Rotational energy and psuedoforce

[Nowhere357]

Newton's first law (Every object in a state of uniform motion tends to remain in that state of motion unless an external force is applied to it) tells me that rotating objects tend to keep rotating.

Assume a rotating space station designed to deliver 1g to people in the outer rim.

Newton's second law (The relationship between an object's mass m, its acceleration a, and the applied force F is F = ma. Acceleration and force are vectors; in this law the direction of the force vector is the same as the direction of the acceleration vector) and the fact that gravity is an acceleration, tell me that force is being delivered to those people.

In this case, the acceleration is delivered by the 'psuedo' centripedal force.

The people experience a real acceleration - a real force. Conservation of energy laws tell me that force cannot be free.

Where does it come from?

If a person on the station were to walk over a hole in the floor, they would 'drop' into space in a straight line away from the station. Like shooting a bullet from a gun. Would the station then have less rotational energy?

[Santas little helper]

Newton's first law (Every object in a state of uniform motion tends to remain in that state of motion
unless an external force is applied
to it) tells me that rotating objects tend to keep rotating.

Uniform motion means in a straight line and with constant speed.So you cannot conclude just from this that
rotating objects keep rotating.In fact unless there are external forces they will not keep rotating.

Newton's second law (The relationship between an object's mass m, its acceleration a, and the applied force F is F = ma. Acceleration
and force are vectors; in this law the direction of the force vector is the same as the direction of the acceleration vector) and the fact
that gravity is an acceleration, tell me that force is being delivered to those people.

When you say gravity you mean the artificial gravity produced by the station's rotation, right ?

In this case, the acceleration is delivered by the 'psuedo' centripedal force

Why pseudo ? It seems real to me.

Where does it come from ?

From the floor of the station obviously.The same way the force from the floor of the room you are
right now prevents you from moving towards the center of the earth.

If a person on the station were to walk over a hole in the floor, they would 'drop' into space in a straight
line away from the station.

No, the person would move along a tangent to the station's rotational orbit.

Would the station then have less rotational energy ?

Yes, it would.The combined system station-human would have the same total mechanical energy since no energy
was gained or lost.Since the human has some kinetic energy, the kinetic energy of the station would be less.
This without taking into account the gravitational attraction between the moving away human and the station.

Another interesting question is whether the station would keep rotating at the same speed.I believe it
would but not sure.

[Tetlepanquetzatzin]

Originally posted by Nowhere357
Newton's first law (Every object in a state of uniform motion tends to remain in that state of motion unless an external force is applied to it) tells me that rotating objects tend to keep rotating.

Newton's first law tells you that the center of mass of a body that is subject to no external forces will continue to move with the same velocity vector.
In this case, the acceleration is delivered by the 'psuedo' centripedal force.

The people experience a real acceleration - a real force. Conservation of energy laws tell me that force cannot be free.

Where does it come from?
The name of the ficticious force is centrifugal force or Coriolis force. Newton's laws of motion are only valid in non-accelerated reference frames. For qualititative purposes the Coriolis force may be thought of as an extra term that must be added to Newton's laws when they are used in rotating frames.

Someone who is standing on the (inside of) the outer rim of the rotating space station is rotating with the same speed as the space station. (Note, however, that it would certainly be possible float freely without rotation inside the space station provided that you don't get hit by any walls as the space station rotates around you.) If there were no external forces, the guy would move in a straight line. This does not happen because the normal force from the floor prevents radial acceleration.
Originally posted by Santas little helper

Another interesting question is whether the station would keep rotating at the same speed.I believe it would but not sure.
It depends on how you lose contact with the space station. If you jump out you will likely exert a force and moment on the space station and it's angular velocity will change. But if you exit cleanly the angular velocity will be left unchanged.

[DaveGE]

Originally posted by Nowhere357 The people experience a real acceleration - a real force. Conservation of energy laws tell me that force cannot be free.
[/B] Mmm, not quite. Conservation of energy says *energy* is not free. Force is not energy. Energy is force times distance, or more precisely the dot-product of the force vector and the displacement vector. In the case of the rotating station, the force is always perpendicular to the direction of displacement, and so the dot-product is always zero--there is no work being done.

