Gravity

[Jaaanosik]

If what you say is true and there is only one centripetal force a[cp] from B1 to A then how do you analyze B1 cycloid curved path?

(Please Note: This uploaded content is no longer available.)

Where is your centripetal acceleration - 90 degrees to the velocity vector? What’s going on?

 

[hecd2]

The reactive centrifugal force is opposite to the centripetal force causing curved path. It’s always 90 degrees to the velocity vector.
The centrifugal force can not be ignored when acceleration analysis is done in any inertial reference frame.

Rubbish and nonsense. Rubbish and nonsense in several respects:

First of all the reactive centrifugal force doesn’t act on the body B1 or B2 in an inertial frame. The acceleration of B1 and B2 is in the direction of A (in all inertial frames) and is caused by the centripetal force acting on B1 and B2 also in the direction of A (in all inertial frames). This is the *only *force acting on B1 and B2 as seen from an inertial frame. It’s what causes the acceleration of B1 and B2 (in the case of circular motion the acceleration results in a change in the direction of the velocity vector but not its magnitude).

Secondly there is a reactive force associated with the circular motion of B1 and B2 but that force is acting not on the balls themselves but on the central axle and they are equal and opposite in direction to the centripetal forces (ie they act along the line from A to B1/B2).

Thirdly the fictitious centrifugal force is absolutely ignored in inertial frames - think about the case where the frame is at rest with respect to the circular motion. The centripetal (real) force and centrifugal (fictitious) force are equal and opposite. If you add them you get zero force on B1 and B2 and therefore zero accelertion and they will fly off in a staight line. Which they don’t.

Finally your statement that the centripetal force is always at 90 degrees to velocity is trivially wrong. It is the fact that the force is not always perpendicular to the velocity as viewed from an inertial frame moving with respect to the circular motion that results in work being done on B1 and B2 in your “ground” frame. Could this be your fundamental mistake? It’s obvious that the centripetal force is perpendicular to the velocity in a frame at rest with respect to the axle; it is also the case that the acceleration and the centripetal (and reactive force) always point along the line joining B1/B2 to A (as all inertial observers will agree). But in the ground frame the x-component of the velocity of B1/B2 has an additional factor v so it’s obvious that, except at two points in the revolution, the acceleration (and centripetal force) will not be perpendicular to the velocity and so the centripetal force can do work.

I have repeated this now possibly four times and you still don’t get it. I see you have decided to ignore my advice to think about the previous scenario until you understand it, and instead you have intyroduced another scenario, still trying to prove me wrong, and thus making more elementary mistakes and learning no physics.

 

[hecd2]

If what you say is true and there is only one centripetal force a[cp] from B1 to A then how do you analyze B1 cycloid curved path?

http://theelectromagneticnatureofthings.com/img/i2/atan01.png

Where is your centripetal acceleration - 90 degrees to the velocity vector? What’s going on?

Sigh! For the sixth time, in your “ground frame” the centripetal acceleration is not at 90 degrees to the velocity vector. The centripetal acceleration always points at the centre of the circle.

Here’s an exercise for you. It’s an easy exercise because the answers are all already posted in this thread:

1. Write down the parametric equations of the cycloid path
2. Differentiate them, producing the equations decribing the velocity of a point on the cycloid (you’ve done this already).
3. Differentiate again to find the acceleration of a point on the cycloid (I did this for you in a previous post). Plot the acceleration as a function of time.
4. Note that the velocity of the inertial frame drops out indicating that the acceleration is the same in all inertial frames and points at the centre of the circle.

Repeat after me: The acceleration vector of body moving in a circle when view from an inertial frame moving with respect to the centre of the circle is ***not ***at 90 degrees to the velocity vector (except instantaneously at two points on the circle).

 

[hecd2]

But in the ground frame the x-component of the velocity of B1/B2 has an additional **factor **v so it’s obvious that, except at two points in the revolution, the acceleration (and centripetal force) will not be perpendicular to the velocity and so the centripetal force can do work.
.

This should read: But in the ground frame the x-component of the velocity of B1/B2 has an additional **component **v so it’s obvious that, except at two points in the revolution, the acceleration (and centripetal force) will not be perpendicular to the velocity and so the centripetal force can do work.

It’s not a factor.

 

[Jaaanosik]

So you agree that there is a work done in the ground reference frame. Correct?
When a work is done it means there is ‘flow’ of energy from one system to another.
Is the steel ball gaining energy all the time? To the infinity?
… or it’s returning it to axle and the outer system on the down side of the cycloid?

Do we have a periodic ‘pumping of energy’ up and down or not?

The answer is obvious. Why do you say that there is no oscillation?

 

[hecd2]

So you agree that there is a work done in the ground reference frame. Correct?

