So we're arguing the very clear definition of gravity vs the effects of gravity of a given planetary body acting upon an object when accounting for effects that would negate the perceived effects of gravity at a given location?
No, they both agree on that. What they're arguing is the existance of the centripetal force due to earth's rotation.
Which ,by the way:
Quote:
There simply is no force trying to throw things off the surface of the Earth because the earth is spinning
Quote:
the Earth is an oblate spheroid bulging towards the equator, not a true sphere,
Explain to me how those two do not contradict each other?
No, they both agree on that. What they're arguing is the existance of the centripetal force due to earth's rotation.
I know - it's embarrassing that this has gone on for so long, but it's hard to drop. And just to clarify, the centripetal force is not due to the earth's rotation, it is required to keep an object on the equator moving in a circular path, and is provided by a (small) part of the gravitational attraction between the object and the earth. The remainder of the gravitational attraction is what we perceive and measure as weight, and is where the discussion started - with someone or other correctly mentioning that the rotation of the earth reduces weight at the equator relative to at the poles.
The argument began when another poster disputed that, but I thought at first that he was really just trying to point out, as Soli summarized, that the magnitude of the gravitational field vector at a point in space is independent of the frame of reference of the observer, provided that frame of reference is steady and non-acceleratory. That, too, is correct, but as an observer standing on the equator, that condition is not satisfied because your frame of reference is accelerating (centripetally) and the measured gravitational attraction (weight) is reduced accordingly. However, it then turned out that he was actually wrongly disputing the very existence of centripetal acceleration in steady circular motion.
Or possibly he just responded in too much of a hurry and didn't really mean to say that. If he replies again we may find out.
I love this forum because of the very diverse and in-depth conversations(arguments) that often occur. I am a member of many forums yet this only happens on AI's forum. It's great! I certainly learned something about gravity from this thread.
I love this forum because of the very diverse and in-depth conversations(arguments) that often occur. I am a member of many forums yet this only happens on AI's forum. It's great! I certainly learned something about gravity from this thread.
I agree. I think it is a reflection of the diverse range of expertise of posters here. I wish I could contribute more to the forum's core subject areas, but mostly I end up just listening and learning.
I love this forum because of the very diverse and in-depth conversations(arguments) that often occur. I am a member of many forums yet this only happens on AI's forum. It's great! I certainly learned something about gravity from this thread.
Dont get me wrong, thats great. It just feels like in this case, theyre starting to argue for the sake of arguing (at least thats the impression i get from the last few posts). Im glad you learned something from it tho.
Dont get me wrong, thats great. It just feels like in this case, theyre starting to argue for the sake of arguing (at least thats the impression i get from the last few posts). Im glad you learned something from it tho.
I'd like to claim that's not the case - I certainly hope I'm not doing that. From my perspective, differences in opinion are just fine and I have no problem letting them go, but fundamental arguments on fact or reasoning (or both in this case), especially in my own field, will keep me coming back until a resolution is reached, or until it becomes clear that a resolution is not the goal of the discussion.
I'd like to claim that's not the case - I certainly hope I'm not doing that. From my perspective, differences in opinion are just fine and I have no problem letting them go, but fundamental arguments on fact or reasoning (or both in this case), especially in my own field, will keep me coming back until a resolution is reached, or until it becomes clear that a resolution is not the goal of the discussion.
From my perspective, he already stated that the earth budges out at the equator. So he already implies that he understands that there is a centripetal force; unless he thinks theres a different explaination for the fact that the earth is not a sphere (which i would be interested to hear). So when he then throws out the "no such thing as centripetal force" it feels like hes just doing it to continue the argument.
With regards to the original confusion, i really think we should just set a hard definition of gravitational force, at least for this thread. Because it can get confusing when different posters refer to gravity to mean different things. Personally i perfer gravity to refer to solely the force from the fact that there is mass i.e. GMm/r^2. And we refer to the total force @ the earth's surface as just that, or F(es).
From my perspective, he already stated that the earth budges out at the equator. So he already implies that he understands that there is a centripetal force; unless he thinks theres a different explaination for the fact that the earth is not a sphere (which i would be interested to hear). So when he then throws out the "no such thing as centripetal force" it feels like hes just doing it to continue the argument.
Yes - I missed that point in your previous post, and of course you are correct - that is the reason for the oblate shape. However, I'm not sure that he realizes that. I also don't think he was denying the existence of centripetal acceleration just to prolong the argument, because that assertion is too demonstrably wrong - I think he just got confused somehow. Pretty determinedly confused I admit.
Quote:
Originally Posted by majjo
With regards to the original confusion, i really think we should just set a hard definition of gravitational force, at least for this thread. Because it can get confusing when different posters refer to gravity to mean different things. Personally i perfer gravity to refer to solely the force from the fact that there is mass i.e. GMm/r^2. And we refer to the total force @ the earth's surface as just that, or F(es).
