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I have a physics professor for whom I'd like to build a machine. Featured in a problem he is notorious for assigning students, originally from Kleppner and Kolenkow's "Mechanics," the machine causes all three blocks to accelerate, including the large block M1. A picture is featured below.

This problem features frictionless surfaces, massless ropes, and massless frictionless pulleys. I'd like to build the machine for the professor as a thank you gift, but I ask if anybody has suggestions for how to get relatively close to the frictionless and massless constraints above with objects I can realistically build with, e.g. teflon and steel for low-friction surfaces.

Thanks for any advice and suggestions!

enter image description here

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    $\begingroup$ Can you confirm if the interface between M1 and M2 is also designated as frictionless? You'd get a more dramatic forwards motion of M1 if there were no friction with the table, but some between M1 and M2. $\endgroup$ Dec 5 '17 at 16:13
  • $\begingroup$ @JonathanRSwift But that may not be in the spirit of the original problem. I see conservation of momentum going crazy here as M2 moves to the right, which would lead to m1 going left, except that m3 is constrained to not move horizontally relative to m1, and thus the momentum vectors get really interesting. $\endgroup$ Dec 5 '17 at 18:38
  • $\begingroup$ BTW, this problem, and solution hints, can be found at scienceanswers.wordpress.com/tag/kleppner , Problem 2.19 and 2.20 $\endgroup$ Dec 5 '17 at 18:40
  • $\begingroup$ I loved this page, even think to use some problems to build physics apparatus for student projects. $\endgroup$
    – Katarina
    Dec 7 '17 at 18:44
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I think I would be tempted to put both M2 and M3 on rails with wheels if they are inset into the respective blocks you should be able to keep a pretty small gap between the respective blocks which should preserve the visual effect, minimise friction and also has the advantage that you are constraining the movement to a single plane and you could also included stops and buffers to make it all easy to reset.

you could even use sets of three wheels like roller-coaster.

This approach would probably also make it easier to build as you could fabricate the blocks and use off the shelf precision ground steel stock for the rails and not have to worry about machining super flat surfaces.

maybe use small Vee pulleys for wheels running on round silver steel rails.

M1 could just go on fairly big wheels, again if they are inside a hollow block the exterior faces could sit just above the surface it sits on.

I would probably use something like dyneema cord for the string as it has a very high strength to weight ratio and is more flexible than steel cable or wire and so behaves well with small pulley systems and can be knotted. It's used for things like fishing line and climbing accessory cord and is available in a wide variety fo diameters and colours fairly cheaply

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Some issues are considered in other answers, I'll just add that you can cut part of the surface of M1 which is in contact with M2 and M3 and build in small cylinders so that you get rolling in stead of sliding, friction should be very small. Something similar to roller conveyor like in the link just be careful to align them with the surface to hide them, should be simple enough.

also maybe think about creating exchangeable parts so your professor can show students what happens when friction is or isn't acting.

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The massless ropes won't be a problem. Just use steel cable. Close enough. Get good ball bearings for the pulley and that should also be close enough. The problem will be the interface between m1 and m2. Even Teflon is probably too much friction. Probably the easiest is to just put wheels on m2 so that it rolls on m1 instead of sliding.

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  • $\begingroup$ Vertical wall friction around M3 is the most significant problem to be alleviated $\endgroup$ Dec 5 '17 at 18:41
  • $\begingroup$ @CarlWitthoft I suggested using cylinders, it could be done on the vertical surface as well. $\endgroup$
    – Katarina
    Dec 7 '17 at 12:39

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