Good heavens! Do you actually read what I write? I think you can find the answer to that question repeated many times if you read back.

When a work is done it means there is ‘flow’ of energy from one system to another.

In this case, we are looking at B1 or B2 in isolation. We can look at the whole system together if you’d like, but since we are looking at B1 or B2 in isolation, what we see is an external force acting on B1 or B2 doing work and cycling the energy when we observe B1 or B2 from an inertial frame moving with respect to the system.

Is the steel ball gaining energy all the time? To the infinity?
… or it’s returning it to axle and the outer system on the down side of the cycloid?

If we look at B1 or B2 in isolation we see a sinusoidal change in kinetic energy over time in inertial frames moving with respect to the system.

Do we have a periodic ‘pumping of energy’ up and down or not?

The energy of B1 and B2 varies sinusoidally (and in anti-phase) over time as a consequence of external forces as observed from these inertial frames.

Note that the kinetic energy of the whole system B1, B2 and all the peripheral apparatus, being a kinematically complete system, is *constant *in *all *inertial frames.

The answer is obvious. Why do you say that there is no oscillation?

Because, as I have taught you multiple times already, there is no harmonic oscillator with a natural frequency and the consequence of resonance in this system. The cycling of kinetic energy of B1 or B2 considered on their own is a consequence of observing them as a kinematically incomplete system so that external forces act on them.

As I have explained before a mechanical oscillator requires a restoring force which is monotonic function of displacement which we don’t have in this (or the previous) arrangement. What this means is that there is a force acting on the mass which is greater the further the mass is displaced from its rest position and acting in the direction of the rest position.

So for example, a simple pendulum, a mass attached to a wall by a spring, a guitar string are all harmonic oscillators and have all the properties of an oscillator:

1. They have a unique rest position where the restoring force is zero: the pendulum hanging straight down, the mass positioned where the spring is neither compressed nor exteded, the guitar string making a straight line between its fixed points. The systems you describe do not have a unique rest position.
2. When displaced from the rest position they all experience forces acting towards the rest position and these forces become greater, the greater the displacement. The pendulum is displaced away from the vertical, gravity acts to restore it to the vertical; the spring tension acts on the displaced mass; as does the guitar string tension on the plucked string. Your systems do not have a rest position, so they can’t be displaced from a thing they don’t have, and further there is no restoring force.
3. Harmonic oscillators all have natural frequencies (which are invariant in Galilean transformation - ie the natural frequency is the same in all inertial frames). The pendulum swings with a single frequency and period depending on its length and the force of gravity, as the does the mass on the spring depending on the mass and the spring rate (the basis of a balance wheel in a watch), as does the guitar string which vibrates with a frequency determined by its mass per unit length, its length and its tension. Your systems don’t have any such natural frequency. You can revolve them at any angular frequency you like.
4. Harmonic oscillators resonate when forced at their natural frequency. Your systems don’t have a natural frequency so they don’t resonate.

The fact of observing a sinusoidal change in kinetic energy (caused by an external force) in some inertial frames is insufficient to make your systems oscillators.

The reason why I say the two systems you have descibed aren’t oscillators is because they aren’t. Do you understand now?

 

[hecd2]

Jano, in this discussion you have not stuck to one point but you’ve introduced new topics and bounced all over the place in your vain attempts to prove Newtonian mechanics inconsistent. I can’t continue to keep teaching you the same things over and over again so I need you now to acknowledge the following that I have explained in some detail in the thread. If you can’t or won’t then I’m afraid we’ll have to draw this to a close.

1. Do you now understand that kinetic energy is not Galilean invariant? Yes or no?
2. Do you now understand that for a kinematically complete system without external forces, energy is a Galilean invariant? Yes or no?
3. Do you now understand that external forces can do different work in different inertial frames? Yes or no?
4. Do you now understand that for a system subject to external forces, kinetic energy can be constant or time-varying in different inertial frames? Yes or no?
5. Do you now understand that a system consisting of body rotating in a circle on a massless infinitely stiff string is kinematically incomplete and subject to external forces? Yes or no?
6. Do you now understand that, as far at the revolving body as an isolated system goes, the centripetal force is an external force? Yes or no?
7. Do you now understand that all observers in all inertial frames agree that the only force acting on the body is the centripetal force and they agree on its magnitude and direction, and that direction is to the centre of the circle? Yes or no?
8. Do you now understand that the centrifugal force on the body is not a force in inertial frames and is not included in inertial frame calculations; that the centrifugal force is an artefact of rotating frames (ie non inertial frames) and is observed only in frames rotating with the body? Yes or no?
9. Do you now understand that in the frame at rest with respect to the circle the centripetal force always acts perpendicularly to the velocity of the body and therefore does no work; and therefore the kinetic energy of the body remains constant? Yes or no?
10. Do you now understand that in inertial frames moving with respect to the circle, the velocity is not perpendicular to the centripetal force and therefore work is done and the kinetic energy varies in time? Yes or no?
11. Do you now understand that there is no acceleration a[ct] as observed from an inertial frame, because the only acceleration of the body in all inertial frames is the centripetal acceleration a[cp] which all inertial observers agree on (acceleration is Galilean invariant)? Yes or no?
12. Do you now understand that there are no oscillators, no natural frequency and no resonance in any system you have described? Yes or no?
13. Do you now understand that there is no physical consequence of an inertial frame moving with respect to the circle at the same speed as the linear speed of rotation and that uniform motion cannot be detected by any physical effect on the dynamical behaviour of the system? Yes or no?

