|W. Kevin Vicklund
Joined: Oct. 2005
Warning! Math ahead!
The Ghost wrote:
|Sorry - Wally world is closed. The moose outside should have told you how to plug numbers into your own formulae! At this time we are around theta=Pi/2. Now, this gives a value r''=-A, consistent with the accelerating universe the big clang theory fails to explain|
That having been said, there is still some explaining to do. Wally mentioned is there is zero velocity there is no Doppler effect. Given his formulae, but not his plug-ins, that is what we would have r'=0. Now, if you would only scroll down to my follow-up post you would be aware that this quinitessence sphere has an intial velocity that I will derive shortly, so, no, the velocity is not zero at this point, and we have a Doppler shift!
In light of the fact that Wally is one of the brightest evolutionists here, and the best he can do is botched calculations grounded in poor reading comprehension and quote-mining. No wonder ordinary people don't take your religion seriously!
Gee, you don't even make it one paragraph without an egregious math error. Please review your math. After substituting for x in my original post, the relevant equation should read
r" = -Asin(theta-pi/2)
As indicated, currently theta should equal approximately pi/2. So if we substitute for theta, we get the following:
r" = -Asin(pi/2-pi/2)
r" = -Asin(0)
r" = 0
When castigating someone for poor math skills, GoP, make sure that your corrections are in fact correct! Perhaps you don't understand the concept of phase shifting - that's the reason I included the -pi/2 factor in my original calculations. As an electrical engineer, I like to put everything in terms of sin and let phase shifts take care of the rest.
Now, I did make two minor mistakes in my original post which I noticed later that day. I wondered if GoP would catch them, so I let them stand (they didn't alter my point). First, I incorrectly defined theta as t/T, when it should be (2pi/T)t. Second, the derivations should properly read:
r = R + Asin(x)
r' = Ax'cos(x)
r" = -Ax'x'sin(x)
and x' is 2pi/T. Turns out that GoP completely overlooked the actual errors and invented one instead. Finally, I did manage to miss GoP's second post where he talks about there being an initial velocity - I am fully responsible for any misunderstandings that arose as a result. So let's do a quick revision in light of that. First, I want to point out that my original equation is equivalent to GoP's equation when the sphere is compressed and released from rest, allowing for sign conventions, phase shifts, and axis shifts. So let's rework my equation to account for an initial velocity. Since I found how to get iB to use Symbol font, I will use the greek letters instead of spelling them out.
First, the definitions. Instead of using theta, I will use wt, where w is 2p/T, or the frequency in radians. f is the phase shift. If your computer does not have Symbol font installed, my post will likely not show up properly. A is the amplitude of the vibration.
My revised equations are now:
r = R - Asin(wt+f)
r' = -Awcos(wt+f)
r" = Awwsin(wt+f)
So how does this compare with GoP's final equation?
I introduce the constant R to accomodate the observer at the center of the sphere-GoP's observer is instead located on the radius about which the sphere is oscillating (the steady-state value). Strictly speaking, his ODE solution should have included this factor (or rather, he assigned the constant a value of zero, I assigned it a value of R - there are an infinite number of possibilities, each corresponding to an observer's position). So that leaves my sin and his cos+sin functions. (Note: I use a negative value for the amplitude because the system is starting from compression)
First, the trivial case where the initial velocity r'(0)=0 and the initial state is compression. In this case, our functions become (mine first, GoP's second)
r = R - Asin(wt+f)
r = r(0)cos(wt)
Now, I can convert cos(wt) into -sin(wt-p/2). This gives me
r = R - Asin(wt+f)
r = -r(0)sin(wt-p/2)
Thus, if we set A=-r(0) (because our sign conventions are opposite) and f=-p/2, my original equation is shown to be equivalent to his equation.
Now for the case if there is initial velocity r'(0) and an initial compressed state. We note that GoP's cos and sin functions have the same frequency w, which allows us to perform a phasor analysis. Since this involves imaginary numbers, I'll just give the results - I don't want ectoplasm oozing out of the computer as Paley tries to grasp the concepts involved. What a phasor analysis does is permit us to convert two sinusoids with the same frequency into a single sinusoid of the form Bsin(wt+q). The variables are determined as follows
B^2 = r(0)^2 +(r'(0)/w)^2
q = arctan((r'(0)/w)/r(0))
(For those familiar with phasors r = r'(0)/w + jr(0))
Our equations as modified:
r = R - Asin(wt+f)
r = Bsin(wt+q)
Again, if A=-B (difference in sign convention) and f=q, the equations are equivalent.
So what we see is that by having an initial velocity, we merely introduce a phase shift and an increase in amplitude to the oscillation (ie, A>r(0) for non-zero r'(0)). This does not violate the fundamental relationship I earlier elucidated, the point of which GoP seemed to miss. If accelleration is near the maximum, velocity is near zero. If velocity is near maximum, accelleration is near zero. Velocity and accelleration can not simultaneously be both positive and increasing. In his original post, GoP said that the accelleration is at a maximum, which would imply that velocity should be near zero. Obviously, that is wrong, and that is what I was pointing out. To rephrase what I also said, if GoP's model is true, given current observations, we must currently be in the first quadrant of the rebound from compression. If GoP is correct about initial velocity, compression, and the time spans involved, that implies that the initial velocity must have been towards compression.
In short, GoP's model has some contradictions between what he has said about it and what it actually implies. The most likely scenario (if the model was correct) given current observations is that we are almost at maximum red-shift and at nearly zero accelleration (incidentally, that puts r(t) near the steady-state point). That accelleration should therefore soon become negative. Values for r(0), r'(0), and w should let us determine how soon we should expect to see this change-over.
So who exactly has "botched calculations grounded in poor reading comprehension and quote-mining" Ghosty? BTW, your speculation about my first name is incorrect, though if you play around with the Wally World phrase long enough you might get the right one.