# Lift as a consequence of streamline arguments

1. Dec 24, 2004

### arildno

A recently closed thread dealt with the generation of aerodynamic lift, in particular tied to a severe and justified criticism of the "equal transit time"-principle, which often is also called the "Bernoulli effect".

However, from what I read, no one showed how streamline arguments can properly be used, but only derailed them on the mistaken assumption that such arguments necessarily had to rely on the fallacious "equal transit time"-principle.

This is why I reopen this issue.

1) 2-D theory
I will focus on the lift in the 2-D case.
Clearly, reality is always 3-D, but if the wing is sufficiently long, we may expect a region of local 2-D flow about the wing (strip theory).

2) The upper&lower curves constituting the wing are streamlines in the fluid.
This is certainly not necessarily the case; separation of streamlines from the wing will occur at high speeds or too sharp curvatures in the wing's profile.

However, the streamline assumption is an important special case, and my argument will also give a simple description of why separation might occur under high speeds/sharp curvatures.

3) Wing's rest frame analysis
This is rather conventional, and we make the following initial assumptions:
a) Far upstream, the air velocity is $$U\vec[i}$$
b) The induced disturbance of the fluid velocity due to the presence of the wing is restricted to a zone about the wing and downstream indicated by some finite vertical distance measured from the wing.
That is, if you go vertically upwards or downwards some distance, you'll end up in the undisturbed free stream (horizontal) velocity region.
c) Since the buoyancy force of air is negligible compared the weight of the wing, we'll neglect it from now on.

4)Zero angle of attack
For simplicity, we will assume that the chord connecting nose and trailing edge of the wing is practically horizontally aligned.

5) Crocco's theorem.
Given that the upper surface of the wing is CURVED (downwards), we should ask ourselves:
Since the fluid particles following the wing experience centripetal acceleration, what must the VERTICAL pressure distribution be above the wing, in order to produce that acceleration?

(NOTE: We are therefore interested in the pressure distribution ACROSS streamlines, rather than ALONG. Crocco's theorem is the analogue integration of Newton's 2.law across streamlines, whereas Bernoulli's equation is valid along the streamline)
Clearly, since we have downwards curvature on all the streamlines above the wing, up to the region where we end up in the straight-line free stream-region, we must have that a typical measure of the pressure on the upper surface of the wing, $$p_{u}$$ should be LESS than the free stream pressure $$p_{0}$$, that is:
$$p_{u}<{p}_{0}$$

Now, assume that we have a slight positive curvature on the LOWER surface of the wing.
Proceeding vertically downwards to the free-stream, we see that the associated measure of the pressure, $$p_{l}$$ also fulfills the inequality:
$$p_{l}<{p}_{0}$$
(In the case of negative/zero curvature on the downside, the proper measure of pressure on the lower surface would typically be greater or equal to the free-stream pressure).

If now we assume the upside's curvature is stronger than the downside's curvature, it is reasonable that $$p_{u}\leq{p}_{l}$$ as long as:
The typical air velocity on the upside is either of the same order as the air velocity on the downside, or greater.

By this argument, the inequality $$p_{u}\leq{p}_{l}$$ looks quite natural, and we see that the pressure distribution necessary for the centripetal acceleration associated with stream-line movement, will typically generate a LIFT.

No use has been made of the fallacious "equal-transit-time" principle here.

We also see that the streamline behaviour places a demand after a specific pressure distribution, and it is natural to expect that if the required centripetal acceleration is "too large" for the fluid to cope with, SEPARATION will occur.
But since the required centripetal acceleration increases with fluid velocity and curvature, it is natural to assume that stream-line behaviour only will occur if these parameters are sufficiently small.

We might also use streamline arguments to show that the lift will be proportional to the product of the fluid density, air velocity and CIRCULATION about the wing (i.e, in accordance with Kutta-Jakowski's theorem).

Last edited: Dec 25, 2004
2. Dec 25, 2004

### alexbib

why do you say that the equal transit-time principle is fallacious? Is it experimentally false that the air above takes the same time to go across the wing as the air below?