If you won’t answer these or if the answer to more than one or two is ‘no’, I can’t help you and we might as well pack it in.

 

[hecd2]

Referring to 2) above, it should of course read:
2) do you now understand that for a kinematically complete system without external forces, energy is constant in all inertial frames.

Of course, energy is not a Galilean invariant which I correctly stated in 1) so I don’t know what got into me when I got to 2)

 

[Jaaanosik]

Hecd2,
I still don’t know your first name.
Patience is the most important word here.

I understand and I agree with everything what you say but it stands on a false assumption that the inertial forces can be ignored.

Having said that, do you understand that when placeholder is dropped B2 ball gains freedom of motion because of bearings?

Do you understand that B2 ball is the body doing work?

Do you understand that B2 ball is pulling the rest of the system in the ground reference frame?

Do you understand that B2 ball is pulling in the a[ct] direction?

So how can you say that a[ct] does not exist?

Do you understand that B2 a[cp] is a reaction force?

B2 ball is like a flywheel. It has a stored energy that is being released by flying off in the a[ct] direction and a[cp] is a reaction force.

The centrifugal forces a[ct] are real, a[t] is real, that’s how B1 ball accelerates faster than expected (precession) and B2 ball slows down faster than expected.
The satellite flyby anomalies could be a prove of that.

It all comes down for you to show that when B2 does the work it does not pull in any direction because a[ct] does not exist.
Good luck!

 

[hecd2]

I understand and I agree with everything what you say but it stands on a false assumption that the inertial forces can be ignored.

So you don’t agree with everything I explained to you. I need you to specifically tell which of the statements you don’t agree with before we can continue. That way I’ll know where your error is and how to improve your understanding.

 

[Jaaanosik]

So you don’t agree with everything I explained to you. I need you to specifically tell which of the statements you don’t agree with before we can continue. That way I’ll know where your error is and how to improve your understanding.

Please, read my latest post for the reason why B2 a[ct] is real force doing work.
If that force does not exist then how B2 is going to lose its stored energy, how it’s going to do any work?

 

[hecd2]

Please, read my latest post for the reason why B2 a[ct] is real force doing work.
If that force does not exist then how B2 is going to lose its stored energy, how it’s going to do any work?

When you tell me specifically which of the 12 statements you disagree with we can continue. Otherwise this is my last post in this thread.

 

[Jaaanosik]

…
By the way, you show an acceleration a[ct] in your diagram above which doesn’t exist.
…

This quote…

 

[hecd2]

This quote…

What’s so difficult about saying which of 1 to 12 you agree with and which ones you disagree with?

Goodbye.

 

[Jaaanosik]

What’s so difficult about saying which of 1 to 12 you agree with and which ones you disagree with?

Goodbye.

I agree with all the 12 points if the assumption is that a[ct] does not exist.
If the a[ct] does exist then almost all of your 12 points are going to break.

I’ll repeat:

It all comes down for you to show that when B2 does the work it does not pull in any direction because a[ct] does not exist.
Good luck!
…
Please, read my latest post for the reason why B2 a[ct] is real force doing work.
If that force does not exist then how B2 is going to lose its stored energy, how it’s going to do any work?

… and really, good luck with that!

 

[hecd2]

I agree with all the 12 points if the assumption is that a[ct] does not exist.
If the a[ct] does exist then almost all of your 12 points are going to break.

I’ll repeat:

… and really, good luck with that!

So if and when I demonstrate that a[ct] depicted in your diagrams does not exist in inertial frames you’ll accept statement 1 - 12 without exception? Only a simple yes or no answer please.

 

[Jaaanosik]

So if and when I demonstrate that a[ct] depicted in your diagrams does not exist in inertial frames you’ll accept statement 1 - 12 without exception? Only a simple yes or no answer please.

Yes.
You need to show me a solid logical proof how B2 ball does mechanical work without a[ct].
The centripetal and centrifugal forces are inseparable. You can not have one without the other.