No problem there, except, as I mentioned earlier, the original discussion was about the reduction in weight of an object at the equator relative to the poles. A discussion of just the gravitational field strength does not fully address that effect, because it is partly due to the oblate shape of the earth, so that you are farther from the center of mass of the earth when at the equator (and the field strength is inversely proportional to r²), and partly due to the centripetal acceleration at the equator. In fact the effects are almost equal.
And just to nip in the bud any further question about centripetal acceleration, here is an a priori derivation of centripetal acceleration for motion in a circle, of radius r, at a constant angular velocity, ω. Working within the formatting options available, omitting the italics for variables convention for readability and using bold face to denote vectors, and using polar coordinates (r, φ) for simplicity, the circular motion is defined in terms of the unit x and y vectors (x, y) by:
r = r(ysinφ + xcosφ) . [1]
For constant motion where the angular velocity dφ/dt is ω and φ(t) = ωt, we can write the time-dependent position vector r(t) as
r(t) = r(ysinωt + xcosωt) . [2]
The velocity vector, v(t) is dr/dt, so differentiating [2] w.r.t. t we get
v(t) = ωr(ycosωt - xsinωt) . [3]
The acceleration vector, a(t), is dv/dt, so differentiating again w.r.t. t, we get
a(t) = - ω²r(ysinωt + xcosωt) . [4]
Substituting r(t) for r(ysinωt + xcosωt) from [1] in the R.H.S. of [4] we get
a(t) = - ω²r(t) , [5]
which is a centripetal acceleration of constant magnitude ω²r parallel to the r vector and directed towards (hence the -ve sign) the center of the circle.
So we're arguing the very clear definition of gravity vs the effects of gravity of a given planetary body acting upon an object when accounting for effects that would negate the perceived effects of gravity at a given location?
Some mess like that. No reasoning with Wikipedia Physicists.
And just to nip in the bud any further question about centripetal acceleration, here is an a priori derivation of centripetal acceleration for motion in a circle, of radius r, at a constant angular velocity, ω.
Which is pretty meaningless because you cannot compare there results of a Cartesian reference frame directly to results of the polar coordinate reference frame. Pick the right reference frame and the problems get simple, pick the wrong one and you spend a lot of effort constructing virtual vectors to balance things out.
Yes - I missed that point in your previous post, and of course you are correct - that is the reason for the oblate shape. However, I'm not sure that he realizes that. I also don't think he was denying the existence of centripetal acceleration just to prolong the argument, because that assertion is too demonstrably wrong - I think he just got confused somehow. Pretty determinedly confused I admit.
What you keep missing is that centripetal acceleration is not gravity, nor any component of gravity! You can keep on for ages about all kinds of fruffery, but gravity is gravity.
So maybe you should lose the judgements of who thinks what about what causes what and get back to actually getting your basic terminology correct. 'Cause if you can't get the basics right there's no place to go.
Which is pretty meaningless because you cannot compare there results of a Cartesian reference frame directly to results of the polar coordinate reference frame. Pick the right reference frame and the problems get simple, pick the wrong one and you spend a lot of effort constructing virtual vectors to balance things out.
Bullshit - of course you can. I'm starting to doubt whether you even know the meaning of the technical terms you are throwing around. Do you actually have any physics or mathematics training or are you just trying, unsuccessfully, to pass yourself off as knowledgeable? Observations in any inertial frame of reference are identical - that is the definition of an inertial frame of reference. You are confusing inertial and non-inertial frames of reference. The coordinate system, polar, cartesian, spherical, or other arbitrary system, within an inertial frame of reference is doubly irrelevant. Didn't you know that?
Either way, now you are just obfuscating. Are you, or are you not, still denying the existence of centripetal acceleration for constant circular motion, as you did in post 117 and used as your basis to discard my entire analysis as flawed? Actually don't bother to answer that. You haven't come up with a cogent or accurately reasoned argument since this discussion started, and I can conclude with reasonable certainty that you are not having this discussion in good faith.
What you keep missing is that centripetal acceleration is not gravity, nor any component of gravity! You can keep on for ages about all kinds of fruffery, but gravity is gravity.
So maybe you should lose the judgements of who thinks what about what causes what and get back to actually getting your basic terminology correct. 'Cause if you can't get the basics right there's no place to go.
No one, as far as I can tell, has been suggesting any such equality - that is another of your diversions. All that was being discussed was the variation in weight of an object with latitude, and its causes - at least until you showed up and derailed the whole subject with your nonsense. Nice work, if that is what you intended.