3. Dec 25, 2004

### pervect

Staff Emeritus
Both experiment and the correct flow equations (the Euler equations) show that the equal transit time explanation is wrong. See for instance

http://www.grc.nasa.gov/WWW/K-12/airplane/wrong1.html

(which, however, disucsses the theory more than the measurements)

4. Dec 26, 2004

### Clausius2

Arildno,

After being read you're analysis I have little to discuss. As I said before I'm not an expert on lift (you seem to have read a bit more about that by the way). Take a look at this:

That was my explanation to one guy who questioned about the famous principle. That principle is used when people want to compare upper and lower velocities. This only can be done by continuity. People usually adopt an equal transit time and different path lenght in each side, so that they apply bernoulli equation and relates both pressures. But we know from Kelvin theorem that just behind the airfoil must be a counterspinning vortex. Therefore, two molecules just at the airfoil front do not reach the rear part just at the same time. Moreover, Euler equations predict a tangential discontinuity (not real) at the rear.

What is actually true is that the responsible of lift are pressure forces over the wing. After all, upper pressure must be less or equal to lower pressure in order to sustain the wing. How is it reached that pressure difference?. The effect of loosing a part of static pressure due to the curvature (and subsequent flow acceleration) is right. But quantifying the lift generated by that method is a bit difficult in my opinion. Behind your argument is Bernoulli equation too.

As you have mentioned it, the responsible of lift is the Circulation:

$$\oint_{wing} \overline{u}\cdot\overline{dl}$$

A net circulation of the flow will generate lift. The problem here is to find a physical meaning for the circulation, which doesn't appear to be an easy thing for me. Your explanation about curvature is closely related to circulation. At first sight the non zero circulation of the flow would mean a global unbalance in velocity along the wing, and therefore a difference of pressures. That global unbalance is caused mainly by the wing shape.

This is my poor collaboration to this subject. :uhh:

EDIT:
Upper and lower wing lines are always stream lines:

$$u=\frac{\partial \psi}{\partial y}$$

$$v=-\frac{\partial \psi}{\partial x}$$ in a local curvilinear system (x,y). and psi=stream function.

If there is no suction trough the wing surfice, v=0 and

$$\psi=constant$$ along the wing profile. (x local direction).

Last edited: Dec 26, 2004
5. Dec 26, 2004

### arildno

I will continue with a few remarks to fill out my own account, prior to going into an exchange of ideas and replies (which I'll get to in a while):
1)Recapitulation of first post
The difference in curvature in the wing profile plus the observation of upper&lower surfaces as streamlines, ought us to expect that the required centripetal acceleration on the upper side is somewhat larger than that on the underside, since c.a. is proportional to curvature (the proportionality factor being the squared speed).

Given this rough, first idea, it seems plausible to conclude that the vertical pressure drop on the upside (measured from the region of free-stream velocity) is greater than the pressure drop on the downside.
Hence, it follows that by these rough arguments, a lower (measure of) pressure on the
upside compared to the downside is to be expected, i.e, the conditions for lift is present.
These ideas relies on Crocco's theorem, rather than Bernoulli.
In particular, these ideas are certainly consistent with a failure of the "equal-transit-time"-principle, in that the upper velocity may well be greater than the lower; such a situation would simply sharpen the need for stronger centr. acc. on the upside (rather than acting against these ideas).
However, even in the (unphysical, as we shall see presently) case of strictly equal velocities, the stronger upside curvature necessitates a stronger centripetal accceleration there, and hence, a larger vertical pressue drop on the upside.

Now, let connect these initial ideas to Bernoulli considerations along the respective streamlines.
Since we have been led to expect a lower pressure at the upside, Bernoulli predicts a higher speed on the upside than the downside, which is in perfect accord with our initial ideas that the centripetal acceleration on the upside is probably greater than on the underside.

Note:
This should NOT be considered as an explanation of the generation of lift, but rather as clarifying the various elements present in an interconnected, lift-sustaining whole.

2) Bernoulli's equation and the presence of circulation:
Bernoulli's equation is an eminent strarting point in order to make the relationship between lift and circulation clear on a simple level.
(With a simple level, I mean to emphasize the difference between this approach and the subtle use of solving the problem with the aid of (for simplicity..) potential theory (i.e, a rigourous derivation of Kutta-Jalowski's theorem))

$$\vec{v}=(U+u)\vec{i}+v\vec{j}$$
where (u,v) is the induced velocity field due to the presence of the wing, and (U,0) the free-stream velocity field.

We make the assumptions that the induced velocity field is rather small in comparison to U, and that the wing is very thin.

