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23 November 2024 22:33
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Question |
| Asked by: |
Jesse |
| Subject: |
conservation of energy in a gyroscope |
| Question: |
I am trying to understand the workings of a gyroscope and I'm having trouble with this thought experiment. Can someone help me out?
a gyroscope is set up with one end in a fixed position, yet the whole apparatus is able rotate about that point in any direction(up,down,clockwise,couter-clockwise).
at start the gyroscope is brought up to speed and set with some angle to the ground, lets say 45 deg. above level and held so that it will not precess...
so the starting state has energy equal to the potential energy of the apparatus due to it's height above the ground plus the rotational energy in the flywheel.
When the gyroscope is released, gravity will cause it to precess, presumably without lowering the apparatus below it's angle of 45deg.(?)
in this state the height of the apparatus stays the same, so there is no change in potential energy. I assume then that the total rotational energy stays the same as well, though this time it is divided between the energy in the flywheel and the rotational energy of the entire apparatus.(?)
Finally a force is applied to the apparatus to rotate it in the direction opposite to that caused by procession. This causes the gyroscope to rotate down toward the ground. lets say the force is applied until the apparatus is at 15deg. relative to the ground, and then the force is removed. When this happens the gyroscope will go back to precessing and the system will be in most respects similar to state 2. however, some unknown quantity of energy has been added by the application of force, and the apparatus is now lower than it was, so some of the potential energy also has to be accounted for.
Where does this energy go?
The only thing I can think of is that the flywheel would have the gain some energy, but everything I've read assures me that that is impossible.
hope this is clear, and thanks in advance to anyone who can shed some light on it.
-Jesse |
| Date: |
23 November 2004
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Answers (Ordered by Date)
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| Answer: |
Nitro MacMad - 25/11/2004 06:29:31
| | | Dear Jesse,
Nice question, though I regret, assuming you are asking if the precessional motion of a gyro in the system you describe could produce “over unity” “perpetual motion” and such like, the answer (in the form you describe at least) is no.
The following is not easy so steel yourself, make yourself comfortable and have the courage to ask further of others on this site if my version stumps you like it nearly stumps me.
Nitro’s first law must never be forgotten (“a gyro will precess every damn force applied to change a gyro’s axial angle - not just the force that you first thought of”). You must also never forget that Newton’s third law still applies to anything that does not have enough spin speed in relation to its precessing force to have Nitro’s first law apply.
Assuming your gyro’s “fixed position” allows, the gyro axial angle to freely “hinge” vertically and rotate horizontally…...
1. The 45 deg. angle at which your gyro is released will be maintained with little or no starting drop. This is provided that the relationship between the gyro’s spin/mass is high enough and its lever length (from pivot to gyro centre) is short enough to cause a precessing force that is within the precessing ability that is created by the above spin/mass. Otherwise the gyro will vacillate between being a gyro and a non rotating mass (Nutation) thus, as spin reduces (or the precessing force increases), it would behave more like a non spinning mass and less like a gyro.
The only cause of the slow droop of the classic Eiffel tower precessing gyro’s angle is friction on the horizontal axis. This is mostly from friction between the ball on the end of the gyro’s cage and the top of the toy tower - though a tiny amount of air friction on the horizontal plane exists, and minutely affects it as well.
Imagine a frictional force arrow pointing against the direction of precession and (using Nitro’s law and the fact that the leading edge of rotation of a precessing gyro gives the direction of its 90deg. displaced precession) it will be seen that the tiny friction of the bearing on the horizontal plane will be precessed into an equally tiny downward motion gradually causing the gyro to droop.
Assuming a “perfect gyro” (Oh! I wish!) With gyro bearing friction and air friction magically non-existent - the precession caused by a force applied to “gravity precess” or as above, “frictionally reverse precess” or (its mirror) to “force precess” the gyro’s axis will not slow or speed the spin speed of the gyro… thus….