Basics? What would you know about basics? In this field, virtually nothing, as you keep demonstrating. You haven't engaged technically with a single point that I have made (apart from your lame and flawed attempt to dispute centripetal acceleration) - just dismissed them and then made up seemingly random combinations of technical terms that presumably you are used to passing off as intelligent comment. I'm sorry, but you are a ridiculously frustrating person to argue with.
Explain to me how those two do not contradict each other?
I flipped the words centrifugal and centripetal when responding to a post about centrifugal force reducing the feeling of gravity (@mstones post). A terminology faux pas, but the math was always and still is still correct when you unflip my boo-boo term. How nobody else realized that and mentioned it then this is surprising having a look back at the original post.
We can both agree that @mstones quoted virtual centrifugal force absolutely cannot reduce the force of gravity because it doesn't actually exist, not to mention the fact that the virtual centrifugal vector points in a direction that would add to the apparent gravity vector, contradicting a lessening.
As for the rest, it's simple. Gravity is gravity. I don't care about centripetal forces as a component of gravity, because they aren't -- that has been my whole point. Yes they are why the earth accreted the way is did, but they are NOT a component of gravity. Period. Ever. (combined forces apparent gravity != gravity)
I flipped the words centrifugal and centripetal when responding to a post about centrifugal force reducing the feeling of gravity (@mstones post). A terminology faux pas, but the math was always and still is still correct when you unflip my boo-boo term. How nobody else realized that and mentioned it then this is surprising having a look back at the original post.
OK this is better. Let's have one more attempt to see if we can reconcile this. I have no problem forgiving a mistyped word, but I was referring to post 117 where you said:
"Bad assumptions. You are trying to measure from the wrong frame and wrong coordinate system to make that direct comparison. When the frame of reference is the center of rotation of the Earth there is no radial acceleration of an object on it's surface because the Earth is for all intent and purposes rotating at constant velocity (on a scale of billions of years)."
You used that to dispute the simple derivation of weight at the equator that I posted in post 116. However, the only assumptions that I made were the equations of motion for simple, constant, circular motion, which I then derived from first principles in post 131.
Further, those equations are independent of the frame of reference, provided that it is an inertial frame of reference. You said "when the frame of reference is the center of rotation of the earth". That would be an inertial frame of reference if it does not rotate with the earth, or a non-inertial frame of reference if it does rotate with the earth. Since you ruled out the existence of centrifugal forces (which only exist in non-inertial rotating frames of reference), I concluded that you too were referring to the non-rotating inertial frame in which my analysis was conducted.
Quote:
Originally Posted by Hiro
We can both agree that @mstones quoted virtual centrifugal force absolutely cannot reduce the force of gravity because it doesn't actually exist, not to mention the fact that the virtual centrifugal vector points in a direction that would add to the apparent gravity vector, contradicting a lessening.
I agree that in the non-rotating, inertial frame of reference of the center of the earth, there is no centrifugal force - but two caveats:
In the rotating, non-inertial frame of reference that is the apparent frame of reference to an observer on the surface of the earth, that force does exist, and the analysis in that frame of reference does produce exactly the same result.
In that non-inertial frame of reference, the centrifugal force vector points outwards (+ve r direction), so it is in opposition to the gravitational force vector that points towards the center of the earth (-ve r direction) and would reduce it.
However, let's stay in the inertial frame of reference where we don't have a centrifugal force.
Quote:
Originally Posted by Hiro
As for the rest, it's simple. Gravity is gravity. I don't care about centripetal forces as a component of gravity, because they aren't -- that has been my whole point. Yes they are why the earth accreted the way is did, but they are NOT a component of gravity. Period. Ever. (combined forces apparent gravity != gravity)
Well yes, I agree, but that's not what you said earlier. And it is not actually that simple, because we were discussing g, our measure of gravity on earth. I made that distinction clear in #116, and yet you still dismissed the analysis as flawed for a bunch of other spurious reasons, most notably claiming (see above) that there was no centripetal acceleration on the equator.
So - let me just restate the argument one more time to make sure that we are not just talking about different things. Maybe you can identify precisely where you disagree, if you still do.
The original discussion was about variation in weight, W, with latitude, where the weight of an object of mass m is defined by
W = mg . [1]
From my derivation in #131 that I won't write out again, the centripetal acceleration, a, of an object on the equator, equatorial radius R, angular velocity of rotation ω, is
a = - ω²R . [2]
The force, F, exerted on the object due to the gravitational attraction of the earth, mass M, is given by
F = - GMm/R² . [3]
W, the weight of the object, is by definition the normal reaction between the object and the surface of the earth, and since the resultant force on the object must equal the product of its mass and acceleration,
F + W = ma . [4]
Substituting for W, a and F from [1], [2] and [3] respectively, we get
- GMm/R² + mg = - aω²R . [5]
Rearranging, and dividing through by m,
g = GM/R² - ω²R . [6]
Writing this more generally than for just on the equator, for latitude L, since we need two different Rs for the two terms on the RHS, one for distance to the center of the earth, R(L), in the first term, and one for distance to the axis of rotation, r(L), in the second term, and take into account that the acceleration vector is normal to the axis of rotation rather than the surface of the earth
g = GM/R² - ω²rsinL , [7]
or explicitly writing r = RsinL
g = GM/R² - ω²Rsin²L , [7]
OK - so this is the g that we use as acceleration due to gravity on earth - it is what we multiply mass by to get weight, and it is the acceleration towards the earth's surface of an object when dropped - and it varies with latitude, not only because the oblate shape of the earth means that R varies slightly, but also because L goes from 90˚ at the poles to 0˚ at the equator. It is the quantity that was being discussed.
