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Although Traveler has some things wrong in his post (I'll get to that in a bit), he is correct in the sense that if the two balls have different masses they will not hit at exactly the same time. Even crackpots can be kind of right sometimes. While the two balls will have the exact acceleration toward the Earth, the Earth will have different accelerations toward the two balls. The resulting difference in timing will, however, be unmeasurable. What formula is this? I assume A is acceleration and R is distance, but what are L and S? Perhaps you meant [math]a_{\text{rel}} = \frac{G(m_1+m_2)}{r^2}[/math]? (Note the difference in sign.) Units? About 1.2e22 kilograms, or 0.002 earth masses, or a rock with a density of 5.5 g/cm3 and a radius of about 1/8 earth radii. None of the above counterindicates the equivalence principle. The feather and the large rock will both have the same acceleration toward the Earth from the perspective of an inertial observer. The 1 ms difference in the time of collision results Earth from the Earth having different accelerations toward the feather and the rock.

Nobody forced people to buy houses for which the payments represented half of their take-home pay (but people did just that, out of greed), or to refinance those houses every time the assessed value went up so they could take their ski trip to the Alps (and people did just that; here for the sake of gluttony and envy rather than greed). Nobody forced those agents to sell those houses to people who did not qualify to do so, or to convince unqualified buyers to lie, or to point unqualified buyers to people who would document the lie, or work with banks that ignored the false documentation. They did just that, once again out of greed. While this did help fan the flames, I wouldn't have placed this cause nearly so high on list. I suspect a political bias against the home mortgage deduction here. I'm inclined to go with the former -- because the bologna was not "left on the floor next to his bowl". The government put the bolgna on the counter and left the room. My dogs, at least, know that anything on the counter or the table is strictly off-limits, even if I am not in the room guarding it.

The section Bascule quote was from the "The Real Deal" section of the article, which apportioned blame to politicians on both sides of the aisle, but more importantly (I think) apportioned blame to us.

That's a bit like comparing apples to oranges1. A better comparison would be to compare the forces on the Earth due to solar radiation pressure and solar gravity at 1AU:2 [math] \aligned F_{\text{rad}} &= \pi r_e^2 P_{\text{rad}} \approx 5.9\cdot10^8\,\text{newtons} \\ F_{\text{grav}} &= \frac {G m_e m_s}{R_e^2} \approx 3.5\cdot10^{22}\,\text{newtons} \\ \frac{F_{\text{rad}}}{F_{\text{grav}}} &\approx 1.7\cdot10^{-14} \endaligned [/math] Since the energy flux density of solar radiation and gravitation are both inverse square laws, the ratio of the force due to radiation pressure to the force due to gravity will be essentially constant as a function of distance from the Sun.3 Bottom line: Solar radiation pressure is inconsequential to a body the size of the Earth. Footnotes: 1Not an apt analogy, as apples and oranges are actually quite comparable and even quite similar. 2Calculation here. 3Actually, the ratio decreases slightly in close proximity to the Sun as less than half of the Earth will see the full Sun and more than half of the Earth will see at least part of the Sun. Both effects act to effectively reduce the cross section to outward radiation pressure.