2. A precessing gyro can most usefully be regarded as a system in equilibrium. (You need to disregard friction here). Discounting the energy you have put in to spin the gyro you will have put in energy in lifting the gyro to its 45deg. start angle above horizontal. Should you try to make the precessional motion of the gyro to drive something; any energy you take out of the precessional rotation will have the effect of applying a force in the opposite direction to the gyro’s direction of precession. This will (see Nitro’s law about any force applied to alter the axial angle of a precessing gyro’s causing a 90deg.displaced reaction) cause a droop proportional to the energy taken out. If this opposing force is removed after a small droop, the gyro will maintain the caused lower precessional path unless a further force to change the axial angle is applied. Any force applied in opposition to the rotational direction of a precessing gyro, therefore, far from adding energy to the system you describe, actually takes energy from it, causing the angle of the gyro axis to droop as the kinetic that was put in by your lifting the gyro to its 45deg start angle is taken out.
3. The energy (not much as it didn’t take much to raise the gyro 45 deg prior to its release) was taken out by (effectively) pushing against the axis of the gyro in the direction opposite to its direction of precession. This horizontal pushing force is precessed (Nitro’s law again) through 90deg to appear as the downward droop in the gyro axis.
If conversely, the gyro was pushed horizontally in the direction of its precession, (force precession) its axial angle to horizontal would rise (Nitros law again) and you would be then putting energy into the system. Just the same as if you had lifted the axis of a non spinning gyro straight up to 45deg above horizontal and thus added to its “Kinetic energy.
Basically (though I will always try to listen to arguments):- You cannot get something for nothing - with what you describe.
Kind regards
NM
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| Answer: |
Brett Walton - 01/12/2004 14:33:51
| | | I can see where you are coming from with your answer, but I'm still not quite getting my head around it.
When you apply a force, you're not doing it over a distance or times a velocity, since the frame will stay fixed under the application of the torque, so no work is done. From conservation of moment of momentum I can see why the rotor will rise and fall, but in doing so it is gaining or losing potential energy. But no work is being put in, and I can't see where a corresponding rise or fall in kinetic energy takes place.
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| Answer: |
Nitro MacMad - 06/12/2004 19:34:33
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Dear Brett
Sorry not to be clearer. It is easy to confuse force with energy when talking about gyros especially when we talk about “force precessing” a gyro when it should really be called “energy precessing”. Moreover, to add to the confusion, if gimballed with its mass central, a gyro will precessionally move vigorously around its centre of mass from the application of force to alter its axial angle, with very little energy input apparent. Yet you will still have to put energy in to get energy out - as far as I have been able to tell (Damn!).
An example of force precessing is when (assuming the usual arrangement of a gyro precessing happily around its Eiffel tower under gravity) extra “energy” is put into the arrangement by pushing the gyro faster around its precessional path than it would naturally precess.
This extra energy input on the horizontal rotational plane results in the gyro precessing this applied “force” over distance acting to change its happily precessing equilibrium into a 90deg. displaced, vertical torsional reaction which raises its axial angle. By raising its axial angle it raises its centre of mass and this increases its kinetic energy.
If the same amount of energy put in by the “force precessing” were taken out of this precessing gyro arrangement (by making the gyro axis drive something, for example by pushing against your finger) the axial angle would droop back to its starting angle to the horizontal. Thus, in this example at least, energy in certainly seems to equals energy out (Like I said; Damn!).
Kind regards,
NM
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| Answer: |
Jesse - 06/12/2004 20:06:11
| | | thanks a lot for your lucid answers. but there is still one thing I'm not clear on.
Pushing the gyro in the direction it is presessing clearly adds energy to the system, but is it not also possible to add energy to the system by pushing the other way?
I understand that the force of friction is an energy loss because the direction of the force and the direction of motion are opposite, so the outside world is doing negative work on the system. But if the push was hard enough to override the precession and the gyroscope was forced backward, then the force applied to the system would have been exerted in the same (the new) direction of motion, and should therefor be possitive work and a gain in energy for the system.
but if all that is true, where does the energy go? a force agains the direction of pressesion would lower the gyroscope, putting the system in a lower energy state. What am I missing?
thanks again,
-Jesse
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| Answer: |
Dan Harrington - 07/09/2005 00:47:30
| | | You can find some amazing things about gyros with a toy gyro and a chemical balance. Weigh the gyro sitting motionless alongside its pedestal, rotating vertically on its pedestal, and precessing horizontally on its pedestal. Since all weights turn out to be equal the only vertical force acting is gravity. Thus while precessing the gyro is held up by gravity alone. Apparently gravity only traverses an angular momentum along the rotation axis.
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