Guys, I have a question. We can reduce a body to a center of mass for calculations, because the results will be identical, assuming the bodies aren't going to collide. Well, a planet isn't a point in space, it's a whole bunch of particles, each attracting something on the surface. Well, the combined gravity forces of those particles must be the same as the gravity force of the center of mass (on the surface body).
Now, looking at the vertical line, running through the center of mass and the body's center of mass. Now, if we take the farthest points in that line, symmetrical regarding the center of mass and assume they are in the center, the combined forces are the same, because the displacement of both is symmetrical and as such, so are the difference on the force each of them apply on the surface body.
But, on the perpendicular of that line, you have 2 slanted force vectors for each pair of particles, whose horizontal components nullify each other, but the sum of the vertical components isn't the same as if those 2 particles were compressed into the center of mass.
Am I wrong, or is this just ignored?
You're saying that the force vectors due to two particles opposite some plane of symmetry passing through the object the sphere is attracting from each other's horizontal components cancel but their vertical components add up? That's true, and Newton proved that the differential force from each volume element of the sphere acting on the body combines to act exactly as though a point of the total mass of the sphere was acting on the body from the center of the sphere. (as long as you are outside the radius of the sphere)
Well, that's the thing, I'm saying that the vertical components add up but the sum result is smaller than the sum result if the 2 particles were condensed in the center of mass.
That's true, but volume elements closer to the body than the center of mass will have a stronger attraction and elements further will have a weaker attraction and they sort of average out.
But elements that are in a line that is perpendicular to the line that crosses both centers of mass are within the same distance to the other body's center of mass.
Oh wait, are you saying that the elements in that one line don't cancel each other out but the 4 that are symmetrical in the perpendicular line and the line itself do? Like this?
[img]http://i51.tinypic.com/21l4v8h.png[/img]
I don't know why they should, honestly. And even if they did, you still have those two points on the axis of symmetry.
No. Why do you need those to cancel?
The ones on the horizontal axis?
Well, you need the resulting forces on both situations to be equal. And you can separate any "movement" into the center as a horizontal and vertical movement, and since the force b ya certain particle changes if that position changes, you need a symmetrical particle with a symmetrical movement and a symmetrical change in force to balance it out. i don't have problem with it vertically, but I don't get it horizontally.
[IMG]http://i54.tinypic.com/33b37r8.png[/IMG]
If those two particles were at the center of mass, the sum of those two forces would be bigger than that, since they don't lose force horizontally, plus, their distance would even be shorter, for both cases.
Yes they would be bigger if they were on the axis, but they're not on the axis. We don't act like they are to find the total force. We integrate the differential mass elements over the whole volume. Or we do it the easy way and just use Gauss' law.
:psyduck:
Why doesn't my method work then? I mean, we use those additions to calculate the resulting force on a body when there are 3 or 4 interacting bodies, why would that change with 1000? I mean, if we forget about how they're dots and how their mass is infinitely small and all the problems that come from it and just assume they're little lumps of rock wit ha certain mass and small to consider the distance to their centers of mass to the body as their distance to the body, it's just another addition problem.
Right?
Your way DOES work. It's rudimentary integration. But you need to keep a close watch on variables and your geometry and such. Particularly your geometry, because you need to get the integrand into a form you can actually solve without your head exploding. If I was back at school with my notes and a scanner I'd scan my mechanics notes so you could see exactly how it's solved. I don't quite remember, despite how many times I've done it before. I remember with a cylinder, you can work out the force due to a ring, then integrate that to find the force due to a disk, and then integrate that along a line to find the force due to the cylinder, but I've forgotten how a sphere is done.
[editline]30th June 2011[/editline]
OH WAIT I have my mechanics notes in .pdf form. Let me see if I can dig this exact problem up. I'm sure we did it.
Mate, I'll go right around and say that I don't know SHIT about calculus. I still defend with all my heart that the definitions of limit and continuity are wrong. :v:
I think I asked about that on the math thread.
Also, if my method can be used, why do the end up giving two different outcomes? I don't think there's enough geometry or just about any math at all to mess up.
They don't give two different results, but doing it with a method as basic as yours, which is basically eyeballing it, it's pretty much impossible to see how the it's all going to add up in the end. Until you've got a grasp on calculus you're just going to have to trust that all the horizontal components cancel and the vertical ones add up so that it ends up equivalent to if you had all the mass concentrated at the center. Problems like this are why integration was invented. It's a nice formal way to add up infinitely many infinitely small things and get a nice finite result.
Damn, that is unsatisfactory as hell.
But disregarding how to actually get to the final result, I don't get why you need calculus to solve this. I'm not trying to find out what the final gravitational pull is, just that there is no difference between the two. And I can do that if I examine each symmetrical pair. If we move from a planet to a car pulled by horses, I don't know why 2 horses pulling on slanted directions propel the car with the same force as a single horse with both their leg power (:v:) added-up.
If the vertical component on the first situation is 2 (Fg x cos a), and the second one is 2 x fg x cos 0, how can they both give out the same result? Whether the system only has those 2 particles or it has more of them.
The problem is that the pairs of "horses" are differential horses. Their contribution to the pull is infinitely small, but there are infinitely many of them. The sum of their infinite number of infinitesimal contributions to the pull along the axis is the same as the contribution of the finite mass. That's why you need calculus. That's impossible to deal with via pre-calculus methods. Calculus shows us how to treat infinitesimals to get sensible answers out of their use.
[editline]30th June 2011[/editline]
A planet contributes a force on the order of what we see every day. Each piece of it contributes small portion, and they equal the total force when they're summed up.
Bla bla FUCKING bla.
