you have created potential angular momentum which is released when the bob is released - this is same as hitting the pendulum/bob - this is not the same as trying to start it from rest ... your logic is faulty again.
Hold the axle of your spinning bike wheel (one hand supporting each end) so that the axle is horizontal. Now, change the orientation of the axle by rotating it in the horizontal plane. A hefty reaction force will be produced, trying to lift one end of the axle, and lower the other end. These vertical forces on the ends of the axle are at right angles to your horizontal movement of the ends of the axle, and therefore you do no work in rotating the axle in the horizontal plane. If you don't believe me, try it. And actually, this must be the case, because if the axle genuinely resisted your altering of it's orientation (rather than producing a force at right angles) then you would be doing work against that resistance. And then physics would be in real trouble. Where would the work done end up? As explained previously, it can't end up increasing the rotational speed of the wheel, because no torque can be transferred through the bearing to the wheel to speed it up.
The "reaction" force is perpendicular to the movement, so no work is done. However, your arms apply a horisontal force, and there is a horisontal movement (the one that creates the reaction force), so your arms are doing some work.
The work ends up as a rotation of yourself and the planet you stand/sit on.
By using a large gyro you can slow down the rotation of the earth and extract that as energy.
(d) It gains angular momentum as it accelerates towards the bottom of the stroke.
You have utterly failed to answer my question as to where the torque comes from for this angular acceleration, given that nothing physically touches the pendulum to accelerate it, and given that the overhead pivot is frictionless.
Sigh, the potential energy of the lifted bob - gravity accelerates the bob to the bottom - imparting angular momentum from the potential it was at previously - and giving it kinetic energy also...
So work was done by lifting the bob - and then released into the system when the bob is released - no different from hitting the bob to the same height
failure to see this simple fact is also a factor in failure to see the difficulties in starting a swing in a frictionless environment ...
Came a bit late to thread but, IMHO, OP's query is akin to the rocket fallacy, 'In space, what does it push against ?'
If you attach a crank to the top bearing of a slanted gyroscope, so precession winds crank, does work, then the gyroscope RPM gotta slow to provide it.
'Owt for nowt.'
Mechanically, you're looking at something akin to a swash-plate engine, see...
http://www.douglas-self.com/MUSEUM/museum.htm
... but prepare to lose yourself to awe & wonder...
Also, with apologies to several posters, bearings 'bear'. Magnetic, air, sleeve, ball, roller or needle, even recirculating, a bearing is still a bearing.
N
You contradict yourself. First you state "no work is done". Then, in the next sentence you say that "your arms are doing some work"!
I can assure you that no work is done, and that therefore your arms are doing no work. Work is force times displacement, where force is the force (or vector component of a force) in the same direction as the displacement. Therefore, if the force is a right angles to the displacement, as it is in this example, then no work is done, period.
I'm sorry, P99 but, IMHO, as soon as you connect any constraining mechanism to the precessing axle, and begin to extract work from system via that precession, the dynamics are totally changed.
Never mind 'perfect' bearings: If supplying that work does not 'draw down' on the gyro's flywheel spin, it's a 'perpetual motion' machine.
Sorry, as I said in my first post, akin to 'rocket' fallacy.
An external force is needed for the precessing force to exist. If you only support one end of a gyroscope, the external force is the gravity. In that case, any energy extracted from the precessing comes from gravity and the loss of potential energy (the unsupported end of the gyroscope will move closer to ground). The spinning RPM of the flywheel in the gyroscope is not affected.
There are two forces involved. One from your arms and one as a reaction from the gyro. The simplest case is when we keep the gyro (bicycle wheel) spinning axis horizontal during the movement. The reaction force from the gyro is then perpendicular to the movement, so no work is done. The force from your arms is not perpendicular to the movement, so work is done. It may not be obvious where this work is going, but it isn't stored in the gyro. We agree that the gyro will not spin up or down due to external forces if we ignore bearing friction. You are not a fix point in space, so the work goes to a change in the movement of yourself and ultimately to a small change in the movement of the earth.
No. If something moves in the direction you apply force, work is done. If you resist the tilting of the spinning axis or in some other way prevent the tilting, the only movement will be the "axle rotation in the horizontal plane", and since that movement is done by the force from your arms, work is done. The precession force from the gyro tries to tilt the spinning axis and does no work if you prevent the tilting.OK. Let's resolve this. If you are holding the spinning bicycle wheel, then all forces are provided by your arms. Your body and your muscles do no work at all when you rotate the axle in the horizontal plane. Period.