In our inertial frame of reference, no question that it differs from the gravitational field strength (GM/R²) by the centripetal acceleration, as I pointed out in an earlier post when I was trying to reconcile the different views. Alternatively, you can view it as equal to the gravitational field strength in the frame of reference of an observer on the surface of the earth, which as I mentioned earlier, is a rotating, non-inertial frame of reference.
And, as a side note related to that observation, remember that although centripetal acceleration and gravity are different in cause, a central result of Newtonian physics is that gravity and acceleration at a point are indistinguishable - in other words an infinitesimally small* observer cannot tell whether he is accelerating in an inertial frame of reference versus moving at constant velocity in an inertial frame of reference in the presence of a gravitational field.
*Only holds for infinitesimally small, since for an observer of finite extent, gravitational attraction will vary detectably with R, whereas actual acceleration will not.
Quote:
Originally Posted by Hiro
What is so freaking hard about this?
Well - I'd really like to ask you the same question. Are we done?
I love this site! There is no other forum I frequent that can have so many in-depth discussion about so many different subjects. If you can't learn something here you aren't trying.
So - let me just restate the argument one more time to make sure that we are not just talking about different things. Maybe you can identify precisely where you disagree, if you still do.
The original discussion was about variation in weight.
Maybe we can pause right there. Weight and gravity were originally conflated, a problem. All along I have talked only about gravity and the need to respect the difference between gravity and other gravity like forces. Weight is not gravity, weight is the result of a mass accelerated (pulled even if unmoving) by gravity and other forces (centripetal being one) at the Earth's (Moons, etc. ...) surface. That is also roughly the Apparent Gravity discussion on the Wikipedia gravity page which was cited as a counter argument to my calling out the incorrect weight/gravity use, and my counter-counter post that we were in the midst of a bad misinterpretation of gravity and what that wikipage actually said.
All along I have only held that those extra force components of Apparent Gravity are not Gravity, just a composite/resultant force that is very close to the actual force of gravity. That is why I keep saying as far as gravity goes I don't care about the centripetal accelerations.
And that directly extends to weight, which rolls back around to where this all started -- the mess over pounds as mass, pounds as weight (and my snide eventual comment about pounds as currency) and the various early posters seemingly willing to ignore they were dealing with an overloaded term.
The remainder of your post looks mostly OK, up until you call your g the gravity we use on Earth. Your g is not gravity, it is a version of apparent gravity (the interaction between gravity and ideal spherical centripetal acceleration).
It is those little slips in technical terminology precision that end up perpetuating the issue.
Quote:
Originally Posted by muppetry
And, as a side note related to that observation, remember that although centripetal acceleration and gravity are different in cause, a central result of Newtonian physics is that gravity and acceleration at a point are indistinguishable - in other words an infinitesimally small* observer cannot tell whether he is accelerating in an inertial frame of reference versus moving at constant velocity in an inertial frame of reference in the presence of a gravitational field.
That's D'Alambert's Principle (the first part), the second part is explained by Einstein in describing General Relativity. D'Alambert just makes the math easier, and unfortunately easier to safely miss-mash the math together. Convenient for a problem at a time, but the mis-mash methods are not necessarily good for generality. Better to save the vector combinations for as late in the evaluation as humanly possible. General Relativity relates interpretations between frames of reference, not the math in each one (apparently the explanations were a mess before that, but I'm not that old nor a physics historian who keeps track of who though what when).
As for whether or not the gravity and the acceleration-at-a-point are are indistinguishable, that is a 'it depends' statement. To an observer standing on the ground with human senses, yes indistinguishable; to a gravitometer calibrated for lat/long and GPS altitude, no quite distinguishable*. Also they are very distinguishable when you are trying to do physical simulations. If you combine the equations algebraically as you have done the math gets ever more nasty when also doing body-body interactions. When you keep the components separated, use sensible local origins and use geometric methods the math is much simpler (though there are more discrete operations), it's also easier to program and extend for more generality (especially important when you don't know what the next question for the sim will be). Seeing as I'm very involved in the latter issues, the side of 'it depends' I fall on defaults to separate components rather than combined far less flexible solutions (try to dynamically simulate something as simple as long range rocket assisted sensor platform delivery using your combined apparent gravity equation compared to by-component methods).