Now, by Bernoulli, the pressure distribution along streamlines are given by:
$$p=p_{0}+\frac{\rho}{2}U^{2}-\frac{\rho}{2}\vec{v}^{2}\approx{p}_{0}-\rho{u}U$$
according to our assumption of a small disturbance field.

Now, the lift (force per unit length) is given by integrating the vertical pressure component around the wing S:
$$L=-\oint_{S}p\vec{n}\cdot\vec{j}ds\approx\rho{U}\oint_{S}{u}\vec{n}\cdot\vec{j}ds$$
(the constant pressure contribution giving zero).

We now make the pleasing discovery:
$$\vec{n}\cdot\vec{j}=\vec{i}\cdot\vec{s}$$
where $$\vec{s}$$ is the local tangent vector.
Since we also have: $$\vec{s}ds=d\vec{s}$$
we find:
$$L\approx\rho{U}\oint_{S}(u\vec{i})\cdot{d}\vec{s}$$

Now, we make a sleazy use of the idea that the wing is THIN:
We approximate the wing by a simple horizontal line segment L!!
Hence, we get:
$$L\approx\rho{U}\oint_{S}(u\vec{i})\cdot{d}\vec{s}\approx\rho{U}\oint_{L}\vec{v}\cdot{d\vec{x}}\approx\rho{U}\Gamma$$
where
$$\Gamma=\oint_{S}\vec{v}\cdot{d\vec{s}}\approx\oint_{L}\vec{v}\cdot{d\vec{x}}$$
is the circulation about the wing.

The curve integral on the horizontal line segment L $$\oint_{L}(u\vec{i})\cdot{d\vec{x}}$$
can clearly be replaced by $$\oint_{L}\vec{v}\cdot{d\vec{x}}$$
since $$d\vec{x}$$ is normal to the vertical part of the velocity field,
and the integral $$\oint_{L}(U\vec{i})\cdot{d\vec{x}}=0$$
since the curve integral traverses the same line twice.

To the order of this thin-wing approximation then, the horizontal velocity is discontinuous in the vertical coordinate at the horizontal line segment representing the wing, and this discontinuity ensures/is consistent with the presence of a non-zero circulation about the wing.

Bernoulli therefore shows us, that in so far that we have a non-zero lift, there has to be a corresponding non-zero circulation about the wing (or, if you like, vice versa).

I have used some rather sleazy maneuvers here to show the connection between circulation and lift, and a few comments should be made in connection with a proper derivation:
a) The horizontal line segment approximation makes a lot more sense in a (semi-) rigourous derivation, since it is actually an approximation of the mean-camber line of the wing rather than the wing itself (if my memory serves me right).
b) Properly formulated as a boundary-value problem, the boundary demand of zero normal velocity on the wing is of crucial importance; this piece of information haven't been explicitly used in my simplistic version.

c) In accordance with the requirement of conservation of circulation on a material curve (i.e, Kelvin's theorem), a proper treatment will find a counter-spinning vortex at the trailing edge, balancing the net circulation around the wing.

A comment on the "equal-transit-time"-principle:
Assuming the validity of the thin-wing approximation, my treatment shows that to the same order of accuracy, we should expect that principle to be VIOLATED, rather than affirmed (by the discontinuity of the horizontal velocity).

3) Angle of attack&effective curvature:
It is of some interest to see how the angle of attack concept ties in with a measure of curvature of the wing profiles.

Let us for this analysis assume a symmetrically shaped wing about the chord connecting nose and trailing edge.
From the previous discussion, we see that if this wing is horizontally aligned, we cannot really expect the presence of lift, since the curvatures are strictly equal. (The required pressure drops must typically be the same).

Let the tangent vector on the upper side at trailing edge in this (horizontal, no-lift) case be:
$$\vec{t}_{u}=\cos\theta\vec{i}-\sin\theta\vec{j}$$
whence it follows that the tangent vector on the lower side at the trailing edge is:
$$\vec{t}_{l}=\cos\theta\vec{i}+\sin\theta\vec{j}$$

We note therefore, (rather roughly) that the rotational displacement of the incoming flow $$U\vec{i}$$ around a given surface is given by the angle $$\theta$$

Let us now see what happens if we tilt the wing with an angle $$\alpha$$ with respect to the horizontal:
Then, the respective tangent vectors at upper and lower sides satisfy:
$$\vec{t}_{u}=\cos(\theta+\alpha)\vec{i}-\sin(\theta+\alpha)\vec{j}$$
$$\vec{t}_{l}=\cos(\theta-\alpha)\vec{i}+\sin(\theta-\alpha)\vec{j}$$
That is, over the same displacement (from "nose" to trailing edge), the fluid passing along the upper side experience a stronger rotation than that part of the fluid passing on the underside.
(It is readily seen that the downwards displacement of fluid flow is consistent with effective curvature ideas).