No. If something moves in the direction you apply force, work is done. If you resist the tilting of the spinning axis or in some other way prevent the tilting, the only movement will be the "axle rotation in the horizontal plane",and since that movement is done by the force from your arms, work is done. The preces sion force from the gyro tries to tilt the spinning axis and does no work if you prevent the tilting.
Imagine yourself in free space with a spinning gyro. Apply force (torque only if you don't want the gyro to move away) to change the spinning axis of the gyro and let go of it. The gyro will stay in the new spinning axis and you will rotate in the opposite direction of the torque you applied. Compared to the start you are now spinning. Did that require some work or not?
Back here on earth, we are discussing a quite different situation where the experimenter keeps the precession rotation of the axle in the horizontal plane, because he has the essentially infinite mass of the earth that allows him to do so, and thus no work is done.
You mean that the spinning axis will change without the hands applying any force/torque in the direction of the change? That 100% of the force/torque from the hands will be used to resist the precession force? That is not the case.
The precession force is a consequence of a movement created by an external force, This means that the net movement can never be 100% in-line with the precession force. Some work must be done (by the external force).
The discussion is directly related to the possibility of extracting energy from the earth's rotation. A gyroscope on earth with the spinning axis perpendicular to the earth axis will appear to rotate one turn every 24 hours.
Resisting that rotation is equivalent to changing the spinning axis of the bicycle wheel, and we disagree about whether "work" is involved or not.
Ok, now we know in detail why we don't agree.
Your argumentation is in the reverse order:
Since the energy can't go anywhere, no work is done, and therefore the force must be zero.
My argumentation is different:
A force is needed to move anything, work is done, in this case the energy is stored as an acceleration/deceleration of the earth's rotation.
There is no meaning to discuss harvesting energy from the earth's rotation as long as we disagree about this.
Ok, now we know in detail why we don't agree.
Your argumentation is in the reverse order:
Since the energy can't go anywhere, no work is done, and therefore the force must be zero.
My argumentation is different:
A force is needed to move anything, work is done, in this case the energy is stored as an acceleration/deceleration of the earth's rotation.
There is no meaning to discuss harvesting energy from the earth's rotation as long as we disagree about this.
Ok, now we know in detail why we don't agree.
Your argumentation is in the reverse order:
Since the energy can't go anywhere, no work is done, and therefore the force must be zero.
My argumentation is different:
A force is needed to move anything, work is done, in this case the energy is stored as an acceleration/deceleration of the earth's rotation.
There is no meaning to discuss harvesting energy from the earth's rotation as long as we disagree about this.
So how about extracting energy from the precession of the earth's axis of rotation? Is that possible? Are those gyro exerciser balls good for something after all
But alas, any earth-bound machine or idea to extract energy from the earth's rotation will fail. Always.
Things can be confusing but I am not here to add more.
In Cartesian space, you have motion on a straight line, kinetic energy is (1/2)m.v^2. This v is nothing but dx/dt.
Lagrange made lots of simplifications to the Newton's laws of motion. If we use a polar coordinate set for our measurements, dr/dt is similar to velocity but d(theta)/dt is angular velocity.
If r is constant (just assume for the time being), the kinetic energy is (1/2)I*w^2 where I has taken over m and omega has taken over from v. How?
Lagrange introduced the concepts of generalized coordinates and generalized momenta. In this case the generalized coordinate is theta and the generalized momentum is I*w
I see lots of confusion because you are going from Cartesian and polar coordinates without respect for the proper transformations. The transformation is provided by the Jacobian (it is not really that messy)
Precession of the earth's axis is rather small (think about 25000 years? a few degrees) but it is theoretically possible to extract energy from this precession.
Just like the precession of the top is caused by the gravitational field (yes, torque is the analog of force in polar coordinate), the precession of the earth's axis is caused by perturbations of other planets. Fortunately these are also periodic forces and it is not difficult to study these effects.
Just like the tides in the seas cause a friction slow down the rotation of the earth.
You can harvest the energy in the tides and that works. As you extract this energy, the angular momentum of the earth-moon system is converted into some useful energy (which was otherwise dissipated as heat). The result will be slowing down of both earth and moon.
Again, I repeat, this would not have been possible if earth were a rigid body.
And yes, at least in principle, you can extract energy from the earth's precession which, like the tides, is as a result of gravitational interaction with external masses. Pity about the 25,000 year period though.
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