Quote:
Originally Posted by muppetry
Well - I'd really like to ask you the same question. Are we done?
We should be, it's just a case of say what we mean, mean what we say. Not something almost and incredibly close to like we mean or say. It's just about maintaining discipline when using terminology and I see great benefit with precise use of terminology.
* The recently reaching lunar orbit gravity mapping mission on the moon will be measuring actual Moon gravity by using differences in the orbital mechanics of two satellites in the same orbit causing varying distances between the spacecraft and measuring those relative distances via femtosecond accurate timing methods. That will allow the team to measure local orbital perturbations via the satellites relatively varying centripetal accelerations and then use the information to isolate the actual gravitation component for a given surface grid location. Again very discriminating despite the individual spacecraft effectively "feeling" a unified gravitational and orbital centripetal acceleration.
Comments
So we're arguing the very clear definition of gravity vs the effects of gravity of a given planetary body acting upon an object when accounting for effects that would negate the perceived effects of gravity at a given location?
No, they both agree on that. What they're arguing is the existance of the centripetal force due to earth's rotation.
Which ,by the way:
There simply is no force trying to throw things off the surface of the Earth because the earth is spinning
the Earth is an oblate spheroid bulging towards the equator, not a true sphere,
Explain to me how those two do not contradict each other?
Oh god, this thread is still going?
No, they both agree on that. What they're arguing is the existance of the centripetal force due to earth's rotation.
I know - it's embarrassing that this has gone on for so long, but it's hard to drop. And just to clarify, the centripetal force is not due to the earth's rotation, it is required to keep an object on the equator moving in a circular path, and is provided by a (small) part of the gravitational attraction between the object and the earth. The remainder of the gravitational attraction is what we perceive and measure as weight, and is where the discussion started - with someone or other correctly mentioning that the rotation of the earth reduces weight at the equator relative to at the poles.
The argument began when another poster disputed that, but I thought at first that he was really just trying to point out, as Soli summarized, that the magnitude of the gravitational field vector at a point in space is independent of the frame of reference of the observer, provided that frame of reference is steady and non-acceleratory. That, too, is correct, but as an observer standing on the equator, that condition is not satisfied because your frame of reference is accelerating (centripetally) and the measured gravitational attraction (weight) is reduced accordingly. However, it then turned out that he was actually wrongly disputing the very existence of centripetal acceleration in steady circular motion.
Or possibly he just responded in too much of a hurry and didn't really mean to say that. If he replies again we may find out.
Oh god, this thread is still going?
I love this forum because of the very diverse and in-depth conversations(arguments) that often occur. I am a member of many forums yet this only happens on AI's forum. It's great! I certainly learned something about gravity from this thread.
I love this forum because of the very diverse and in-depth conversations(arguments) that often occur. I am a member of many forums yet this only happens on AI's forum. It's great! I certainly learned something about gravity from this thread.
I agree. I think it is a reflection of the diverse range of expertise of posters here. I wish I could contribute more to the forum's core subject areas, but mostly I end up just listening and learning.
Oh god, this thread is still going?
For good reason, I think. Besies, it's not the only thread lingering on. Check here
I love this forum because of the very diverse and in-depth conversations(arguments) that often occur. I am a member of many forums yet this only happens on AI's forum. It's great! I certainly learned something about gravity from this thread.
Dont get me wrong, thats great. It just feels like in this case, theyre starting to argue for the sake of arguing (at least thats the impression i get from the last few posts). Im glad you learned something from it tho.
Dont get me wrong, thats great. It just feels like in this case, theyre starting to argue for the sake of arguing (at least thats the impression i get from the last few posts). Im glad you learned something from it tho.
I'd like to claim that's not the case - I certainly hope I'm not doing that. From my perspective, differences in opinion are just fine and I have no problem letting them go, but fundamental arguments on fact or reasoning (or both in this case), especially in my own field, will keep me coming back until a resolution is reached, or until it becomes clear that a resolution is not the goal of the discussion.
I'd like to claim that's not the case - I certainly hope I'm not doing that. From my perspective, differences in opinion are just fine and I have no problem letting them go, but fundamental arguments on fact or reasoning (or both in this case), especially in my own field, will keep me coming back until a resolution is reached, or until it becomes clear that a resolution is not the goal of the discussion.
From my perspective, he already stated that the earth budges out at the equator. So he already implies that he understands that there is a centripetal force; unless he thinks theres a different explaination for the fact that the earth is not a sphere (which i would be interested to hear). So when he then throws out the "no such thing as centripetal force" it feels like hes just doing it to continue the argument.
