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Post by Progenitor A on Mar 21, 2011 13:58:20 GMT 1
If we traverse a circle at constant speed, we do indeed get a sinusoid as the x or y component of the motion. If we do the same for an ellipse, the first point is WHAT speed is constant? If we use the parametric equation for an ellipse, we get: x = acos theta, y = b sin theta So if angular speed is constant, theta = omega t, we still get a sinusoid. If we project onto some direction other than the major and minor axes (x and y axes in this case), then we just get a combination of cos theta and sin theta, which is STILL just a sinusoid, albeit with the phase shifted. So what remains? We could try and traverse an ellipse at constant linear speed (which isn't what planets do BTW), but which will probably involve some messy maths with elliptic integrals............
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Post by speakertoanimals on Mar 21, 2011 15:15:40 GMT 1
You obviously have a different idea as to what innumeracy is compared to most people. Or is it you just will try to have a pop at me, whatever I say.................
What in my message did you find confusing? x = acos theta, y = b sin theta is indeed the parametric equation of an ellipse, most people could have a stab at theta being angle, hence theta = omega t and omega being angular speed isn't HARD..........
Instantaneous value of WHAT, for starters. The angle isn't necessarily the angle made with the x-axis anyway. If we trace x coordinate wrt angle parameter, we just get a sinusoid, x = a cos theta, or for y-axis, y = b sin theta.
If the axes aren't aligned with the axes of the ellipse, then we get a more complicated expression, but its still a sinusoid.
And the parametric form of the equation tells us we won't get much else anyway!
In fact, we could have derived this result pretty much without algebra, just knowing that we take a point moving around a circle (sinusoid on each axis), then we SCALE one direction to get an ellipse. Hence if we scale y to make the circle into an ellipse, that isn't going to change the sinusoid for x-cpt, or y-cpt, just change amplitude. And for rotated ellipse (ie no longer aligned with axes), it is just going to be adding two sinusoids with different amplitudes and phases, which is STILL a sinusoid.
The ONLY thing left to play with is plotting coordinate against time, and adding in some freaky way that the angle is traced with time, but that is getting even more complicated, as I said.
You can't convert sinusoids into anything much else just by scaling one axis, which is the transition from a cirlce to an ellipse. QED.
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Post by carnyx on Mar 21, 2011 15:31:55 GMT 1
@sta
I am interested in what you say about planets not moving at a constant linear speed ........
Could you expand on that? Does the earth experience an acceleration and deceleration on it's elliptical orbit around the sun?
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Post by Progenitor A on Mar 21, 2011 15:41:36 GMT 1
You obviously have a different idea as to what innumeracy is compared to most people. Or is it you just will try to have a pop at me, whatever I say.................
What in my message did you find confusing? x = acos theta, y = b sin theta is indeed the parametric equation of an ellipse, most people could have a stab at theta being angle, hence theta = omega t and omega being angular speed isn't HARD..........[/quote]
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Post by speakertoanimals on Mar 21, 2011 15:57:06 GMT 1
If my conclusion is wrong, why don't YOU try and show where it is, rather than just blustering..........................Or explain how you can get anything other than a sinusoid out of an ellipse.................
This is one of Keplers laws of planetary motion, rather ad hoc laws about how planets moved before we had Newtonian gravity to give actual explanation of orbits.
There are two ways to think of how a planet moves, the LINEAR speed (miles per hour), and the angular speed about the sun (how many degrees per day, looking from the position of the sun).
Turns out that the closer planets are to the sun, the faster they move, both in terms of linear speed, and in terms of angular speed. So, the actual relation is that the line joining the planet to the sun traces out equal areas in equal times.
The reason for this -- think about angular momentum, which depends on mass (constant), linear speed, and distance. In terms of distance and angular speed, it is distance from sun squared multiplied by angular speed. Hence the smaller the distance, the greater the angular speed.
Related to (but not quite the same as) the fact that a skater with approximately constant angular momentum spins faster if they pull their arms in.
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Post by speakertoanimals on Mar 21, 2011 16:07:42 GMT 1
What is a circle? X^2 + y^2 is a constant. Can also write this as:
x^2/a^2 + y^2/a^2 = 1, where a is the radius.
To get from a circle to the ellipse is simple, we just write instead:
x^2/a^2 + y^2/b^2 = 1, where we have scaled y, but not x. Acords with intuition that an ellipse is just a stretched circle.
In terms of angle, we hopefully can remember that sin^2 theta + cos^2 theta = 1 whatever theta.
Hence for a circle, we have the equation in terms of angle parameter:
x = a cos theta, y = a sin theta.
For ellipse, corresponding equation is: x = a cos theta, y = b sin theta, note a and b appear for x and y, not SAME value as for circle. Hence both sinusoids. This is the usual parametric equation for an ellipse.
Unlike a circle, actual angle of line joining point on ellipse to origin isn't just theta any more, but some other angle phi where tan phi = (b/a) tan theta.
What other useful angles can we define for the ellipse? We could take angle formed by lines joining point in ellipse to the two foci, but algebra becomes more complicated, and nowhere near as simple as the simple canonical form given above.
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Post by carnyx on Mar 21, 2011 18:18:26 GMT 1
@sta,
Does it accelerate and decelerate linearly along it's flightpath?
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Post by carnyx on Mar 21, 2011 19:40:04 GMT 1
STA mentioned the idea that the projections of an ellipse will produce major and minor axis sinusoids. But, there is the problem of what to take as the origin of the maxor axis. Which focus do you use?
Would there be a kind of permanent positive or negative offset? Could this bias have a physical manifestation?