 

[hecd2]

hecd2:

So if and when I demonstrate that a[ct] depicted in your diagrams does not exist in inertial frames you’ll accept statement 1 - 12 without exception? Only a simple yes or no answer please.

Yes.
You need to show me a solid logical proof how B2 ball does mechanical work without a[ct].
The centripetal and centrifugal forces are inseparable. You can not have one without the other.

Ok, so we’ll begin by answering Jano’s earlier post. The “logical proof” that he requested will then follow.

…do you understand that when placeholder is dropped B2 ball gains freedom of motion because of bearings?

Yes

Do you understand that B2 ball is the body doing work?

No, this is not correct - if you consider B1 and B2 separately, work is being done on both by the forces acting in the direction of the centre of the circle. The work being done on B1 and B2 has equal magnitude and opposite sign.

Do you understand that B2 ball is pulling the rest of the system in the ground reference frame?

There is a reaction force on the central axle equal in magnitude to ma[cp] (where m is the mass of B2) and in the opposite dircetion, i.e. pointing from the centre to B2.

Do you understand that B2 ball is pulling in the a[ct] direction?

It’s not

So how can you say that a[ct] does not exist?

Because it doesn’t exist in an inertial frame.

Do you understand that B2 a[cp] is a reaction force?

It’s not a force, it’s an acceleration

B2 ball is like a flywheel. It has a stored energy that is being released by flying off in the a[ct] direction and a[cp] is a reaction force.

Wrong. First of all a[ct] and a[cp] are accelerations not forces, and the only accelerations of B1 and B2 in any inertial frame are the centripetal accelerations directed to the centre of the circle.

The centrifugal forces a[ct] are real, a[t] is real, that’s how B1 ball accelerates faster than expected (precession) and B2 ball slows down faster than expected.
The satellite flyby anomalies could be a prove of that.

Nonsense. Neither B1 nor B2 accelerate or decelerate “faster than expected”. The acceleration of B1 and B2 as observed from any inertial frame has *constant *magnitude and is directed to the centre of the circle. a[ct] does not exist in inertial frames and therefore its resultant with a[cp], a[t] also doesn’t exist.

It all comes down for you to show that when B2 does the work it does not pull in any direction because a[ct] does not exist.
Good luck!

I don’t need good luck to show that a[ct] doesn’t exist. In the next post, I will *prove *it. However the next post won’t be completed for a short while, as I have to use an equation editor to draft the maths and then find a way of displaying that on this forum.

 

[hecd2]

Ok, so in this post I will present the *proof *that Jano has requested. Having done so, I remind him that he then agrees with all 12 statements I made in post #127.

I hope that Jano (and Tomdstone if he’s still following along) appreciate the inordinate effort that I have put in to educate Jano. Creating this post has not been the work of five minutes as CAF does not render LaTex and so I have to make and host images of the maths and then get these into the post.

Let’s start. The first thing to be noted is that the acceleration of a body is defined unambiguously as a change in the velocity vector of that body. In other words, if the velocity can be described as an analytical function of time, then you can calculate the acceleration as a function of time by differentiating the velocity with respect to time. OK so far?

The same thing of course is true for velocity - the velocity is defined as the change in the position vector as a function of time and the velocity can be calculated as a function of time by differentiating the position with respect to time. So, if you have an analytical function for the position of a body, you can calculate its velocity and acceleration. OK?

Let’s look first at the ball B1 (this also applies to the body on the end of the string) and I’ve tried to stick to the same notation as Jano does to avoid confusion:



So the key thing to inspect here are the equations for acceleration - they tell us exactly the acceleration of a body moving in a circle around a point as seen from an inertial frame moving at velocity V in the plane of the circle. The first thing that you notice is that the acceleration does not include V - in other words the acceleration of such a body is the same in all inertial frames - it is independent of V. (This result holds for all inertial frames not just for frames moving in the x-direction - the velocity of the inertial frame, being constant, always disappears in the equations for acceleration.)

Then consider the form of the equations for acceleration - they are the parametric form of a circle centred on the origin, 180 degrees out of phase with the circle describing the position of B1 - in other words, the vector for acceleration always has the same amplitude (r*omega^2), and always points to the centre of the circle. This is true regardless of the value of V. QED.

In truth, we are done here - we have proven that the acceleration of B1 is a[cp] and there is no a[ct] observed in any inertial frame. However, I’ll create and post some more later tonight or tomorrow, drawing out some more detail about the scenario that includes B1 and B2, eg specifically calculating the work done on those two bodies.

 

[Tomdstone]

I hope that Jano (and Tomdstone if he’s still following along) appreciate the inordinate effort that I have put in to educate Jano.

Definitely appreciated.