With regards to the original confusion, i really think we should just set a hard definition of gravitational force, at least for this thread. Because it can get confusing when different posters refer to gravity to mean different things. Personally i perfer gravity to refer to solely the force from the fact that there is mass i.e. GMm/r^2. And we refer to the total force @ the earth's surface as just that, or F(es).
From my perspective, he already stated that the earth budges out at the equator. So he already implies that he understands that there is a centripetal force; unless he thinks theres a different explaination for the fact that the earth is not a sphere (which i would be interested to hear). So when he then throws out the "no such thing as centripetal force" it feels like hes just doing it to continue the argument.
Yes - I missed that point in your previous post, and of course you are correct - that is the reason for the oblate shape. However, I'm not sure that he realizes that. I also don't think he was denying the existence of centripetal acceleration just to prolong the argument, because that assertion is too demonstrably wrong - I think he just got confused somehow. Pretty determinedly confused I admit.
With regards to the original confusion, i really think we should just set a hard definition of gravitational force, at least for this thread. Because it can get confusing when different posters refer to gravity to mean different things. Personally i perfer gravity to refer to solely the force from the fact that there is mass i.e. GMm/r^2. And we refer to the total force @ the earth's surface as just that, or F(es).
No problem there, except, as I mentioned earlier, the original discussion was about the reduction in weight of an object at the equator relative to the poles. A discussion of just the gravitational field strength does not fully address that effect, because it is partly due to the oblate shape of the earth, so that you are farther from the center of mass of the earth when at the equator (and the field strength is inversely proportional to r²), and partly due to the centripetal acceleration at the equator. In fact the effects are almost equal.
So we're arguing the very clear definition of gravity vs the effects of gravity of a given planetary body acting upon an object when accounting for effects that would negate the perceived effects of gravity at a given location?
Some mess like that. No reasoning with Wikipedia Physicists.
And just to nip in the bud any further question about centripetal acceleration, here is an a priori derivation of centripetal acceleration for motion in a circle, of radius r, at a constant angular velocity, ω.
<snipped entirely valid rectangular reference plane cycloid motion derivation>
Which is pretty meaningless because you cannot compare there results of a Cartesian reference frame directly to results of the polar coordinate reference frame. Pick the right reference frame and the problems get simple, pick the wrong one and you spend a lot of effort constructing virtual vectors to balance things out.
Yes - I missed that point in your previous post, and of course you are correct - that is the reason for the oblate shape. However, I'm not sure that he realizes that. I also don't think he was denying the existence of centripetal acceleration just to prolong the argument, because that assertion is too demonstrably wrong - I think he just got confused somehow. Pretty determinedly confused I admit.
What you keep missing is that centripetal acceleration is not gravity, nor any component of gravity! You can keep on for ages about all kinds of fruffery, but gravity is gravity.
So maybe you should lose the judgements of who thinks what about what causes what and get back to actually getting your basic terminology correct. 'Cause if you can't get the basics right there's no place to go.
Which is pretty meaningless because you cannot compare there results of a Cartesian reference frame directly to results of the polar coordinate reference frame. Pick the right reference frame and the problems get simple, pick the wrong one and you spend a lot of effort constructing virtual vectors to balance things out.
Bullshit - of course you can. I'm starting to doubt whether you even know the meaning of the technical terms you are throwing around. Do you actually have any physics or mathematics training or are you just trying, unsuccessfully, to pass yourself off as knowledgeable? Observations in any inertial frame of reference are identical - that is the definition of an inertial frame of reference. You are confusing inertial and non-inertial frames of reference. The coordinate system, polar, cartesian, spherical, or other arbitrary system, within an inertial frame of reference is doubly irrelevant. Didn't you know that?
Either way, now you are just obfuscating. Are you, or are you not, still denying the existence of centripetal acceleration for constant circular motion, as you did in post 117 and used as your basis to discard my entire analysis as flawed? Actually don't bother to answer that. You haven't come up with a cogent or accurately reasoned argument since this discussion started, and I can conclude with reasonable certainty that you are not having this discussion in good faith.
What you keep missing is that centripetal acceleration is not gravity, nor any component of gravity! You can keep on for ages about all kinds of fruffery, but gravity is gravity.
So maybe you should lose the judgements of who thinks what about what causes what and get back to actually getting your basic terminology correct. 'Cause if you can't get the basics right there's no place to go.
No one, as far as I can tell, has been suggesting any such equality - that is another of your diversions. All that was being discussed was the variation in weight of an object with latitude, and its causes - at least until you showed up and derailed the whole subject with your nonsense. Nice work, if that is what you intended.
Basics? What would you know about basics? In this field, virtually nothing, as you keep demonstrating. You haven't engaged technically with a single point that I have made (apart from your lame and flawed attempt to dispute centripetal acceleration) - just dismissed them and then made up seemingly random combinations of technical terms that presumably you are used to passing off as intelligent comment. I'm sorry, but you are a ridiculously frustrating person to argue with.
