Straight lines

[Loudmouth]

Post #33. There is no acceleration in the phase plane - just constant velocity ... no different than a plane flying along a geodesic.

The acceleration happens when the spring starts moving and accelerates the body.

 

[Resha Caner]

The acceleration happens when the spring starts moving and accelerates the body.

I'll just wait to see if anyone else has something to offer.

 

[Michael]

Why do objects in motion travel in a straight line when force is absent?

Objects in motion tend to stay in (that same) motion unless something acts upon it. I would agree with LM's answer:

http://www.christianforums.com/threads/straight-lines.7949664/page-2#post-69701780

In the absence of any force or geometric effect from gravity, there's no geometric considerations or other forces acting on the motion of the body, and therefore it's going to move in a "straight" line that is consistent with the geometry of spacetime. If spacetime isn't curved by any other mass in the area, the body in motion will simply conserve it's momentum and it will move in the same direction that it's already moving. Unless spacetime is 'bent' by gravity, or some other force acts on the body, it will simply move in a straight line. If however spacetime is 'warped' (as in our solar system) a body in motion (like the Earth) simply follows the local spacetime curvature in the area as it circles the sun.

 

[Resha Caner]

... it's going to move in a "straight" line that is consistent with the geometry of spacetime ...

Have you read posts #30-35?

 

[[serious]]

Have you read posts #30-35?

Regardless of whether it's intuitive, it's observed.

A good deal of modern physics aren't intuitive, but they are still born out by testing.

If you are looking to build a model with a noninertial reference frame, it's entirely possible, but it makes all the math harder and doesn't make anything easier.

 

[essentialsaltes]

I feel gravity.

No, you don't. Maybe you feel your chair pushing on you, preventing you from being in freefall.

Accelerometers register the effects.

Nope, they read zero in freefall. "An accelerometer measures proper acceleration, which is the acceleration it experiences relative to freefall". "In relativity theory, proper acceleration[1] is the physical acceleration (i.e., measurable acceleration as by an accelerometer) experienced by an object. It is thus acceleration relative to a free-fall, or inertial, observer who is momentarily at rest relative to the object being measured. Gravitation therefore does not cause proper acceleration."

I didn't mean to imply a coordinate system where the object isn't moving. In a phase plane the object is moving at constant velocity around the circle, as measured by the change in the phase angle.

#1 - You've transformed to phase space, but when you say 'constant velocity', that is not what we typically mean by velocity, because velocity is essentially one of your new coordinates. And if the coordinate moves in a circle then the velocity is changing, thus an accleration.

#2 - If we're treating phase space as real, then moving in a circle can't be done at constant velocity, only constant speed. If the direction is changing, then there is an accleration, as with a circular orbit.

 

[Resha Caner]

Nope, they read zero in freefall. "An accelerometer measures proper acceleration, which is the acceleration it experiences relative to freefall". "In relativity theory, proper acceleration[1] is the physical acceleration (i.e., measurable acceleration as by an accelerometer) experienced by an object. It is thus acceleration relative to a free-fall, or inertial, observer who is momentarily at rest relative to the object being measured. Gravitation therefore does not cause proper acceleration."

OK. I'll concede I spoke erroneously. However, the wikipedia references were not consistent, and it took some time for me to sort out what was being said and comparing it to what you have said. In addition I think we have some semantic/definitional issues here.

Bear with me then. I'm going to make some very nit-picky distinctions, but I think it's necessary at this point to be sure I understand you. An accelerometer doesn't directly measure proper acceleration. What happens (for piezoelectric accelerometers) is the deflection of a crystal that creates a charge, etc. ... or we could schematically think of it as measuring the distance my hypothetical spring deflects. We then associate the deflection with a force (through f = kx as I mentioned earlier). And, because we propose that elastic and inertial forces have a one-to-one equivalence relation, we associate the force with an acceleration (through f = ma).

I go through all that because in one place wikipedia says accelerometers measure proper acceleration, and in another place it says they measure proper force. Depending on the semantic context, either is basically correct. In other places within the wikipedia references it also speaks of weight and (snort!) gravitational force. Again, it seems all just a matter of semantics.

All in all, what I interpret them to be saying is that we actually have 2 masses connected by a spring. However, in a free state gravity is acting equally on both masses. Because of that, there is no relative deflection between the two masses due to gravity. Hence, the spring produces no force and we detect nothing in our measurements. It is only when we restrain one of the masses (apply a force to it) that deflection occurs and we detect something that can be interpreted as acceleration.

I still maintain that we have multiple options as to how to define what has occurred. But, I'll not push against chosen convention. That would only make this messier than it needs to be. I believe I now understand what you mean.