Thus, we should expect that an effective curvature difference is typically proportional to the angle of attack (one might make a bit more detailed arguments here..).

Which of these concepts we would like to use when concerned with lift, ought therefore largely to be a matter of preference; the ideas developed by one technique should be translatable into corresponding ideas by the other.

Last edited: Dec 26, 2004
6. Dec 26, 2004

### arildno

Clausius2:
Just one comment to this:
This is untrue! (EDIT: See addendum)
In the stationary, case, we have:
$$\frac{D\vec{v}}{dt}=(\vec{v}\cdot\nabla)\vec{v}=\nabla(\frac{1}{2}\vec{v}^{2})+\vec{c}\times\vec{v},\vec{c}\equiv\nabla\times\vec{v}$$
Note:
I think I've got the right sign on the cross-product, but a possible sign error here is of no consequence further on.

The field formulation of the momentum equation can therefore be written as:
$$\nabla(\frac{1}{2}\vec{v}^{2}+\frac{p}{\rho}+gz)+\vec{c}\times\vec{v}=\vec{0}$$
Let now $$d\vec{s}$$ be tangent to a streamline S.
Hence, $$d\vec{s}$$ is of course parallell to $$\vec{v}$$, and Bernoulli's equation is simply the curve integral along S between two arbitrary points $$\vec{x}_{1},\vec{x}_{0}$$ of the tangential component of the momentum equation:
$$\oint_{S}(\nabla (\frac{1}{2}\vec{v}^{2}+\frac{p}{\rho}+gz)+\vec{c}\times\vec{v})\cdot{d}\vec{s}=\oint_{S}\nabla (\frac{1}{2}\vec{v}^{2}+\frac{p}{\rho}+gz)\cdot{d}\vec{s}=(\frac{1}{2}\vec{v}^{2}+\frac{p}{\rho}+gz)\mid_{\vec{x}_{1}}-(\frac{1}{2}\vec{v}^{2}+\frac{p}{\rho}+gz)\mid_{\vec{x}_{0}}=0$$

Hence, Bernoulli's equation is valid even if $$\vec{c}\neq\vec{0}$$

Kelvin's theorem is valid for inviscid (not necessarily irrotational) flow, and the production of the counter-balancing vortex can be seen to conform to this.

Alternatively, using potential (that is, irrotational) theory on the lift phenomenon, generates a singularity of circulation strength equal to the circulation about the wing.
The production of singularities means of course, that the potential theory has been pushed to its maximum usefulness, where vortices appear as singularities whose effects cancel each other so that irrotationality is (just barely) preserved.

In Norway, the pressure distribution in the fluid domain for irrotational/potential flow (stationary or not) goes by the name of "Euler's equation for the pressure".
But, as I now recall, this equation is often referred to as "Bernoulli's equation".
I now think it is this "Bernoulli's equation" you were referring to, rather than the equation for the pressure distribution in the stationary case along streamlines.
In that case, you're of course correct in saying that we should expect some trouble on the trailing edge.

Finally, I'd like to say that I think it is conceptually somewhat easier to realize the existence of a non-zero circulation about the wing as a result of a pressure difference (i.e, lift conditions), than the other way around. In that case, it is, in effect Bernoulli's equation which predicts a significant velocity difference between top and bottom, i.e, net circulation.

I know that this flies in the face of the standard interpretation, but I, for one, feel more comfortable with this version.

Last edited: Dec 26, 2004
7. Dec 26, 2004

### krab

So do I. And BTW, I want to thank you for your lucid and constructive analysis. The other thread provided a lot of heat and no light. I must say the Mentors in that case were no help whatever.

8. Dec 26, 2004

### Integral

Staff Emeritus
I have very limited knowledge of this topic. If I don't know about something I tend to stay quite and watch. Clearly the previous thread was not going in a reasonable direction, which is why I finally locked it. This thread on the other hand has been on a much better track. I also appreciate the careful development, unfortunately, I am still out of my league in its evaluation. Thank you all for providing a lucid approach to the problem.