The idiom is "apples and oranges", which are actually quite comparable and even quite similar. Anyhow, back on topic. What CaptainPanic is alluding to here is an extremely powerful tool called "dimensional analysis". Links: Wikipedia article on dimensional analysis, and a tutorial from the University of Guelph. So, using dimensional analysis, what must the gravitational constant look like dimensionally? Ignoring the direction of the force, Newton's universal law of gravity is that [math]F = \frac{G\, m_1 m_2}{r_{12}^2}[/math] or [math]G = \frac{F\, r_{12}^2}{m_1 m_2}[/math] I'll define the units operator [math]\mathcal{U}(x)[/math] and the dimensional operator [math]\mathcal{D}(x)[/math] as follows: [math]\mathcal{U}(x)[/math]. The units operator strips away the specific numbers from a quantity, leaving only the units. For example, if m1 = 5.9742×1024 kilograms and m2 = 1.2 ounces, U(m1) = kilograms and U(m2) = ounces. [math]\mathcal{D}(x)[/math]. The dimensional operator abstracts away the specific units, leaving only the dimensions themselves. There are only a handful of physical dimensions: mass, length, time, electric current, and temperature (some use electric charge instead of current). In the above example, while the masses have two different units, they both have the same dimension, which is of course mass. . Applying dimensional analysis to Newton's equation for gravity, [math] \mathcal{D}(G) = \mathcal{D}\left(\frac{F\, r_{12}^2}{m_1 m_2}\right) = \frac{\mathcal{D}(F)\, \mathcal{D}(r_{12}^2)}{\mathcal{D}(m_1)\mathcal{D}(m_2)}[/math] On the right hand side, [math]\mathcal{D}(F)[/math]. Force is mass times acceleration, which respectively have dimensions of mass and length/time/time, so [math]\mathcal{D}(F)[/math] is mass*length/time/time, or mass*length*time-2, or even more compactly, MLT-2. [math]\mathcal{D}(r_{12}^2)[/math]. The distance between the two objects has units of length. The quantity in question is the square of this distance, so [math]\mathcal{D}(r_{12}^2)[/math] is length*length, or more compactly, L2. [math]\mathcal{D}(m_1)\mathcal{D}(m_2)[/math]. Each term in the denominator is a mass, so the dimensions of the denominator are mass*mass, or more compactly, M2. Putting it all together, the left-hand side has dimensions (mass*length/time/time)*(length*length)/(mass*mass), or (MLT-2)*(L2)/(M2). Collecting like quantities, this becomes (length*length*length)/(mass)/(time*time), or L3M-1T-2. These are the physical dimensions of the left-hand side of the above equation. The only thing on the right-hand side of the above equation is the gravitational constant itself, so the physical dimensions of the gravitational constant must be (length*length*length)/(mass)/(time*time), or L3M-1T-2. Physicists almost exclusively use the international system of measurements (SI), aka the metric system. In SI, the units of length, mass, and time are meters (m), seconds (sec), and kilograms (kg). Thus in SI units, the gravitational constant has units of m3kg-1sec-2. If you use different units you will get a different numeric value for G, but it will still have the same physical dimensions. Astronomers, for example, occasionally use astronomic units instead of meters and days instead of seconds. To them, [math]G=(1.48789\pm0.00015)\times10^{-34}\,\text{AU}^3/\text{kg}/\text{day}^2[/math].

Our best model to date, general relativity, indicates that the matter collapses into a singularity. In every other branch of physics where singularities have arisen in some theory, physicists interpret the singularity as indicative of a flaw in the theory. The same is true of black holes forming singularities in general relativity. Einstein himself rejected the concept of a singularity. I am going to reiterate what I said in my first response: We do not know what happens inside a black hole. This is from Wikipedia: The appearance of singularities in general relativity is commonly perceived as signaling the breakdown of the theory. This breakdown is not unexpected, as it occurs in a situation where quantum mechanical effects should become important, since densities are high and particle interactions should thus play a role. Unfortunately, to date it has not been possible to combine quantum and gravitation effects in a single theory. It is however quite generally expected that a theory of quantum gravity will feature black holes without singularities. They might well do that. Other physicists have conjectured that the formation of a black hole triggers the formation of a new universe. For example, see http://www.npr.org/templates/story/story.php?storyId=6545246.

Lucapsa: I merged your post into this existing thread on this topic. Now to address some of the questions you raised. Caveat: I am not a cosmologist. My answers may well be incorrect. Black holes most likely have been observed. We do not, however, know exactly what they are. In other domains in physics, the presence of singularities is indicative of a problem with the physical model rather than the physical system being modeled. That general relativity predicts black holes, i.e. volumes where spacetime is so highly curved that even light cannot escape, is well-accepted. That GR further predicts that the material forms a mathematical singularity is not so well-accepted. Many physicists see this as indicative that GR itself is flawed. The first thing to remember is that, IMHO, this article is highly conjectural. It hints at some future tests. Another thing to remember is that we do not yet know the ultimate fate of the universe. Last I read, the jury was still out. Just because our universe might not collapse in on itself does not necessarily mean our universe is "the end of the line." Other physicists have conjectured that the formation of a black hole may lead to the creation of a universe. If that is the case, our universe may well have already spawned a whole slew of universes. That implicitly assumes things are conserved from one universe to another. What makes you think they are? The conservation laws derive from Noether's first theorem. For example, conservation of energy results from the homogeneity of time. Time is not homogeneous across the "bounce". Bojowald similarly implies that the second law of thermodynamics does not apply across the "bounce". There are some signs that the speed of light does vary slightly with wavelength. For example, see this article, Gamma Ray Delay May Be Sign of 'New Physics'.