#1 - You've transformed to phase space, but when you say 'constant velocity', that is not what we typically mean by velocity, because velocity is essentially one of your new coordinates. And if the coordinate moves in a circle then the velocity is changing, thus an accleration.

#2 - If we're treating phase space as real, then moving in a circle can't be done at constant velocity, only constant speed. If the direction is changing, then there is an accleration, as with a circular orbit.

I did think of what you're saying here, and I understand your point. However, per your example with the planes flying a geodesic (a great circle on a sphere), aren't you saying they fly at constant velocity, and therefore have no proper acceleration?

Isn't that because "space" has supposedly been curved? I realize my earlier explanation was muddled. Even though the motion is drawn on a flat phase plane, I could likewise assume a curved space. So, is there some space such that the circle sketched by the "phase plane" motion follows a geodesic?

 

[essentialsaltes]

An accelerometer doesn't directly measure proper acceleration.

Oh, I think it does.

we could schematically think of it as measuring the distance my hypothetical spring deflects. We then associate the deflection with a force (through f = kx as I mentioned earlier).

Indeed, yes. I mean, basically, an accelerometer is a spring scale. We have a spring of length L, and we hang a mass off the end of it. If we do this near the surface of the earth, the spring stretches out by some amount. This is a nonzero reading. The accelerometer measures it. This is what we call weight. Now, in many circumstances, we 'reset' this and say it's zero. That there is no acceleration. But literally, the spring stretches. There is a nonzero measurement.

But if you were in a freefalling elevator with a scale, the scale would register zero, if you 'stood' on it. When you are only influenced by gravity (which is not a force) the accelerometer measures zero acceleration. Nothing stretches out the spring from its normal length.

I did think of what you're saying here, and I understand your point. However, per your example with the planes flying a geodesic (a great circle on a sphere), aren't you saying they fly at constant velocity, and therefore have no proper acceleration?

It's just an analogy. They are flying the geodesics over a 2-sphere. If they measured their motion against the stars, they would see that their velocity changes. Their velocity is not constant.

 

[Resha Caner]

It's just an analogy. They are flying the geodesics over a 2-sphere. If they measured their motion against the stars, they would see that their velocity changes. Their velocity is not constant.

Then you've managed to confuse me again. Let's put the airplanes aside.

One of the new things I took from your wikipedia references was that the inertial frame minimizes the number of forces we need to consider. "Fictitious" forces such as the coriolis disappear (I always hated dealing with the coriolis force anyway, so I'm happy to join that bandwagon and call it fictitious). The inertial frame is the reference frame with the simplest, most parsimonious physics. Yes?

So, I'm still thinking the spring-mass motion takes the form of a circle. If a circle is to be a geodesic, doesn't that mean it would need to be the great circle of a sphere? So, if I draw the spring-mass motion as a great circle on a spherical space, and that motion is constant, does the motion in that space represent unforced motion?

If not, is there some other space ... elliptical, hyperbolic, whatever?

 

[Kylie]

A spinning wheel though...

A point on the outside edge of a spinning wheel DOES have a force acting on it. The fact that it is attached to a fixed axis of rotation.

 

[[serious]]

Then you've managed to confuse me again. Let's put the airplanes aside.

One of the new things I took from your wikipedia references was that the inertial frame minimizes the number of forces we need to consider. "Fictitious" forces such as the coriolis disappear (I always hated dealing with the coriolis force anyway, so I'm happy to join that bandwagon and call it fictitious). The inertial frame is the reference frame with the simplest, most parsimonious physics. Yes?

So, I'm still thinking the spring-mass motion takes the form of a circle. If a circle is to be a geodesic, doesn't that mean it would need to be the great circle of a sphere? So, if I draw the spring-mass motion as a great circle on a spherical space, and that motion is constant, does the motion in that space represent unforced motion?

If not, is there some other space ... elliptical, hyperbolic, whatever?

Orbital paths are eliptucal, not circular. A circular orbit is a special case where both focal points have the same coordinates. If you are looking at doing something with an object in orbit being your reference point, you will have to account for both changes in position, direction, AND changes in speed at different parts of the orbit.

 

[essentialsaltes]

The inertial frame is the reference frame with the simplest, most parsimonious physics. Yes?

That's not the goal. An inertial frame has no proper acceleration.

So, I'm still thinking the spring-mass motion takes the form of a circle.

A mass bobbing up and down on a spring moves in one dimension. If you make a 2 dimensional space with the physical coordinate on one axis, and the velocity on another axis, and scale it right, the trajectory would appear as a circle.

