Modifications of Classical Equations?

[Sobeita]

Hello. I'm new here to the Physics Forum, but I was hoping I could contribute.

While I was in high school, my physics teacher told me that gravity at one Earth's radius was about -9.8 m/s^2. As it turns out, 'negative means down', and 'down' is in the direction of the Earth's core. Isn't 'down' more or less arbitrary? If we were solving a problem on the surface of Mars, we wouldn't use a frame of reference based on the Earth's surface; more importantly, if we were to compare the sun to the center of the earth, we might incorrectly arrive at the conclusion that the sun were orbiting the earth. If we have graduated beyond the notion of geocentrism, then it's important to update our methods.

Gravitational force, as we all know, is not a uniform force pulling all matter in one direction. With that in mind, I can confidently say that gravity is not negative. If gravity were negative where you sat in New York, then using the same frame of reference - your body - then Australians would fall into space; or, more accurately, Australian gravity would have to be defined as positive in respect to your body.

Now, look at the equation for gravitational force. Where, in that equation, is direction defined? The gravitational constant is positive, masses are positive, and distance is fundamentally absolute. The equation addresses only magnitude. If you were to calculate the gravitational force on your body where you are sitting, using this equation, you would determine that you should be collapsed against your ceiling!

What I have done is redesigned a few of the classical equations to account for the frame of reference. I'll post two of my equations, first with mathematical notation and then explain it in case you don't understand or can't see the symbols.

1. Angle of incidence.
a) Traditional equation: ϴr = ϴi
b) Modified equation: ϴr = 2ϴp - ϴi

Okay, in what universe does a reflected ray come out at exactly the same angle it enters at? The implications of this equation are disastrous; worse yet, it's authorized by the State of New York, and most likely every other state (although I'm not sure of that). When you actually use the traditional equation properly, it requires you to first measure an angle in degrees (except your measurement is 180 degrees wrong, since an incident ray pointing down and left at 225 degrees will be measured as 45 degrees) and then to mirror your frame of reference in the middle of the equation.

In my equation, Angle P is the angle of the plane. Flat, level ground, which is not inclined, would be considered 0 degrees. Angle I is the incident angle: a ray pointing straight down is 270 degrees. Angle R is obviously the reflected angle - the previous ray should reflect straight back up, at 90 degrees. All angles are measured in the same frame of reference.

Not only does this equation solve the frame of reference problem and unify the angles of measurement, but it introduces the possibility of an angled plane. This is essential for almost any realistic calculation - like tipped mirrors, sunlight reflecting off of a car, etc. - so you aren't forced to "rotate your graph paper", so to speak. I applied this equation when I was creating a few Flash demos - http://www.soulfox.com/flash/gravitation2.php" was my best and most recent - and you can see how well it works.

2. Gravitational Force.
a) Traditional equation: Fg = G M1 M2 / r^2
b) My equation: Fg[x,y,z] = G M1 M2 [x2-x1, y2-y1, z2-z1] / r^3

My equation uses two bodies: body 1 has the properties M1, x1, y1, z1, and body 2 has the properties M2, x2, y2, z2. The equation introduces direction and splits the force into three dimensions using a trigonometric equation based on the Pythagorean theorem. If the system is two dimensional, or even one dimensional, the unused variables, Z or both Y and Z, are 0.

So...

My equations are designed to correct flaws with the traditional equations that made them unintuitive and illogical, and to make them more applicable in programmatic applications. I have successfully used both of my equations without error in Flash programming. I have several examples of the reflection equation, and one example of the gravitation equation, uploaded on my website (the same website in the link earlier in this post).

What do you think? Is this worth looking into? I want to redesign all of the equations to correct flaws like this, and to include frame of reference.

 

[Mapes]

Hi Sobeita, welcome to PF. It looks like you're reconciling certain equations to fit your personal coordinate system and conventions. This is fine, but it doesn't mean the original equations are flawed. It is generally understood, for example, that a gravitational acceleration of -9.8 m s^-1 is only meaningful in a coordinate system where the positive axis points at the sky. You have to understand that using this radial coordinate system means that the direction of the axis is location-dependent.

Similarly, the Snell's Law equation is only meaningful when accompanied by a diagram showing how the angles are measured, but the general convention is that the angles are measured from two different places. Again, it's OK that you'd like to use your own personal convention (to measure the angles from the same axis, and I think you mean $\pi$ instead of $2\pi$), but it's likely to confuse others, and doesn't mean that there's a flaw with the original convention.

