio . 2 to 1.
1/2 foot down on one side and 1/2 foot up on the other is still a mechanical advantage of one. The distance put into the machine is the distance out. The force pulling down will equal the force pulling up if the person is just hanging there. The force required to lift an object with a rope and a fixed pulley will be a bit greater than the weight of the object lifted, due to friction.
Isilder, to get a the 2:1 reduction you mention, one must use a movable pulley. A rope tied to the ceiling, run under a pulley attached to the monkey guy, and then back over another pulley hooked to the ceiling would provide that. The second pulley only provides a change of direction so the rope can be pulled down. Another way would be to have someone else sitting in the rafters pulling up on the rope after it goes under the monkey guy’s belt pulley. Somebody pulling himself up by using a rope and a single fixed pulley will need to expend the same energy and force needed to climb a single rope. If the OP says it’s easier, it’s because the setup allows a better technique, not because of a mechanical advantage.
If you start with 100 feet of rope, you’re hanging 50 feet below it. If you pull it all the way through so that you reach the pulley, you’ve pulled 100 feet of rope, and only risen 50 feet.
Contrast that with someone on the ground doing the pulling. He pulls 50 feet of rope to raise the hanging person 50 feet. That’s why the person hanging has a 2 to 1 mechanical advantage.
Your drawing is flawed. The rope with the word me would be four dashes up compared to the other side, not two. While the left moves down two dashes, the right moves up two. 2+2=4. Try it and see. I did (using a string with one inch marks), because at first glance your diagram seems right, but it’s not.
I count a total of 12 dashes worth of rope, in both pictures. I also count two dashes worth of rope hanging below me in the right picture. Am I mistaken on either of those counts?
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