Explain to me how those two do not contradict each other?
I flipped the words centrifugal and centripetal when responding to a post about centrifugal force reducing the feeling of gravity (@mstones post). A terminology faux pas, but the math was always and still is still correct when you unflip my boo-boo term. How nobody else realized that and mentioned it then this is surprising having a look back at the original post.
We can both agree that @mstones quoted virtual centrifugal force absolutely cannot reduce the force of gravity because it doesn't actually exist, not to mention the fact that the virtual centrifugal vector points in a direction that would add to the apparent gravity vector, contradicting a lessening.
As for the rest, it's simple. Gravity is gravity. I don't care about centripetal forces as a component of gravity, because they aren't -- that has been my whole point. Yes they are why the earth accreted the way is did, but they are NOT a component of gravity. Period. Ever. (combined forces apparent gravity != gravity)
What is so freaking hard about this?
I flipped the words centrifugal and centripetal when responding to a post about centrifugal force reducing the feeling of gravity (@mstones post). A terminology faux pas, but the math was always and still is still correct when you unflip my boo-boo term. How nobody else realized that and mentioned it then this is surprising having a look back at the original post.
OK this is better. Let's have one more attempt to see if we can reconcile this. I have no problem forgiving a mistyped word, but I was referring to post 117 where you said: You used that to dispute the simple derivation of weight at the equator that I posted in post 116. However, the only assumptions that I made were the equations of motion for simple, constant, circular motion, which I then derived from first principles in post 131.
Further, those equations are independent of the frame of reference, provided that it is an inertial frame of reference. You said "when the frame of reference is the center of rotation of the earth". That would be an inertial frame of reference if it does not rotate with the earth, or a non-inertial frame of reference if it does rotate with the earth. Since you ruled out the existence of centrifugal forces (which only exist in non-inertial rotating frames of reference), I concluded that you too were referring to the non-rotating inertial frame in which my analysis was conducted.
We can both agree that @mstones quoted virtual centrifugal force absolutely cannot reduce the force of gravity because it doesn't actually exist, not to mention the fact that the virtual centrifugal vector points in a direction that would add to the apparent gravity vector, contradicting a lessening.
I agree that in the non-rotating, inertial frame of reference of the center of the earth, there is no centrifugal force - but two caveats:
- In the rotating, non-inertial frame of reference that is the apparent frame of reference to an observer on the surface of the earth, that force does exist, and the analysis in that frame of reference does produce exactly the same result.
- In that non-inertial frame of reference, the centrifugal force vector points outwards (+ve r direction), so it is in opposition to the gravitational force vector that points towards the center of the earth (-ve r direction) and would reduce it.
However, let's stay in the inertial frame of reference where we don't have a centrifugal force.As for the rest, it's simple. Gravity is gravity. I don't care about centripetal forces as a component of gravity, because they aren't -- that has been my whole point. Yes they are why the earth accreted the way is did, but they are NOT a component of gravity. Period. Ever. (combined forces apparent gravity != gravity)
Well yes, I agree, but that's not what you said earlier. And it is not actually that simple, because we were discussing g, our measure of gravity on earth. I made that distinction clear in #116, and yet you still dismissed the analysis as flawed for a bunch of other spurious reasons, most notably claiming (see above) that there was no centripetal acceleration on the equator.
So - let me just restate the argument one more time to make sure that we are not just talking about different things. Maybe you can identify precisely where you disagree, if you still do.
The original discussion was about variation in weight, W, with latitude, where the weight of an object of mass m is defined by From my derivation in #131 that I won't write out again, the centripetal acceleration, a, of an object on the equator, equatorial radius R, angular velocity of rotation ω, is The force, F, exerted on the object due to the gravitational attraction of the earth, mass M, is given by W, the weight of the object, is by definition the normal reaction between the object and the surface of the earth, and since the resultant force on the object must equal the product of its mass and acceleration, Substituting for W, a and F from [1], [2] and [3] respectively, we get Rearranging, and dividing through by m, Writing this more generally than for just on the equator, for latitude L, since we need two different Rs for the two terms on the RHS, one for distance to the center of the earth, R(L), in the first term, and one for distance to the axis of rotation, r(L), in the second term, and take into account that the acceleration vector is normal to the axis of rotation rather than the surface of the earth or explicitly writing r = RsinL OK - so this is the g that we use as acceleration due to gravity on earth - it is what we multiply mass by to get weight, and it is the acceleration towards the earth's surface of an object when dropped - and it varies with latitude, not only because the oblate shape of the earth means that R varies slightly, but also because L goes from 90˚ at the poles to 0˚ at the equator. It is the quantity that was being discussed.
