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Whenever I pull an object from a rope or cable, the weight never really feel the same depending on the rope length, and I would like to understand why, and learn more about the maths behind this.

So, first let's say there is an object of weight m at the end of a rope of length l. This rope is vertical and needs to be pulled up. In this case, how are all the variables related?

In the second case, what and how would the relations change if the rope goes over some kind of bar so that the rope needs to be pulled down to bring the object up?

In other words, how does the length of the rope impact the required force to pull the object at its end?

EDIT: In both cases the rope is very light and does not change much with the length of the rope. For simplification we can say the weigth of the rope is always the same.

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  • $\begingroup$ IDK what kind of "rope or cable" you are using and I don't know the weight of your object, but real rope has mass. The lines used by mountain climbers or arborists can weigh several kilograms each. As you raise the object, you also are raising the section of rope that connects your hand to the object. The length of that section, and therefore its mass, becomes less as you raise it. Depending on the relative weights of the rope and the object, this may or may not be an effect that you can feel. $\endgroup$
    – Solomon Slow
    Commented Sep 2 at 15:16
  • $\begingroup$ @SolomonSlow the object could be any weight, and considering the rope is light enough to not really have an impact $\endgroup$
    – Apo
    Commented Sep 3 at 0:39
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    $\begingroup$ You said, "Whenever I pull an object..." That caused me to think that you were talking about an experience you had with a particular rope and a particular object. Sorry for my mistake. If you ignore the weight of the rope, then the length should not matter at all. $\endgroup$
    – Solomon Slow
    Commented Sep 3 at 11:25

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The human body is a terrible tool with which to understand Physics. It contains countless complications which turn simple equations into complex detailed patterns -- all in the name of being more efficient. Be very wary of using your intuition about the human body to intuit about simple machines.

In both of your cases, you must pull with the same force. However, you will find that doing the pulling uses different muscles in the two cases, and different muscles have different strengths. You may get the feeling like one of them is harder than the other, when they both require the same force.

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    $\begingroup$ And in this case, when pulling the rope down you are using your own weight to help to some extent, while that's no help when pulling up. $\endgroup$
    – gidds
    Commented Sep 2 at 13:50
  • $\begingroup$ @gidds I thought about that one, and it's an example of how tricky the human body can be. Your arms have to pull the same in either case, but the rest of your body feels less force. However, you also have to stavaliz yourself more in that case, which is very difficult even though the forces are smaller because you use different muscles $\endgroup$
    – Cort Ammon
    Commented Sep 2 at 17:43
  • $\begingroup$ @CortAmmon But what if the rope is longer and the object with the same weight, would the same force be required to pull the rope? (considering a very lightweight rope, so that its own mass does not really have an impact) $\endgroup$
    – Apo
    Commented Sep 3 at 0:36
  • $\begingroup$ @Apo Let me turn that around: why would it require any different amount of force? $\endgroup$
    – Cort Ammon
    Commented Sep 3 at 5:33
  • $\begingroup$ The only time the story will change is when they introduce pulleys. They do introduce mechanical advantage. $\endgroup$
    – Cort Ammon
    Commented Sep 3 at 5:35

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