It depends on what you mean by "random function". If, on one hand, you mean taking a forming a variable Z by adding random variable X to a strictly deterministic variable Y, then Z will indeed be a random variable. If, on the other hand, you mean taking a function that represents a probability distribution such as [math]f(x)=\frac 1 {\sqrt{2\pi}}\exp\left(-\,\frac {x^2} 2\right)[/math] and forming some other function [math]h(x)=f(x)+g(x)[/math] where g(x) is some other function, then no, the new function h(x) most likely is not a "random function". It is nonsense, kind of like asking "what is ten meters plus five seconds".

We survived. Lost power around 6 PM last Friday and only regained power around 5 PM today. I can not only sleep through a hurricane, I can sleep through a tornado! However, the dead calm of the eye is another story. I woke up around 4 AM Saturday when the eye went over and went back to bed after seeing the southern eye wall start to hit. That side of the storm was particularly nasty, spinning off a lot of mini-tornados as a side benefit. One of them hit our neighborhood. The resulting mess took days to clean up. No loss of human life, but many trees were not so lucky.

In another thread I wrote Dang. Not this time. The forecast for Ike hasn't been all that great. The projected landfall area has been moving northward all week, starting from south of Corpus Christi. It now looks like Ike will hit Houston 45 hours from now. The areas under mandatory evacuation yesterday might well be in the clear while the area likely to suffer has barely begun evacuating. The evacuations for Galveston County (where I live) as of Sept. 10 were voluntary and were limited to a few very flood prone areas. There were no evacuation notices for Harris County, period. Mandatory notices will come out this morning, but there won't be enough time to evacuate 1 million+ people in front of a Cat 3 / Cat 4 hurricane. This is not good, not good at all. I got most of the wood up yesterday; I have another hour or so to finish. I hope it wasn't all in vain. Wish me luck.

Accelerating with respect to what? If the object isn't accelerating with respect to the surface, it depends on whether the gravitational body is rotating. For example, apparent and actual weight are not equal-but-opposite on the Earth. The orientation of the object is irrelevant, so long as the object is touching the surface.

Not quite. The planet is rotating. Think about it this way: If the Earth were rotating 17 times faster than it is now, objects at the equator would fly off into orbit. In other words, their apparent weight would become zero. Another way to look at it: Astronauts in the Shuttle in orbit around the Earth do have near-zero apparent weight. (That's why we call it "weightlessness".) On the other hand, their actual weight, tautologically defined as mass times the acceleration due to gravity, is about 10% or so smaller than their actual weight on the surface of the Earth. While the Earth is rather unlikely to rotating 17 times than it is now, the same is not true for asteroids. One side effect of solar radiation pressure is that radiation pressure can induce a torque on the asteroid. This second-order effect is called the Yarkovsky-O'Keefe-Radzievskii-Paddack effect, or YORP effect for short. Under the right conditions the Sun can slowly make an asteroid spin faster and faster. The Sun can make an asteroid throw itself apart! Two articles in Science on 1999 KW4: S.J. Ostro et. al., "Radar Imaging of Binary Near-Earth Asteroid (66391) 1999 KW4," Science 314(5803), 1276-1280 (24 Nov 2006) http://echo.jpl.nasa.gov/~ostro/kw4/ostro+kw4.pdf. From the article, Together, Alpha's size, shape, spin, density, and porosity reveal it to be an unconsolidated gravitational aggregate close to its breakup spin rate, suggesting that KW4’s origin involved spin-up and disruptive mass shedding of a loosely bound precursor object. The disruption may have been caused by tidal effects of a close encounter with a planet or by torques due to anisotropic thermal radiation of absorbed sunlight (the YORP effect). Also see http://echo.jpl.nasa.gov/~ostro/kw4/index.html. S.C. Lowry, et. al. "Direct detection of the asteroidal YORP effect," Science 316 (5822), 272-4 (13 Apr 2007) From the article, The Yarkovsky-O'Keefe-Radzievskii-Paddack (YORP) effect is believed to alter the spin states of small bodies in the solar system. However, evidence for the effect has so far been indirect. We report precise optical photometric observations of a small near-Earth asteroid, (54509) 2000 PH5, acquired over 4 years. We found that the asteroid has been continuously increasing its rotation rate. We simulated the asteroid's close Earth approaches from 2001 to 2005, showing that gravitational torques cannot explain the observed spin rate increase.