9. Dec 27, 2004

### FUNKER

Also with this argument NASA website had the 5 most common fallacious thoeries of lift and the equal time theory was the number 1, just check the websiteskies

10. Dec 27, 2004

### Clausius2

Yes, you're right. The way you derived Bernoulli equation was one of the questions of some exam I did some time ago... :yuck: . In fact I said it wrong. I appreciate too your effort at clearing it up. It can be considered as your gift to us for Christmas, isn't it? .

I agree with all your arguments, so I will try to summarize:

-Lift force is closely related to circulation of velocity. The circulation is closely related too with the wing shape, in particular with its assymetrical shape or at least with the attack angle if symmetrical (at this point I don't agree with the "centripetal force" you referred to, because it doesn't appear at all in the formulation as you may see). The circulation represents an unbalance of velocities, which ultimately causes a pressure difference as Bernoulli yields. But the principle of equal transit times has no sense.

-That principle is not valid due to the fact of the discontinuity at the trailing edge (in the case of vanishing viscosity) caused by Kelvin's Theorem. The flow remains irrotational except in the weak, where a counter-spinning vortex is formed in order to conservate the zero initial circulation.

What do you think?.

EDIT: as a cautionary note, and adding my ignorance to the pool, I have to say that potential theory is not my best, as it has been demonstrated in this thread. In fact, my proffessor never taught us nothing of potential flow, so the little I know is in my own. On the other hand, our knowledge in compressible flow is higher than the mean. As it is said in Spain, you cannot have all things in this life (that's spanglish ).

Last edited: Dec 27, 2004
11. Dec 27, 2004

### 4newton

You need to look at lift on the atomic scale in order to understand the mechanism that results in lift. The shape of the wing is the important part. The molecules of air going over the top of the wing are attracted to the surface of the wing (gravitational force). As the molecules are diverted part of there momentum is transferred to the wing and the velocity of the air molecule is slowed down. The wing is shaped to try and maintain optimum distance between the air and the wing to provide the greatest lift. The attachment shows this function.

If the wing is not curved as also shown in the attachment the air is only able to transfer momentum to the surface for a very short distance. If pressure differential resulted in lift this would provide the most lift the fact that it provides little or no lift is proof that lift has nothing to do with pressure differential.

The velocity difference between the top and the bottom of the wing is the result of the transfer of momentum to the wing. Turbulence in back of the wing is the result of the slowing of the air over the top of the wing. The most turbulence is when the plane is landing and the velocity differential is the greatest.
Many small planes get knocked out of the sky when trying to land behind a large slow moving plane.

In icing conditions it takes only a small amount of frost to decrease the lift on the wing. This is due to a loss of laminar flow over the surface with an effective larger separation of the air and the wing with a small turbulence at the surface of the ice.

The problem a pilot has in icing conditions is that he is able to achieve lift off with the help of ground effect but as he tries to pull up he finds he has no more lift or a lot less than he is use to. Instinct tells him to pull up more. All during this time he will not have a stall warning. Stall warning only tells you when the air over the leading edge of the wing is turbulent. He will continue to increase the angel of attack until the plane has no lift from the wing or he does get turbulence on the leading edge of the wing and stalls.

#### Attached Files:

• ###### lift001.pdf
File size:
14.1 KB
Views:
64
12. Dec 27, 2004

### Clausius2

That's not necessary at all. Navier-Stokes equation gives us several information without the need of looking at atomic scales.

That's false. Gravitational force is negligible at this flow regimes. The Froud number is usually so large as to neglect the Buoyancy terms.

It has no sense. As far as I am concerned, none engineer thinks of that when designing a wing. The airfoil shape is more related to Arildno's explanation.

The unique forces acting on the wing are pressure forces. Therfore, differential pressure is the ultimate responsible of lift. When separation occurs at high Reynolds# or high attack angles, such differential pressure dissapears and the boundary layer becomes disrupted. Boundary layer attachment provides the pressure transmission to the body. Once the separation has overcome, pressure at the cross layer section is not uniform and might be fluctuant.

