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January 22

Does a second pulley make something heavy easier to lift?

Apologies, I've never been any good at math, so this may be a really easy question. It's not for homework or anything, I just got to thinking and don't have the materials on hand to test it at my desk.

Take this quick sketch. The man is using a pulley to hold an X-pound weight perfectly still. If I remember high-school physics correctly, the man must be exerting X pounds of force.

Now take this sketch. He's holding the same weight, but with two pulleys in between him and the weight. Is the weight any easier for him to hold? If so, is it possible to calculate how much easier?

Final question: If the pulleys were instead unyielding (like horizontal poles covered in grippy rubber), would that change the ease with which he could hold up the weight? --Aabicus (talk) 09:17, 22 January 2020 (UTC)[reply]

In both your images, the man has to pull with force X unless the second small circle imparts lots of friction. To convert to a pull of X/2 or X/3 or less, see Pulley. -- SGBailey (talk) 09:43, 22 January 2020 (UTC)[reply]

~~Friction doesn't matter if he's holding it perfectly still.~~ The pulleys just redirect the force, so the amount is still X, as you said.

In that linked article, specifically see the section Pulley#Rope and pulley systems. --142.112.159.101 (talk) 19:49, 22 January 2020 (UTC)[reply]

Of course it matters. Static friction is a thing. If the weight is X, he can keep it perfectly still by applying a force less than X. The tension in the rope then is different on the two sides of the pulley, and static friction along the pulley keeps everything in place despite the difference. -- Meni Rosenfeld (talk) 01:37, 23 January 2020 (UTC)[reply]

Whoops! Thanks. --142.112.159.101 (talk) 06:41, 23 January 2020 (UTC)[reply]

@Meni Rosenfeld: That is true if you're talking about keeping the weigh still, as Aabicus described it above (“[the man holds] an X-pound weight perfectly still”). It's because the friction works againt the movement, so it adds to the force applied, thus reducing it. However, if you consider lifting the weight, as OP said in the title (“make something heavy easier to lift”), then things work another way: the friction works against lifting and makes it more difficult, one needs to apply a force equal the sum of weight and friction to move the weight upwards. --CiaPan (talk) 10:22, 23 January 2020 (UTC)[reply]

You can use pulleys to make heavy things easier to lift with a block and tackle, which is an arrangement of the pulleys so that pulling the rope 2 feet lifts the object only 1 foot. That amplifies the amount of force reaching the object. Maybe that's what you were thinking of. See the article for explanation. 2601:648:8202:96B0:0:0:0:4FFF (talk) 22:34, 22 January 2020 (UTC)[reply]

See also Capstan equation, for a related scenario. AndrewWTaylor (talk) 09:20, 24 January 2020 (UTC)[reply]

Is eccentricity in disconnected graphs infinite?

Please see Talk:Distance (graph theory)#Is eccentricity in disconnected graphs infinite? --CiaPan (talk) 10:42, 22 January 2020 (UTC)[reply]