In our inertial frame of reference, no question that it differs from the gravitational field strength (GM/R²) by the centripetal acceleration, as I pointed out in an earlier post when I was trying to reconcile the different views. Alternatively, you can view it as equal to the gravitational field strength in the frame of reference of an observer on the surface of the earth, which as I mentioned earlier, is a rotating, non-inertial frame of reference.
And, as a side note related to that observation, remember that although centripetal acceleration and gravity are different in cause, a central result of Newtonian physics is that gravity and acceleration at a point are indistinguishable - in other words an infinitesimally small* observer cannot tell whether he is accelerating in an inertial frame of reference versus moving at constant velocity in an inertial frame of reference in the presence of a gravitational field.
*Only holds for infinitesimally small, since for an observer of finite extent, gravitational attraction will vary detectably with R, whereas actual acceleration will not.
What is so freaking hard about this?
Well - I'd really like to ask you the same question. Are we done?
So - let me just restate the argument one more time to make sure that we are not just talking about different things. Maybe you can identify precisely where you disagree, if you still do.
The original discussion was about variation in weight.
Maybe we can pause right there. Weight and gravity were originally conflated, a problem. All along I have talked only about gravity and the need to respect the difference between gravity and other gravity like forces. Weight is not gravity, weight is the result of a mass accelerated (pulled even if unmoving) by gravity and other forces (centripetal being one) at the Earth's (Moons, etc. ...) surface. That is also roughly the Apparent Gravity discussion on the Wikipedia gravity page which was cited as a counter argument to my calling out the incorrect weight/gravity use, and my counter-counter post that we were in the midst of a bad misinterpretation of gravity and what that wikipage actually said.
All along I have only held that those extra force components of Apparent Gravity are not Gravity, just a composite/resultant force that is very close to the actual force of gravity. That is why I keep saying as far as gravity goes I don't care about the centripetal accelerations.
And that directly extends to weight, which rolls back around to where this all started -- the mess over pounds as mass, pounds as weight (and my snide eventual comment about pounds as currency) and the various early posters seemingly willing to ignore they were dealing with an overloaded term.
The remainder of your post looks mostly OK, up until you call your g the gravity we use on Earth. Your g is not gravity, it is a version of apparent gravity (the interaction between gravity and ideal spherical centripetal acceleration).
It is those little slips in technical terminology precision that end up perpetuating the issue.
And, as a side note related to that observation, remember that although centripetal acceleration and gravity are different in cause, a central result of Newtonian physics is that gravity and acceleration at a point are indistinguishable - in other words an infinitesimally small* observer cannot tell whether he is accelerating in an inertial frame of reference versus moving at constant velocity in an inertial frame of reference in the presence of a gravitational field.
That's D'Alambert's Principle (the first part), the second part is explained by Einstein in describing General Relativity. D'Alambert just makes the math easier, and unfortunately easier to safely miss-mash the math together. Convenient for a problem at a time, but the mis-mash methods are not necessarily good for generality. Better to save the vector combinations for as late in the evaluation as humanly possible. General Relativity relates interpretations between frames of reference, not the math in each one (apparently the explanations were a mess before that, but I'm not that old nor a physics historian who keeps track of who though what when).
As for whether or not the gravity and the acceleration-at-a-point are are indistinguishable, that is a 'it depends' statement. To an observer standing on the ground with human senses, yes indistinguishable; to a gravitometer calibrated for lat/long and GPS altitude, no quite distinguishable*. Also they are very distinguishable when you are trying to do physical simulations. If you combine the equations algebraically as you have done the math gets ever more nasty when also doing body-body interactions. When you keep the components separated, use sensible local origins and use geometric methods the math is much simpler (though there are more discrete operations), it's also easier to program and extend for more generality (especially important when you don't know what the next question for the sim will be). Seeing as I'm very involved in the latter issues, the side of 'it depends' I fall on defaults to separate components rather than combined far less flexible solutions (try to dynamically simulate something as simple as long range rocket assisted sensor platform delivery using your combined apparent gravity equation compared to by-component methods).
Well - I'd really like to ask you the same question. Are we done?
We should be, it's just a case of say what we mean, mean what we say. Not something almost and incredibly close to like we mean or say. It's just about maintaining discipline when using terminology and I see great benefit with precise use of terminology.
* The recently reaching lunar orbit gravity mapping mission on the moon will be measuring actual Moon gravity by using differences in the orbital mechanics of two satellites in the same orbit causing varying distances between the spacecraft and measuring those relative distances via femtosecond accurate timing methods. That will allow the team to measure local orbital perturbations via the satellites relatively varying centripetal accelerations and then use the information to isolate the actual gravitation component for a given surface grid location. Again very discriminating despite the individual spacecraft effectively "feeling" a unified gravitational and orbital centripetal acceleration.