Turbulence? Who has talked about turbulence?. Potential theory is capable of explaining lift without turbulence. I don't think the most turbulence is reached when the plane is landing, moreover I think due to the ground effect the layer is proner to be attached. Moreover, turbulence has nothing to do with slowing down any flow. It is a problem of stability. What you actually meant was separation of the boundary layer, which has NOTHING to do with turbulence.

Last edited: Dec 27, 2004
13. Dec 27, 2004

### |2eason

Congratulations! You alone have uncovered this huge conspiracy. Physicists have tried to keep this a secret for decades now. If the public knew that things could defie gravity by using gravity, we would all wave our hands and be sucked into outer space. :rofl:

14. Dec 27, 2004

### arildno

Now, I'll get back to various objections&replies given so far, but I would like to commend pervect for the excellent link provided.

My analysis is fully in accordance with what is presented there; what I've given is essentially the local analysis consistent with the global TURNING of the flow given there.
I'll prepare some more comments in a while.

15. Dec 27, 2004

### arildno

4newton:
Please remember that the "velocity field" and "pressure" are large-scale structures, compared to molecular dimensions.
From a molecular point of view the "velocity field" should ALWAYS be understood as an a quantity averaged over a humungous number of individual molecules.

Navier-Stokes/Euler, are properly seen from the molecular point of view, as equations predicting the evolution of this averaged quantity, rather than predicting the velocity profile of an individual molecule.

"Pressure" is a large-scale variable closely related to the average collision rates between molecules (i.e, how strongly molecules bump into each other).

"Pressure", and for that matter "viscous effects" are therefore not unconnected with molecular behaviour (as you seem to imply), but are the net aggregate effects of gazillions of molecular interactions.

What it means that whenever the Euler/Navier-Stokes equations are ACCURATE, is that we have a situation WHERE ALL DOMINANT MOLECULAR EFFECTS HAVE BEEN ACCOUNTED FOR, in our modelling of these in large-scale quantities.

Since aerodynamic modelling has been shown to give extremely accurate results (when using, say the Prandtl approximations), this means that there don't exist other dominant molecular effects than those implicitly contained in our approximations.

Last edited: Dec 27, 2004
16. Dec 27, 2004

### 4newton

Clausius2

There is never a need to look deeper into a problem if you are happy with what you have. As an engineer I am sure you were told that you did not need to understand the physics of a problem to work a solution.

Again you are looking only at the macro understanding of the engineer. You have only two forces in nature that can have an effect on the wing, the gravitational force or a charge force. The air must impart momentum to the wing is some form. We know that the air on the bottom of the wing is not pushing up because the airflow under the wing is not disturbed. We also know that diverted pressure is not able to produce the lift otherwise a wing at a 45-degree angle would produce the most lift.

You may think of forces with different names such as viscosity, boundary layer or what ever but they can only still be the result of mass attraction, gravity.

As you can see from the posts everyone, except me, thinks the concept of lift is well understood. You can see that this well understood concept is different for everyone. Even the NASA link ends with a doubt. Engineers don’t have a problem. The equations work well enough to let man fly. This however does not lead to advancement of new ideas. If my idea is correct then the material of the surface of the wing should have some effect on lift. If the wing is coated with a thin layer of lead or gold the required area for equal lift may be reduced.

Again what is the force of the boundary layer attachment? Remember your choices are limited in nature.

I see you don’t fly an airplane. When landing you extend flaps to create more lift at slow speed. This extends the surface of the wing allowing more lift. More lift imparted to the wing must result in a decrease of the momentum of the air. The momentum of the air is a function of the velocity of the air over the wing. The more momentum the air gives up the slower the air flows over the wing and the greater the velocity difference when the air from the top of the wing meets the air from the bottom.

When ice is on the wing there is a small turbulence at the surface of the ice, this brakes the boundary layer.
The ground effect I am talking about is the extra lift a wing gets from air between the ground and the wing.

17. Dec 27, 2004

### 4newton

|2eason

thank you for your humor. I of course find it funny for different reasons.

18. Dec 27, 2004

### 4newton

arildno

I am not implying that any of fluid dynamics is incorrect. I am saying that you ignore the molecular mechanism and the basic force of nature in your consideration and as a result you arrive at an incorrect reason for the function. This requires you to develop a complicated reason that has nothing to do with the result. The theories that you are using already understand the mass attraction as the force involved. You do not need a run on theory.

19. Dec 27, 2004

4newton: