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Energy Conservation Paradox

  1. Jul 19, 2014 #1
    Of course I am sure this is not a paradox but rather a problem in my understanding of the situation.
    But this question is bogging me, and I haven't found a satisfying explanation yet.

    Suppose you are floating in space, in an inertial frame. An object much more massive than you is moving with constant velocity at your arm's reach. You grab hold of the object, moving along with it.

    From the perspective of someone staying in the first inertial frame, the massive object transmitted a small part of its kinetic energy to accelerate you to its speed, and energy is conserved.

    From your perspective, you are not accelerating, instead the object deccelerates until its velocity is zero. It must have lost an enourmous amount of kinetic energy. The only force which could have done any work on this object is your hand's contact force, but (unless you are Hercules) this force is clearly not enough to stop this object. So how has all this energy been lost?


    I have a possible explanation, which is that when we grab hold of the object, we temporarily enter in a non-inertial reference frame, in which Newton's second law needs a new fictitious force term. This fictitious force will do the major part of the work on the object, and then will vanish when the object stops.

    But I am not sure what to think of forces appearing and disappearing at any instant.
     
  2. jcsd
  3. Jul 19, 2014 #2

    Dale

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    That is the correct explanation. Excellent job!
     
  4. Jul 19, 2014 #3

    TumblingDice

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    Hmmm... Why did you change your perspective from moving with the object to instead, the object stopping? Your entire body would indeed experience acceleration when you grab hold, and an accelerometer would confirm that you accelerated from one inertial frame to another.
     
  5. Jul 19, 2014 #4

    TumblingDice

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    I don't understand. A ficticious force makes the object stop w/o accelerating the observer who grabs hold? Can you explain more? What would the non-inertial ficticious force be called?
     
  6. Jul 19, 2014 #5

    Drakkith

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    How so? You can definitely feel an acceleration when you grab onto the object.
     
  7. Jul 19, 2014 #6
    Well, the reason I said I am not accelerating is that, if my coordinates are centered at my center of mass for example, then obviously my center of mass cannot accelerate in my reference frame. The other parts of my body will probably accelerate, though.

    I suppose the fictitious force and the force of the object deccelerating on me could be responsible for this, but I can't come up with a good argument.
     
  8. Jul 19, 2014 #7

    TumblingDice

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    When you grab hold of the object, your center of mass (and the object's, too) becomes significant. Not only will you receive acceleration of the object, you'll also receive angular momentum and begin rotating about your CoM. If you don't let go, I think you'll become the tail being wagged by the dog. :wink:
     
  9. Jul 19, 2014 #8
    "moving along with it" was a poor choice of words on my part. The second sentence is what I really meant.


    What do you mean by "becomes significant"?
     
  10. Jul 19, 2014 #9

    TumblingDice

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    When you grab hold of the massive object, both of you will experience two types of motion - translational and rotational.
    See here, and also the translational link in it:
    http://en.wikipedia.org/wiki/Rotation

    You've been thinking of the translational side where you both will share the same inertial frame. But, you both will also experience rotational acceleration proportional to your masses and in relation to the CoM of each. If you hold on, you'll become a single "system" that tries to establish circular motion about your combined CoM. As you're not very rigid, this could take some time...
     
  11. Jul 19, 2014 #10

    Dale

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    Both the object and the observer are subject to two forces, the fictitious force and the grabbing force. The net force is zero for the observer, but not for the object.

    Uhh, how about Fred?
     
  12. Jul 19, 2014 #11

    Dale

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    There is no reason to assume any rotation. The OP said nothing about rotation.
     
  13. Jul 19, 2014 #12

    Dale

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    Again, your instincts seem to be spot on. Your reasoning is correct, so I am not sure what kind of argument you are looking for to gain confidence in your correct reasoning.
     
  14. Jul 19, 2014 #13

    TumblingDice

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    I think you left out the somersault and the cartwheel forces. :smile:

    This just doesn't seem realistic. The observer who grabs on is going to undergo acceleration in a variety of directions. I don't see how net force could be observed as zero, but maybe you're speaking in a context I'm not familiar with.

    :rofl:

    I've always been partial to Bob.
     
  15. Jul 19, 2014 #14

    TumblingDice

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    The OP mentioned grabbing on with his hand. His CoM is much nearer to his bellybutton. He's going to go through more than translational acceleration.
     
  16. Jul 19, 2014 #15

    Dale

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    Please don't unnecessarily complicate the scenario. There is only one real force mentioned in the OP, and no rotational motion mentioned. Bringing them in is not helpful.
     
  17. Jul 19, 2014 #16

    TumblingDice

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    O.K. The description is valid in the unique, imaginary case that the object is able to move through the observers CoM. I understand the value of keeping experiments and explanations simple, however if I had posed this scenario, I would have wanted to know that there are significant ways that what was described as could be observed/explained would not be the case. At the least, I'd want the scenario 'adjusted' to provide a better understanding.
     
  18. Jul 20, 2014 #17

    TumblingDice

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    Boorglar,

    I'd like to offer to you what I would have liked to hear in these replies. Drakkith came on board in the same direction I began with, the observation of acceleration. We can still follow through with this and keep with DaleSpam's 'Wile E. Coyote' outlook.

    If you know your mass, you should be able to use Newton's second law as part of a calculation to determine the force that was applied. To do so you'd grab the object with one hand while holding an accelerometer in your other hand. If you keep your hands at rest with each other, you'll observe acceleration on the accelerometer. As Drakkith pointed out, you'll feel it, too, (just as if you accelerated in a car when the light turned green, your body fluids and masses will initially/inertially resist the acceleration).

    But more significantly, and I hope I'm correct here, you could use the observed acceleration on the accelerometer together with your mass to calculate the force applied using Newton's second law

    I'd like to believe this description is more appropriate to the spirit of the OP and Newton's second law than explaining it off with true acceleration as a fictitious, non-inertial force. I'm open to criticism and hoping I will be corrected where wrong, or put on track if this is viewed as causing confusion
     
  19. Jul 20, 2014 #18

    A.T.

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    First of all: "fictitious force" = "inertial force" (not "non-inertial force")

    The OP specifically asks about the energy in rest frame of the person, which is non-inertial, and therefore implies inertial forces. I don't see you addressing the work done on the big mass in that reference frame at all.

    The OP has already answered his question himself correctly.
     
  20. Jul 20, 2014 #19

    Nugatory

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    No law says that the center of mass of an object has to be inside the object, so there's nothing strange or unphysical about having the center of mass of one object pass through another object without colliding.

    It's easy to construct realistic geometries in which there are no rotational effects at all and the center of mass of the object moves directly through the center of mass of the observer. For example, the object could be a tube aligned with the direction of travel, the observer moves along the centerline of the tube and uses his two hands to grab two handholds on opposite sides of the interior of the tube.
     
  21. Jul 20, 2014 #20

    Nugatory

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    Yes, but what is that measured ("feeling" the acceleration is equivalent to measuring it with an accelerometer) acceleration telling us? It is telling us that the frame in which we are at rest is not inertial, so if we choose to work in that frame we should expect fictional forces to appear. And indeed they do: the second derivative of the object's position in this frame is non-zero, and despite the force we feel the second derivative of our position is zero.

    It's easier to analyze the motion of the observer and the object if we choose a frame in which the object is at rest (if its mass is large compared with that of the observer) or the center of mass of the object/observer system is at rest. But we are not required to use the "easy" frame, and the original question was whether energy will be conserved if we don't.
     
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