The idea of conservation of angular momentum in an elliptical orbit, as observed by equal areas in equal time is interesting because it applies from EITHER foci, but not to the centre of the figure itself.
If you look at a moon orbiting a planet in a elliptical orbit, the planet will be at one focus, with nothing at the other focus, but the moon will swing around the empty focus AS IF a planet was there.
But when you consider that one focus is 'empty'.. the spectacle of a great satellite apparently under the influence of a mere point in space, changing speed and direction quite radically as if there were a large mass there, just has to be a bizarre ..... the very example of an invisible twin!
Now, if you get in your spaceship and move your point of view to look at the moon-planet combo sideways on, or even from the top down, you will see this crazy assymetry, with the planet offset at ONE of the focus points but the moon describing an ellipse about both foci.
But here is the freaky bit; if you move your spaceship to where the moon orbit looks like a circle, then both foci merge into one; the planet moves to the centre, and the moon describes a steady circular orbit!
And THAT is what I find interesting about ellipses. We are told that subatomic particles orbit other particles, and so they ought to describe eliipses, too. And if you could imagine such systems generating say EM waves (or e.g. gravity waves) then we ought to see some interesting effects on the waveforms, dependent on the viewing angles.
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Post by speakertoanimals on Mar 22, 2011 3:28:59 GMT 1
To get a sinusoid, you don't take either focus, you take the point in between as the natural centre.
Does the earth experience an acceleration? Of COURSE it does, a force acts therefore it MUST accelerate, either by moving in a circle at constant speed, but continuously changing direction of velocity, OR by changing both the modulus (speed) of the velocity AND its direction. The only way to leave direction unchanged is to fall radially inwards at ever increasing speed.
This empty foci picture is just daft, because of course, ONE of the foci must be non-empty,else the thing would not orbit. The existence of the other seemingly empty focus is just an artifact of the symmetry of the problem, or that you could get the same track for the mass in a different position. The same ellipse, but DIFFERENT motion, in that where the speed is greatest depends on where the central mass actually is, not just where the foci are. So we have greated speed closest to one foci, the one where the mass actually is. So it doesn't swing round the empty focus as if it were not empty, the speeds are wrong!
If we try to view an ellipse from an angle, to try and make it look like a circle, we know it isn't actually one, because the central mass will be in the wrong position, at the centre, rather than one of the foci. Ditto viewing a circle and trying to make it look like an ellipse, the central mass will be in the wrong place to look like a real elliptical orbit. Hence we can't make elliptical orbits look like circular ones, and vice versa.
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Post by Progenitor A on Mar 22, 2011 8:27:05 GMT 1
STA mentioned the idea that the projections of an ellipse will produce major and minor axis sinusoids. But, there is the problem of what to take as the origin of the maxor axis. Which focus do you use? This is nonsense.
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Post by Progenitor A on Mar 22, 2011 8:28:24 GMT 1
To get a sinusoid, you don't take either focus, you take the point in between as the natural centre.
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Post by carnyx on Mar 22, 2011 9:30:45 GMT 1
However, here she is saying that orbiting planets vary their linear speeds, which after being asked twice to clarify and to dilate on whether this means that planets speed up and slow down, ( i.e. that they slow down when going into the bend at the end of the orbit, and speed up as they come out ).
STA has been asked this question twice now, but does not answer it. Ordinarily this could be taken as a signal that grace ought to be extended, in order to preserve the decorum of the discourse. But, STA seems ever to be combative; gladiatorial, and sees all of these discourses as point-scoring exercises. As far as the kind of civilisation that STA woud be most at home with, I suspect it woud be the Roman era, or perhaps one of those religious states of the mid-east where civil discourse is limited to binary dogmatic argument, and they need dictators to run the state.
The point is, cher STA, that most come here not to fence, or to engage in dogmatic-fights, and zero-sum win-lose games; which is merely being barbaric. Rather, the are here for recreational purposes in the broadest sense, and to muse, amuse, and be amused. Which muse is your favourite, I wonder?
(And back to the irrelevancy of barycentric motion, which of course is the reason why there can never be a truly concentric orbital system ... (and which everybody had knowledge of as children when dancing together) .. it is beside the point.)
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Post by carnyx on Mar 22, 2011 14:40:45 GMT 1
jean, your disinterest is an ideal frame of mind from which to contemplate this problem; OTOH STA claims that; And OTOH STA says: Now these views are not unique to STA; they are held by a considerable number of people. But, they are apparently contradictory. If the linear speed of a moon in an elliptical orbit varies assymetrically, then it cannot project a sinusoid, from the major or the minor axis, or from the centroid. But if the linear speed-changes are symmetrical, or the linear speed is in fact constant, there is no contradiction. Can you help to resolve this very interesting problem?
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Post by carnyx on Mar 22, 2011 19:44:39 GMT 1
jean. You are well off-topic with that one. Will you respond to the on-topic question asked of you in #30?
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Post by Joanne Byers on Mar 23, 2011 3:14:56 GMT 1
Simplex,
There are other boards where you can pursue your pre-occupation with the clutch of enemies you love to hate. This board is not going to become the latest that your attention turns into a barren wilderness.
You have nothing to add to the discussion of science topics, indeed you detract from them. Your arrival here is clearly merely to combat personalities. You have been given you the chance to demonstrate your bona fides but this latitude has only encouraged you in the interpersonal needling that is your hallmark.
We discuss science here, mainly. You have contributed little that is on topic and much that is sterile banter merely promoting ill-will so your membership is terminated.
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