[QUOTE=JohnnyMo1;30794275]The problem is that the pairs of "horses" are differential horses. Their contribution to the pull is infinitely small, but there are infinitely many of them. The sum of their infinite number of infinitesimal contributions to the pull along the axis is the same as the contribution of the finite mass. That's why you need calculus. That's impossible to deal with via pre-calculus methods. Calculus shows us how to treat infinitesimals to get sensible answers out of their use.
[editline]30th June 2011[/editline]
A planet contributes a force on the order of what we see every day. Each piece of it contributes small portion, and they equal the total force when they're summed up.[/QUOTE]
I know how 0 x infinity has infinite solutions so we can't use that method we can't be sure that there are as many "horses" on one side as here are in the other?
Well, then, can't we just talk about very small lumps of rock (with finite mass) so small that we can consider their distance to the surface body as the distance between both centers of mass, ignoring the fact that it has multiple bodies inside of it.
[QUOTE=MountainWatcher;30802717]I know how 0 x infinity has infinite solutions so we can't use that method we can't be sure that there are as many "horses" on one side as here are in the other?
Well, then, can't we just talk about very small lumps of rock (with finite mass) so small that we can consider their distance to the surface body as the distance between both centers of mass, ignoring the fact that it has multiple bodies inside of it.[/QUOTE]
It's a fucking NIGHTMARE to solve for the motion (or forces) of any system with any more than two particles using modern day physics. For example the Schrödinger equation only really describes the Hydrogen atom (or Hydrogen like ions) perfectly. The second you introduce a third particle to the system (a second electron to orbit the nucleus) the entire thing goes to hell and you have to start using approximations which slowly get worse as you go through the elements. Even the approximations become wildly wrong when you get to elements with dozens of electrons.
Now, if we can't explain an atom properly when it's system is made up of three separate particles, how the fuck are we meant to explain the gravitational pull of Earth, or another body, particle by particle?
[QUOTE=MountainWatcher;30802717]I know how 0 x infinity has infinite solutions so we can't use that method we can't be sure that there are as many "horses" on one side as here are in the other?[/QUOTE]
That only has one solution: 0. No number times zero is anything but zero.
[editline]30th June 2011[/editline]
Well, technically it doesn't have an exact solution because infinity is not a number but if you take, for instance, the limit as x goes to infinity of x*e^(-x), it's zero.
you guys should build a robot or some shit
I'd rather build a machine intelligence. That would be AWESOME.
that's what i was saying.
a robot.
A robot is not a machine intelligence.
There are two main things I want to see humanity accomplish in the century :
Setting foot on Mars
Creating a sentient machine
Either one will do.
[QUOTE=JohnnyMo1;30806635]That only has one solution: 0. No number times zero is anything but zero.
[editline]30th June 2011[/editline]
Well, technically it doesn't have an exact solution because infinity is not a number but if you take, for instance, the limit as x goes to infinity of x*e^(-x), it's zero.[/QUOTE]
Oh, thank Lord, someone in power who agrees with me. I meant 0+.
Guys, why did you ignore the link I posted ? We already HAVE created a sentient machine. Sort of. It has already made correct thesis' based o ndata he collected based on experiments he projected. Also, learning robot.
I was thinking to myself about the idea of AI a while back and had an odd thought. What if AI isn't possible? At least in the way we're pursuing it. I mean we're trying to create consciousness, intelligence, whatever using regular computers and just trying to up the processing power and speed among other things.
But what if the fundamental properties of carbon and silicon are just too different from one another to even allow intelligence/sentience or an analogue of it to develop into a system that uses silicon? Maybe it's simply not possible the way we're pursuing it.
Found these randomly in the Times (in order from left to right)
[img]http://i51.tinypic.com/nyblle.jpg[/img]
Take that, humanities graduates :v:
I get pretty pissed off at anybody who thinks that their area of interest should be the ONLY ONE allowed to exist or be funded and that anyone else who does anything different than them is a pathetic excuse for a person who has wasted their life. If you think that you're an arrogant asshole and need a HUGE reality check.
Furthermore, to that last letter in particular: philosophy has been INCREDIBLY important to humanity and to talk of it in such low regard is disgusting. It's the base for modern day ethics among countless other important rule sets. It's even responsible for the methodology and mindsets used to tackle logical problems.
that "Superior Science" shorty was interesting..
And frankly it summs up some of my thoughts too, only worded out more professionally
Guys, it suffices to say that philosophy is the stem of all thought. The guy probably meant metaphysics, but even then, it's wrong, metaphysics is of the utmost importance to ethics (and what could be more important to Man as how man should act?).
While the Humanities' worth (except philosophy) pales in regard of that of Science's, it is still needed. You need History as factual data, you need the arts as another way to develop creativity and logical thinking. Metaphors are great for this.
Also, only linguists can develop the perfect language we should be speaking, without a word that isn't necessary to convey any kind of concept.
[QUOTE=sltungle;30836581]I get pretty pissed off at anybody who thinks that their area of interest should be the ONLY ONE allowed to exist or be funded and that anyone else who does anything different than them is a pathetic excuse for a person who has wasted their life. If you think that you're an arrogant asshole and need a HUGE reality check.
Furthermore, to that last letter in particular: philosophy has been INCREDIBLY important to humanity and to talk of it in such low regard is disgusting. It's the base for modern day ethics among countless other important rule sets. It's even responsible for the methodology and mindsets used to tackle logical problems.[/QUOTE]
I don't agree fully with that last guy, I think philosophy, psychology and sociology are important, but I do agree that we need more science graduates in government.
[editline]2nd July 2011[/editline]
It's funny because he's gonna piss off so many people in the Times
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