If a circle is to be a geodesic, doesn't that mean it would need to be the great circle of a sphere? So, if I draw the spring-mass motion as a great circle on a spherical space, and that motion is constant, does the motion in that space represent unforced motion?

You chose a coordinate to make your circle, and you could choose a third coordinate and fit your circle into a sphere. But that third coordinate has no obvious (to me) meaning. Unforced motion would be motion without proper acceleration. As we look at the circle, we see that the velocity coordinate is changing back and forth, so we know the acceleration is non-zero. So we know there is a force at work.

 

[Resha Caner]

That's not the goal. An inertial frame has no proper acceleration.

Yes, I realize that. But it is the result. I'll assume I'm correct in that since you didn't disagree.

Unforced motion would be motion without proper acceleration. As we look at the circle, we see that the velocity coordinate is changing back and forth, so we know the acceleration is non-zero. So we know there is a force at work.

I've disconnected from those associations. It is just a great circle on a sphere with an object that has a constant motion.

You chose a coordinate to make your circle, and you could choose a third coordinate and fit your circle into a sphere. But that third coordinate has no obvious (to me) meaning.

Indeed, I did consider that what I'm proposing may be arbitrary and without meaning. However, after some thought, I see what the 3rd coordinate could possibly mean. I'd have to work through the math (if I'm even capable), but before I make the effort it would be nice to know a bit more.

So ... though it seems you may be losing interest ... how about this approach: forget the spring-mass. It may be creating a paradigm that's standing in the way.

Instead, let's look only at the 2-sphere. That is the "space" in question. And the question is: How can an object move in that space without accelerating?

 

[[serious]]

Instead, let's look only at the 2-sphere. That is the "space" in question. And the question is: How can an object move in that space without accelerating?

The only way I see would be to define the origin as an accelerating body and then look at an object not under the influence of that body. I'm not sure why you would want to though. It doesn't change anything, just makes the math harder.

Are you trying to do something like make a geocentric model the solar system? Even if you do, the earth will still be experiencing acceleration.

 

[essentialsaltes]

Instead, let's look only at the 2-sphere. That is the "space" in question. And the question is: How can an object move in that space without accelerating?

If the 2-sphere is literally the space in question, then objects can move at constant velocity on any great circle route.

(Unlike airplanes which actually live in a 3 dimensional space, and we 'pretend' they are confined to a 2-sphere that matches the surface of the earth.)

 

[Resha Caner]

Are you trying to do something like make a geocentric model the solar system? Even if you do, the earth will still be experiencing acceleration.

No, I'm not interested in astronomy. My interests are in machine performance & control - specifically in nonlinear behaviors.

If the 2-sphere is literally the space in question, then objects can move at constant velocity on any great circle route.

Cool. So now let's apply a force. Given our space, do all forces have to be tangent to the 2-sphere? I mean, the object can't leave space. So, if we applied a force normal to the sphere, nothing would happen. Yes? So, in essence, normal forces don't exist in our space ...

Unless ...

Can we think of normal forces as a "pressure" on space? is it possible for the shape of space to change? For example, as the earth orbits the sun, the curvature of space due to its gravity moves with it. Yes? So gravity is like a pressure on space. We could likewise have a pressure that expands or contracts the sphere. Does that work?

 

[essentialsaltes]

Given our space, do all forces have to be tangent to the 2-sphere? I mean, the object can't leave space. So, if we applied a force normal to the sphere, nothing would happen. Yes?

There is no place for those forces to live. You said the space was a 2-sphere, so if you now want to embed it in a 3-dimensional space and have forces pointing in other directions, you can do that, but you're changing the space.

Can we think of normal forces as a "pressure" on space? is it possible for the shape of space to change? For example, as the earth orbits the sun, the curvature of space due to its gravity moves with it. Yes? So gravity is like a pressure on space. We could likewise have a pressure that expands or contracts the sphere. Does that work?

Work? I dunno. But yes, you could give your space some dynamics. But these are not forces on your object.

 

[Resha Caner]

Work? I dunno. But yes, you could give your space some dynamics. But these are not forces on your object.

Cool. It's even better that the dynamics of the space are not forces on the object. I was just thinking that as gravitational effects on space are proportional to mass, so effects on my space would be proportional to the mass/stiffness ratio.

There is no place for those forces to live. You said the space was a 2-sphere, so if you now want to embed it in a 3-dimensional space and have forces pointing in other directions, you can do that, but you're changing the space.

No, I was just asking. I began to realize that even though I could claim this space is similar to some mechanical system, it could also be similar to other things. In essence, the space doesn't have all the right constraints to yield a particular solution. So, rather than going to a 3-space, I was actually wondering if I would need to go to a 1-sphere.