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Simple Torque from Gravity Problem

  1. Feb 11, 2012 #1
    I can't figure this one out, but I bet someone here can really quickly. I hope I can explain it right.

    In 2D, if a shaft mounted wheel is connected to a weightless arm and at the end of the arm is a mass then the wheel will rotate because of gravity until the mass loses it's energy.

    Now lets say we put 8 of these arms around the same wheel and we are given full control over the length of each arm at any given time (dynamically). Then, decrease the length of those on the right side and increase the length of those on the left side so that we still get a torque on the system. If would seem to be possible to change the lengths so that the potential energy would remain constant, yet we would get a net torque. (By having symmetrical lengths across the x axis, but non-symmetrical lengths across the y-axis)

    Since we have full control over the lengths, couldn't the lengths change accordingly as the wheel is rotating as to keep the net torque on the left side of the wheel causing accelerated rotation?
  2. jcsd
  3. Feb 11, 2012 #2


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    welcome to pf!

    hi sacher! welcome to pf! :wink:

    let's start with the simplest case … a wheel with one cylindrical hole across a diameter, and a rod we can slide along it

    with the rod horizontal and with the brake on, we slide the bar to the left, then we release the brake …

    the wheel turns 180°, until the rod is at the right …

    now we apply the brake again, slide the rod through, release the brake, and so ad infinitem :smile:


    i] this assumes no friction … in practice, because of friction, the rod won't quite return to horizontal

    ii] even if we add more rods (at different angles), pushing them out when they're exactly horizontal, the same thing will happen: the wheel will turn to the point where the centre of mass is at the original height

    (and if we push a rod out when it's above the horizontal, then we're supplying energy by doing so, aren't we? :wink:)
  4. Feb 11, 2012 #3
    Yes, but if we are pushing a rod out not only when its horizontal but also pushing an equal and opposite arm outwards (across the x-axis) wont it take in theory zero energy? So, the arm that originally starts on the left side would be at it's longest and then as it rotates around the wheel would be it's shortest on the right side. If this arm had an equal and opposite arm across the x-axis after it rotates 1/8 of the way around the wheel, wouldn't PE stay constant and all it would take in energy is the friction to push the arms in and out? Yet, there is still a net force at all times on the left side that would surely be greater than the resistance from friction.

    If you draw a picture of this scenario where there is an arm that is the longest at the left and there is an arm opposite on the right that is the shortest, and in between these arms the arms are a little longer than the one on the right increasing but never as long as the one on the left, making sure to add arms opposite across the x-axis.

    Now using this picture there is a net torque on the left side. keeping the lengths in this arrangement, adjusting the length dynamically to match this picture at all times as the wheel rotates.

    I just don't see how the center of mass will ever be anywhere other than on the left side. (unlike the example you gave)

    I guess I'm coming to the conclusion that not only is the system able to run indefinitely (without friction) but wouldn't it accelerate?
    Last edited: Feb 11, 2012
  5. Feb 11, 2012 #4


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    it doesn't matter how far left or right it goes …

    what matters is whether it goes up or down, and if it stays where it is, there's no extra energy
    yes, of course if you ignore friction almost any system will run indefinitely

    but it certainly won't accelerate
  6. Feb 11, 2012 #5
    But if the center of mass is to the left of the shaft and the shaft is held stationary, then the wheel will rotate around the shaft.

    I agree that potential energy doesn't increase or decrease here, that's the point. I understand that the total energy can't increase or decrease as this is a law. I just can't explain how this scenario would not accelerate. If you always have the force of gravity, and you always have an object's center of mass off center (in the same direction), how does this create rotation without any energy input?

    Last edited: Feb 11, 2012
  7. Feb 11, 2012 #6
    In order to have gravity affect the rotation of a wheel, it must be out of balance. In order to make a wheel out of balance, only a horizontal mass movement is necessary (almost negligible energy needed to do this)

    At this point, the wheel will rotate and potential energy will be converted to KE. Now, if an equal and opposite mass counteracts the PE lost by increasing PE in the opposite direction (across the x-axis) you would think that the wheel then wouldn't rotate right? I don't think that's the case here because the PE is increased on the same side of the wheel keeping the center of gravity out of balance.
  8. Feb 12, 2012 #7
    The way you have drawn the wheel, it will indeed accelerate until the extended rod is at the bottom, after which the extended rod will slow the wheel just as much and you will have gained nothing.

    The only way to gain something is to retract the rod when it is at the very bottom. However, that requires energy.
  9. Feb 12, 2012 #8
    I dissagree, If everything that was said by Tim is correct, I believe this system will not move.

    Exactly correct, the rods will retract as the wheel rotates around. The rods will change length to always be in the off-round shape as shown in the picture.

    I'm guessing if I went through every increase and decrease in PE somehow this system will cancel out. I just don't see how the wheel will not rotate if the center of gravity is off balance. (Or is it not possible to make the wheel off balance without more energy being putting into the system? In which case it still will not rotate)

    Last edited: Feb 12, 2012
  10. Feb 12, 2012 #9


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    This is on the fringes of a perpetual motion idea. If the wheel is to move against any friction at all or gain / change Kinetic energy then some work will need to be done. Sliding rods etc. will involve work against friction.

    There are an infinite number of arrangements that will not change the PE and some of them will involve setting up an imbalance of torque. Changing the torque situation need have nothing to do with the energy situation - until the wheel is allowed to move. I get the impression that someone is looking to find a Paradox here but, as long as Force and Energy are not confused with each other, there is nothing strange about the set up.
  11. Feb 12, 2012 #10


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    See this page from "the museum of unworkable devices": http://www.lhup.edu/~dsimanek/museum/overbal.htm

    For the explanation of why this doesn't work, go about half way down the page to the paragraph "To illustrate what really happens, consider an experiment that anyone can perform..."
  12. Feb 12, 2012 #11
    Hi sacher,

    Yes, it accelerates because it is gaining rotational kinetic energy. The only valid question is "where is the energy coming from?" The answer is evident from your drawing.

    First, we agree that the wheel is rotating counterclockwise, right?

    Second, note that the weights are being pulled uphill (along the direction of the rod) as they rotate from the 9 o'clock position to the 3 o'clock position. That requires movement against a force, or work. Similarly, they are being pushed uphill (along the direction of the rod) as they rotate from the 3'oclock position to the 9 o'clock position. Again that requires work. So it is no mystery why the wheel gains (kinetic) energy.

    - Regards
  13. Feb 12, 2012 #12


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    Sacher, you are making a classic PMM crackpot error: making the system complicated enough that you are no longer able to analyze it and believing that within your misunderstanding lies perpetual motion. Sorry, but it doesn't and the more complicated version don't say anything different from the simple version you were given at the beginning of the thread. I'll re-state it before locking the thread:

    If you use one rod with two weights, start from horizontal and ignore friction, the wheel will turn exactly 180 degrees before stopping again horizontally.

    Moving the rod in between only makes the scenario more difficult to analyze as you are stealing some of the rotational kinetic energy to convert back to potential energy by lifting the rod. Adding more rods just makes it even more difficult to analyze.

    You cannot defeat conservation of energy by making a device that you are unable to analyze.
    Last edited: Feb 13, 2012
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