(But then why, you ask, does a rocket expend energy when it flies in a circle. Because, I reply, the rocket exhaust is being thrown out the back. All the energy goes into the exhaust. In the case of the station, there's no exhaust.)

[Calzaer]

Why pseudo ? It seems real to me.

The only forces involved are the normal force of the floor holding you in and the linear accellerational force of the space station itself (your mass wants to continue going in a straight tanget line due to inertia, but the floor holds you in).

There's no gap in which to fit centrifugal "force". All forces are accounted for already. There's no mystical force that pushing you out away from the center in a rotational fashion. Just a force pushing you in a straight line and a force keeping you from going in a straight line.

[Santas little helper]

The only forces involved are the normal force of the floor holding you in and the linear
accellerational force of the space station itself
(your mass wants to continue going in a straight tanget line due to inertia,
but the floor holds you in).

Actually I believe the only force is the force of the floor.And this is the centripetal
force in this example.Since 357 said "'psuedo' centripedal force" I was asking why he calls
it pseudo.

[Nowhere357]

Santas little helper
In fact unless there are external forces they will not keep rotating.
Can you elaborate? I thought a rotating object would tend to keep rotating. Spin a marble in space and - with no outside influence - it eventually would stop rotating?

When you say gravity you mean the artificial gravity produced by the station's rotation, right ?
Yes. My conceptual problem is that although it's called "artificial" gravity (which it is) it also is very real acceleration. Where does the energy for this acceleration come from - what is losing energy as a result?

Analogy: if I accelerate a ball by kicking it, the energy came from my foot - which slowed down as a result.

Why pseudo ? It seems real to me.
I was taught (iirc) that centrifigal force is real, the object receiving it wants to take off in a straight line. They are pulled back to the center - yet there is no true force directed towards the center - the "centripedal" force is the resultant of vectors - the vector from the centrifugal force, and the vector from the floor. Note that the floor does not move toward the center. There is no "center-directed" force. Hence centripedal force is a psuedo-force.

From the floor of the station obviously. The same way the force from the floor of the room you are right now prevents you from moving towards the center of the earth.
Then the floor should slow down, shouldn't it? As an opposite reaction to delivering acceleration, just as my foot slows down when it kicked the ball?

Is that the case? Does a rotating object slow itself down?

No, the person would move along a tangent to the station's rotational orbit.
You're correct. Same question though - does this remove rotational energy from the station?

Yes, it would.The combined system station-human would have the same total mechanical energy since no energy was gained or lost. Since the human has some kinetic energy, the kinetic energy of the station would be less.
Hmm. Rotational, mechanical, and kinetic energy all in the same question/answer. I guess this stuff really is not all that easy, so I excuse myself for my ignorance. :)

Another interesting question is whether the station would keep rotating at the same speed. I believe it would but not sure.
This is what I was looking for with that particular question. The station loses rotation energy, but it also loses mass. I guess I need to find the formula to see how this affects the rotational speed.

Anyway, my main thrust is to learn where the energy comes from that is delivered to the accelerated object on the station. If rotating objects tend to slow down, then I think that answers the question. I am not convinced that's the case, however. My understanding is that the rotation of the earth, for example, may slow down, but due to drag caused by the liquid center.

[Shadowy Man]

Originally posted by Santas little helper
Uniform motion means in a straight line and with constant speed.So you cannot conclude just from this that
rotating objects keep rotating.In fact unless there are external forces they will not keep rotating.


Nope. Don't forget conservation of angular momentum.

Newton's laws can be applied to rotating objects, but you replace force with torque, mass with moment of inertia, acceleration with angular acceleration, and velocity with angular velocity.

Angular momentum will be conserved unless acted upon by an external torque.

[Nowhere357]

Tetlepanquetzatzin
Newton's first law tells you that the center of mass of a body that is subject to no external forces will continue to move with the same velocity vector.
The law describes uniform motion. This does not apply to rotation?

Is it the case that a rotating object tends to slow down?

The name of the ficticious force is centrifugal force or Coriolis force. Newton's laws of motion are only valid in non-accelerated reference frames. For qualititative purposes the Coriolis force may be thought of as an extra term that must be added to Newton's laws when they are used in rotating frames.
Centripetal is center-seeking; centrifugal is away-from-center. I'm not sure how that relates to coriolis effect, which is caused by moving reference frame. A rocket leaving earth in a straight line from the center of the earth will appear to move to the right in the northern hemisphere and to the left in the southern hemisphere, when viewed from the earth, due to the rotation of the earth.