(I applaud your efforts to think carefully about equations describing physical effects and the accompanying definitions and conventions, though.)

EDIT: OK, on the $\pi$ vs. $2\pi$ question, I see what you're doing with the angles.

 

[Gerenuk]

As it turns out, 'negative means down', and 'down' is in the direction of the Earth's core. Isn't 'down' more or less arbitrary?

No, not in school. There the teacher tells you how to interpret things.
But of course everything in physics only works, if two people agree on conventions. Some people even use different letters for physics observables.

a) Traditional equation: Fg = G M1 M2 / r^2
b) My equation: Fg[x,y,z] = G M1 M2 [x2-x1, y2-y1, z2-z1] / r^3

The first is a statement about the magnitude of the force. The second takes force as a vector.

You are indeed correct with your less ambigious equation. It would be annoying if the teacher insists on his wording, but he isn't wrong in a way. It's just his convention.

 

[Sobeita]

@Mapes: My personal convention is one that is accepted by programs, which are already typically based on Cartesian coordinates. For example, in Flash, which was my program of choice, each object has two coordinates - X and Y - and a few other properties, like rotation. If I wanted to apply acceleration to an object, I would begin with a value for acceleration, change the velocity value based on that acceleration, and then apply that velocity to the object in respect to the object's origin. It doesn't require any leaps of logic, like the one where you have to flip your protractor over in the Snell's law equation, because computers aren't capable of discerning the intentions behind equations. In fact, if you wanted to program Snell's law literally, an inbound ray would pass directly through the surface.

(I don't know what you meant by pi and 2pi, I don't remember posting anything with that. I typically use degrees in my calculations anyway. Where did you see it?)

@Gerenuk: That may be true, but I'm doing my best to maintain the same conventions within each equation.

The first is a statement about the magnitude of the force. The second takes force as a vector.

That's true, but my teacher - and all of the others I've heard in video lectures - are very inconsistent. They use 'speed' and 'velocity' interchangeably, as well as force as a scalar as well as a vector, and then expect their students to remember the difference when they take tests.

@All: I'm interested in converting the modified Snell's law to accept three dimensions instead of two alone. I believe there are two angles that define the plane, the incident ray, and the reflected ray, each, but I'm not sure how to treat them in a single equation. Do you have any ideas how to go about that?

 

[Mapes]

@All: I'm interested in converting the modified Snell's law to accept three dimensions instead of two alone. I believe there are two angles that define the plane, the incident ray, and the reflected ray, each, but I'm not sure how to treat them in a single equation. Do you have any ideas how to go about that?

But Snell's Law applies to a single plane, the one defined by the incident ray and the surface normal. It's probably easiest to define your coordinate system to naturally include this plane.

 

[Gerenuk]

But Snell's Law applies to a single plane, the one defined by the incident ray and the surface normal. It's probably easiest to define your coordinate system to naturally include this plane.

Yes of course, and he asked how to do that. I suggest
$$n_1\vec{d}_i\times\vec{p}=n_2\vec{d}_o\times\vec{p}$$
where $\vec{p}$ is a normal vector to the plane and $\vec{d}_{i,o}$ are unit directions of incoming and outgoing rays.

Hmm, but what exactly is the task here?

 

[Sobeita]

But Snell's Law applies to a single plane, the one defined by the incident ray and the surface normal. It's probably easiest to define your coordinate system to naturally include this plane.

Well, yes, that's true. Unfortunately, it's difficult to redefine the coordinate system universally to accept planes that spontaneously create and destroy themselves each time an incident ray meets a surface. It's much better to try to reinterpret all of the incoming information.

Hmm, but what exactly is the task here?

The task for me is to redefine our current equations so that they can be adapted into most contemporary programs and applications. My program of choice was Flash MX (I'm not sure if I mentioned that) which had a peculiar idea of how to handle direction, but even that typically managed to achieve decent throughput. I'm planning on learning C++ so I can bypass all of that middle-man programming anyway.

EDIT: Thank you for that equation, I really think that'll help.

 

[Gerenuk]

Oh, before I check my equation it's probably as easy to consult Wikipedia Snells law Section 5 :)