Whereas the fellow on the station who steps over a hole will be flung into space by the centrifugal force, while his station-mates experience artificial gravity due to the centripetal force.

Someone who is standing on the (inside of) the outer rim of the rotating space station is rotating with the same speed as the space station. If there were no external forces, the guy would move in a straight line. This does not happen because the normal force from the floor prevents radial acceleration.
Okay. Does the floor then slow down as an opposite reaction?

It depends on how you lose contact with the space station. If you jump out you will likely exert a force and moment on the space station and it's angular velocity will change. But if you exit cleanly the angular velocity will be left unchanged.
So the station does not slow it's rotational speed when it ejects part of itself? Since the ejected object carries with it some of the energy from the station, where is this loss reflected?

(Note, however, that it would certainly be possible float freely without rotation inside the space station provided that you don't get hit by any walls as the space station rotates around you.)
I have read sf stories where, from floating freely in the hub, the person starts falling towards the rim. I always felt that was wrong. I get a headache trying to visualize the situation, but I think that until he meets a wall (rotating along with the station) he would not be affected by the rotation of the station at all. If he hasn't met a wall, and assuming no air resistance, he could thrust 'down' to the inside of the outer rim and hover there, watching the 'floor' zip by. Okay, I need an aspirin.


Btw I was wondering what your user name meant, so a took a random stab at a google list. I wonder what you think - is this anywhere close?:

Your name of Tetlepanquetzatzin gives you a highly sensitive, idealistic, and intuitive nature. You could be very expressive and creative in the arts, music, or drama. However, there is a lack of practicality and business judgment which could limit your success. You feel and sense much that you do not fully understand, and you can be deeply influenced through the thoughts and feelings of others without realizing just how you are being affected. Others are inclined to take advantage of your generosity and friendliness and then, when there is a lack of reciprocation, you can feel very despondent and disillusioned. Moods are a problem as you can be highly inspired one minute, and the next become quite irritated and annoyed over some ill-timed remark or lack of consideration on the part of someone close to you. You could suffer through nervous breakdowns, as well as disturbed thoughts, poor memory, or disorders in the fluid functions.

:)

[Nowhere357]

DaveGE
Mmm, not quite. Conservation of energy says *energy* is not free. Force is not energy. Energy is force times distance, or more precisely the dot-product of the force vector and the displacement vector. In the case of the rotating station, the force is always perpendicular to the direction of displacement, and so the dot-product is always zero--there is no work being done.
This is getting closer to what I'm looking for. It is indeed my understanding that no work is being done, which leads to my mental block, because things are really accelerated! And isn't work being done when I kick (accelerate) the ball?

I tend to use energy and force as equivalent, of course that is imprecise. However, they are directly related. Doesn't the artificial gravity impart energy to the object? Apparently not, but I don't understand how something can be accelerated without an input of energy.

[Jesse]

Nowhere357:
Centripetal is center-seeking; centrifugal is away-from-center. I'm not sure how that relates to coriolis effect, which is caused by moving reference frame. A rocket leaving earth in a straight line from the center of the earth will appear to move to the right in the northern hemisphere and to the left in the southern hemisphere, when viewed from the earth, due to the rotation of the earth.

Whereas the fellow on the station who steps over a hole will be flung into space by the centrifugal force, while his station-mates experience artificial gravity due to the centripetal force.

The centrifugal force is the "fictitious" force that appears to pull you away from the center in a rotating reference frame, while the centripetal force is the actual force (from within an inertial reference frame where Newton's laws are valid) pulling you towards the center which keeps you rotating instead of flying off on a tangent. If you're standing inside the outer rim of a rotating space station, the "normal force" from the rim acts as a centripetal force, constantly pushing on your feet and preventing you from flying off on a tangent--if you step through a hole in the rim there's no longer any sort of force keeping you on the curved path, so you do fly off on a tangent. On the other hand, people within the rotating reference frame on the rim will consider themselves to be standing still while a force pulls them down, and this is the fictitious centrifugal force. But when they see their neighbor "fall" through the hole, he will not fall straight down from their point of view, but will instead seem to move both down and sideways (think of the tangent line from their point of view), which they would explain in terms of a combination of the fictitious centrifugal force and the equally fictitious coriolis force.

So basically, centripetal = actual inward-pulling force in an inertial reference frame where Newton's laws apply, centrifugal + coriolis = fictitious forces that result from using a rotating reference frame.

[Nowhere357]

Calzaer
The only forces involved are the normal force of the floor holding you in and the linear accellerational force of the space station itself (your mass wants to continue going in a straight tanget line due to inertia, but the floor holds you in).

There's no gap in which to fit centrifugal "force". All forces are accounted for already. There's no mystical force that pushing you out away from the center in a rotational fashion. Just a force pushing you in a straight line and a force keeping you from going in a straight line.
I'm trying to visualize the vectors. There is the vector of the straight line the object 'wants' to move in, and the vector from the floor to the hub (centripetal). The resultant follows the rotation.

Well, I accept that I was wrong about which was the psuedo-force; that would be the centrifugal force. There is no force pushing out from the center.
____________________________

I 'lost' all my science and physics books during my last move. God I miss them. Also missing is my china (from Japan!), my collection of "graphics art novels" and most of my rock climbing equipment. Other stuff also, but these are the ones that still hurt. @#*&%! :(

[Tetlepanquetzatzin]

Some facts from rigid body mechanics may help you, Nowhere357:

* The position and orientation of a rigid body (http://scienceworld.wolfram.com/physics/RigidBody.html) is fully determined by its center of mass and two angles.

* The motion of the center of mass is given by Newton's second law: F = ma, where F is the sum of all external forces on the body.

* The rotation of a rigid body is given by Euler's equation of motion: M = I q, where M is the sum of all external torques about the center of mass, I are the moments of inertia (http://scienceworld.wolfram.com/physics/MomentofInertia.html) about the center of mass, and q is the angular acceleration. In the 2D case there are only one angle and one moment of inertia, so the equation reduces to M = Iq. Note that if there are no external torques, the angular acceleration will be zero and, consequently, the direction and speed of the rotation will not change.

* The work done by a force on a body is given by integrating the dot product of the force and the position displacements, like this

http://scienceworld.wolfram.com/physics/wimg245.gif

Note that if the force is always perpendicular to the position changes the dot product will always be zero. Thus, such a force does no work at all, despite the fact that it accelerates the body.

* The work done by a torque is given by integrating the torque times the angular displacements.

* In the case of central motion -- e.g. planetary motion around the sun or astronauts inside rotating space stations -- there is a radial force that continually accelerates the object towards the rotational axis. This force is called centripetal force and it is a real force, at least in Newton's theory. This should not be mixed up with the centrifugal force, which is a pseudoforce that is introduced in calculations performed in rotating coordinates. In the case of planetary motion the centripetal force is just the gravitational force from the sun and in the case the space station it is the normal force from the floor.


Let us now use these facts to analyze the rotating space station. We consider a cylindrical space station, which rotates with angular velocity w about its axis of symmetry. The space station has radius R and an astronaut is standing on the inside of the cylindrical surface. We assume that R is so large that the astronaut can be considered as a point-particle in comparison. The only forces acting on the astronaut come from the surface on which the astronaut is standing. There will be a normal force, directed radially towards the axis of rotation, that prevents the astronaut from penetrating and passing through the surface. There can also be frictional forces, directed along tangents of the surface. If we assume that the astronaut is already rotating with the same speed as the space station, we may neglect the frictional forces and consider only the radially directed normal force -- the centripetal force in this example.

Our first insight is that since the normal force is directed radially while astronaut always moves along a tangent of the surface, the force and position changes are always perpendicular. This means that the space station does no work on the astronaut. Therefore no energy is lost by the space station.

Our second insight is that, because there are no external forces acting on the total system of astronaut + space station, their combined center of mass will not accelerate.

Our third insight is that the space station can be regarded as a rigid body. Neglecting friction, the only force acting on the space station is the reaction force resulting from the normal force from the space station on the astronaut (Newton's third law). This force is directed radially and therefore will not result in any torque. By Euler's equation of motion, the angular velocity of the space station will stay the same.

Our fourth insight is that if we place the origin of our coordinate system at the combined center of mass, the coordinates of the astronaut (in a plane perpendicular to the symmetry axis of the cylindrical space station) will be given by

x = Rcos(wt)
y = Rsin(wt),

since both the distance to the center of mass and the angular velocity are constant. Differentiating twice gives us the acceleration

x" = -w^2 Rcos(wt)
y" = -w^2 Rsin(wt).

By Newton's second law applied to the astronaut, the x- and y-components of the normal force from the floor are given by -mw^2 Rcos(wt) and -mw^2 Rsin(wt), respectively, where m is the mass of the astronaut.

I hope this helped.

Nowhere357 also asked:
Is it the case that a rotating object tends to slow down?
The answer is no, at least for a rotating rigid body subject to no external forces.
Okay. Does the floor then slow down as an opposite reaction?
Yes, the floor accelerates the guy inwards, and by Newton's third law, the space station is accelerated outwards. This is how the center of mass stays at rest!
So the station does not slow it's rotational speed when it ejects part of itself? Since the ejected object carries with it some of the energy from the station, where is this loss reflected?

It depends on how the part is ejected. You need to do it very smoothly so that no torque acts on the space station. Rotational energy depends on the moments on inertia, which in turn depend on the mass of the system. The loss of mass thus results in loss of rotational energy.
I have read sf stories where, from floating freely in the hub, the person starts falling towards the rim. I always felt that was wrong. I get a headache trying to visualize the situation, but I think that until he meets a wall (rotating along with the station) he would not be affected by the rotation of the station at all. If he hasn't met a wall, and assuming no air resistance, he could thrust 'down' to the inside of the outer rim and hover there, watching the 'floor' zip by. Okay, I need an aspirin.
It depends on the velocity of the person doing the floating. If stand on the rim you will rotate with the same angular velocity as the rim. If you then jump inwards (radially) you will also have a velocity directed in the tangential direction. The total velocity will have a direction somewhere in between. Someone who is not rotating will see you travelling in a straight line until you eventually hit the rim. However, if you didn't have this radial velocity you could float inside the space station without necessarily approaching any walls.

My name means that I got tired of so many Hotmail login names being taken. I assumed -- correctly it would turn out -- that no one had registered Tetlepanquetzatzin. Besides, I think it kind of rolls off the tongue.

[Nowhere357]

Originally posted by Jesse
So basically, centripetal = actual inward-pulling force in an inertial reference frame where Newton's laws apply, centrifugal + coriolis = fictitious forces that result from using a rotating reference frame.
I nodded in agreement with your entire post. The relationship between all these forces is becoming clearer - yet still my mental block remains. The objects on the station are 'kicked' by the floor - why doesn't the floor slow down as a result? :confused:

I need some visuals. Can you recommend any good on-line sites at a late high school or early college level?

[Lobstrosity]

Originally posted by Nowhere357
This is getting closer to what I'm looking for. It is indeed my understanding that no work is being done, which leads to my mental block, because things are really accelerated! And isn't work being done when I kick (accelerate) the ball?

I tend to use energy and force as equivalent, of course that is imprecise. However, they are directly related. Doesn't the artificial gravity impart energy to the object? Apparently not, but I don't understand how something can be accelerated without an input of energy.
I haven't read the full thread, so forgive me if I'm repeating what others have said.

Energy and force are not equivalent, Work = ∫ F ∙ ds. If the force does not displace an object in a direction with a component parallel to that force, that force is doing no work.

Look at it this way: say you have mass m and are on the rim of a rotating space station. The space station has radius R and angular velocity w. From an inertial frame you are rotating and hence have rotational kinetic energy equal to ½Iw², where I is rotational inertia and is equal to mr². Therefore, we have that your energy in the inertial frame is:

T = ½mr²w²
U = 0 (there are no force fields)

This energy is entirely kinetic; there is no potential energy in this frame. You can also note that rw = v (where v is tangential velocity), so we have that the energy is really, as expected, ½mv².

Now, in the rotating frame, nothing is moving at all. You perceive yourself as being completely stationary. Where did the energy go? Well, now the energy is in the form of potential energy. In the rotating frame there is a force that must be added known as the centrifugal force. This force exists only in this non-inertial, rotating frame and serves to keep the physics of this frame isomorphic with the physics one would observe in the inertial, non-rotating frame. The centrifugal force is given by F = -mw x (w x r), which reduces to F = mw²r when w is orthogonal to r. This force, like gravity, results in potential energy given by U = -∫ F ∙ dr = -½mw²r² (if we arbitrarily set U = 0 at r = 0). Therefore, we have that your energy in the rotating frame is:

T = 0 (you're not moving in this frame)
U = -½mr²w²

In one frame your energy is purely kinetic while in the other it's purely potential. In both frames it would take the same amount of minimum energy to reach the center of the station.

[Jesse]

Nowhere357:
I nodded in agreement with your entire post. The relationship between all these forces is becoming clearer - yet still my mental block remains. The objects on the station are 'kicked' by the floor - why doesn't the floor slow down as a result? :confused:

Well, as others have said, if there is no movement along the vector of the force itself, only perpendicular to the vector, then no work is done and whatever is emitting the force does not lose energy. Do you understand how this would be true in the case of the gravitational force on a planet from the sun? If so, the normal force is very similar to this, except with a repulsive force from a ring around the moving object pushing it towards the center of its circle instead of an attractive force on the object pulling it towards the center. Since the normal force is ultimately due to electromagnetic forces between atoms in the floor and atoms in your feet, you might picture a simpler case like a positively-charged area on the floor of a rotating ring with a positively-charged particle suspended above it at a constant height (because the particle's inertia is balanced out by the repulsive electromagnetic force between like charges, in much the same way that the inertia of a planet moving in a circular orbit is balanced by the attractive gravitational force between it and the sun).

Nowhere357:
I need some visuals. Can you recommend any good on-line sites at a late high school or early college level?

What aspect of this problem do you want visuals for?

[Nowhere357]

Lobstrosity and Jesse

On first reading of your posts, I think that comparing the reference frames, and considering the simpler case of fields and particles, are what will help me understand. I need some time to digest.

Jesse, I'm interested in physics primers, of course; also help with conceptually bridging between classical and quantum physics. That is, how effects in the macro-world (such as described by Newton's laws) arise from subatomic activity.

[nermal]

Originally posted by Nowhere357
The objects on the station are 'kicked' by the floor - why doesn't the floor slow down as a result? :confused:



If an object is somewhere in the center line of the torus such that the station rotates, but the object is in a fixed point, then that object is "kicked" as you say into moving, the station's rotation will indeed slow.
If an astronaut is at the hub and climbs to the torus, the rotation will slow, but energy is conserved. If the astronaut makes his way back to the hub (here he is actually "climbing"), the station's rotation will speed back up to its original velocity.
If all objects are fixed within the station, such that it is a rigid body, it will, for all practical purposes, rotate at a continuous velocity.
Don't get too wrapped up in centripedal, centrifugal, and pseudo forces. They serve no purpose but to confuse, and you will never see them again after freshman physics.

(Just for fun, think about this: Centripedal (center seeking) forces are considered real, and centrifugal forces (outward) forces are "pseudo". Tie a brick to a rope, and spin it around. A center seeking force should put slack into the rope, but it doesn't. The rope is kept taught by a "pseudo" force.
To be fair, we call centrifugal "pseudo" because it isn't really a force, it is simply a consequence of inertia.)

Ed

[Lobstrosity]

Originally posted by Nowhere357
Lobstrosity and Jesse

On first reading of your posts, I think that comparing the reference frames, and considering the simpler case of fields and particles, are what will help me understand. I need some time to digest.
Something else for you to consider (and which I believe addresses some of the concerns you have voiced) is a rotating space station with an elevator. Occupants would arrive at the center (r = 0) and then ride a rigid elevator along a radial spoke down/up/over to the inner rim. This would slowly transfer them from a non-rotating inertial frame to the rotating frame of the space station. Now, typically one would assume that the mass of this elevator is negligible compared to the mass of the space station itself, however for fun let's pretend this isn't the case. Let's look at what happens as the elevator descends along the radial spoke from r = 0 to r = R:

In the rotating frame, this elevator is moving parallel to the centrifugal force, which means the centrifugal force is actually doing work on the elevator. If the space station is efficiently constructed then the designers probably use the centrifugal force to actually move the elevator from the center to the rim (with adequate dampers and speed-control, of course). So where does this energy come from? From my earlier derivation, we see that initially the elevator had a total energy of zero (since it has no initial kinetic energy and U(0) = 0 by definition). At the rim, however, it once again has zero kinetic energy (once it comes to a stop) but now U(R) = -½mR²w²--it has lost energy, which means the entire elevator/space station system has lost energy. What happens is that the rotational energy (i.e. potential due to the centrifugal force) of the system has been converted into kinetic energy of the elevator and then that energy is lost to heat/friction as the elevator is brought to a stop at the rim. If the elevator's mass is appreciable compared to the space station, the space station's angular velocity will decrease as the elevator moves towards the rim.

Take home message: in this scenario there is displacement parallel to a "force" (yes, it's fictitious in inertial frames but real enough in non-inertial, rotating ones). This means that the force is doing work, which in turn means that the source of that force will lose energy (i.e. the space station's rate of rotation will slow).

Now, what happens when the elevator rises from the rim back up to the center at r = 0? In this case an engine powering a winch must be used to raise the elevator against the radial centrifugal force. Here the direction of the motion is antiparallel to the direction of the centrifugal force, which means that it is the winch that is doing the work, not the space station's rotation. Here the winch is actually transferring its energy to the space station itself! The ultimate effect is that the space station's rotational velocity will increase as the elevator rises. If the same number of people are going up as initially came down, the station will return to the angular velocity it originally had prior to the original descent.


It's easy to see why this happens from an inertial point of view. In our inertial frame we're not dealing directly with forces so much as with conservation of angular momentum. When the elevator is at the center, it has zero angular momentum. As it moves towards the rim, its angular momentum increases, which means that the angular momemtum of the station must decrease in order to keep the total angular momentum constant (angular momentum is a conserved quantity in the absense of external torques, and there are no external torques in this problem). As the elevator moves from the rim back to the center it relinquishes its angular momentum back to the station. This is the same process one observes when one watches a figure skater performing a tight spin. As they pull their arms in they spin faster and faster (their muscles burn fuel and add energy to the system at this point). As they let their arms (using energy from the system) out they slow down. You might also have discovered this principle playing on a tire swing (at least that's where I first noticed it). If you wind it up and sit on it you will begin to spin. If you tuck in your legs you spin super-fast and get all dizzy and discombobulated and such. During each of these process the energy of the rotating system is not conserved but angular momentum is. It is the addition of pseudo-forces when one enters the rotating frame that provides a way to account for this lost energy in the form of potential energy (and thus energy conservation is happy once again). It just so happens that in a rotating frame you have no concept that you are actually rotating, so you're not actually aware you possess this strange "angular momentum," let alone that it is conserved. It is conservation of angular momentum that is responsible for the oft-baffling Coriolis force.

[Jesse]

Originally posted by Nowhere357
Jesse, I'm interested in physics primers, of course; also help with conceptually bridging between classical and quantum physics. That is, how effects in the macro-world (such as described by Newton's laws) arise from subatomic activity.

When you say you're interested in the bridge between classical and quantum physics, are you just interested in the micro-explanation for things like the normal force from the floor on your feet, or are you interested more generally in how Newton's laws can be derived as the large-scale limit of the laws of quantum mechanics? The second question would take you pretty far afield from the issues discussed on this thread, and it would involve even more hard-to-visualize abstract mathematics than Newtonian physics (see this (http://www.wikipedia.org/wiki/Correspondence_principle) page for a discussion of how quantum rules must reduce to classical ones in the large-scale limit).

Here are some pages that talk briefly about how ordinary forces can be explained in terms of the four "fundamental forces" (gravity, electromagnetism, strong, weak) of nature:

http://www.as.wvu.edu/coll03/phys/www/rotter/phys201/4_Forces_Dynamics/Classifying_Forces.html
http://www.br.psu.edu/faculty/lht1/concepts/friction/
http://www.physics.hereford.ac.uk/fundamental_interactions.htm
http://faculty.uvi.edu/users/dstorm/classes/00phy241/mods/lect_10.html

Here are some intros to Newtonian physics in general:
http://www.glenbrook.k12.il.us/gbssci/phys/Class/newtlaws/newtltoc.html
http://hyperphysics.phy-astr.gsu.edu/hbase/newt.html