Are you being intentionally thick?

I will work in the planet-fixed (i.e., rotating) reference frame. Because this is a rotating frame, a fictitious centrifugal force is needed to make Newton's second law applicable in this frame: [math]\mathbf F_{\text{eff}} = \mathbf F_{\text{inertial}} - m\,\boldsymbol{\omega} \times (\boldsymbol{\omega} \times \mathbf r)[/math] I have left out the Coriolis and rotational acceleration terms because (a) the object is stationary and this has no Coriolis acceleration, and (b) the Earth's rotational acceleration is very, very small and safely can be ignored. The real forces (I am assuming Newtonian mechanics, where gravity is real force) acting on the object are gravity and the normal force. As the object is stationary in the planet-fixed frame, the effective force on the object is identically zero. Putting these together, [math]\mathbf F_{\text{eff}} = 0 = \mathbf F_{\text{grav}} + \mathbf F_{\text{normal}} - m\,\boldsymbol{\omega} \times (\boldsymbol{\omega} \times \mathbf r)[/math] or [math]\mathbf F_{\text{normal}} = -(\mathbf F_{\text{grav}} - m\,\boldsymbol{\omega} \times (\boldsymbol{\omega} \times \mathbf r))[/math] This is what a spring scale measures. (Almost. A spring scale actually measures the displacement of a spring, and the displacement of the provides the normal force in question.) Define the apparent acceleration [math]\mathbf a_{\text{apparent}}[/math] as the acceleration as measured in the rotating frame that would result if the normal force were not present. The only real force besides the normal force is gravity. The apparent force (observer fixed with respect to the rotating frame) in the absence of the normal force is [math]\mathbf F_{\text{apparent}} = \mathbf F_{\text{grav}} - m\,\boldsymbol{\omega} \times (\boldsymbol{\omega} \times \mathbf r)[/math] Dividing by the mass yields the apparent acceleration: [math]\mathbf a_{\text{apparent}} \equiv \frac{1}{m}\,(\mathbf F_{\text{grav}} - m\,\boldsymbol{\omega} \times (\boldsymbol{\omega} \times \mathbf r))[/math] Applying this term to the expression for the normal force, [math]\mathbf F_{\text{normal}} = -m\,\mathbf a_{\text{apparent}}[/math] or [math] m = \frac{||\mathbf F_{\text{normal}}||}{||\mathbf a_{\text{apparent}}||} [/math] This might look like sophistry in that appears that I have rather arbitrary defined an acceleration that lets me compute the mass given the normal force. This is not sophistry. The apparent acceleration is exactly the quantity that accelerometers and gravimeters measure.

As you posted these questions in the homework section, one has to presume that this is homework. We have a policy here at ScienceForums: We don't do your homework for you. We instead help you do your homework for yourself. So, what do you think the answers to your questions are?

Apparent weight is the net force acting on a body less the actual weight, or all of the sum of all forces acting on a body except for gravity. The only forces acting on an object on a spring scale are the normal force and gravity. In other words, the apparent weight of an object sitting on a spring scale is the normal force. Can one use F=m*a to calculate the mass given the apparent weight? Of course. All you need to know is the acceleration due to this force. The acceleration due to gravity can only be computed; there is no way to measure it. The acceleration due to apparent weight, on the other hand, is exactly what accelerometers and gravimeters measure. If you want a precise estimate of the mass you need a precise measurement of apparent weight and a precise gravimeter reading. If less precision is needed, look up the apparent acceleration in your location. If even less precision is needed, use the standard acceleration due to gravity, 9.80665 m/s2.

Accelerometers do not measure the acceleration due to gravity for the simple reason that nothing can directly measure the acceleration due to gravity. This is a direct consequence of the equivalence principle, which has now been shown to be true to within one part in 1013 (see http://physicsworld.com/cws/article/print/21148), making this one of the best (if not the best) verified physical laws. The acceleration due to gravity is the driving factor in determining the trajectory a spacecraft will follow. All measurable forces on the vehicle pale in comparison to gravity. Atmospheric drag is obviously a non-issue beyond a few thousand kilometers from the Earth. Solar radiation pressure is tiny. Spacecraft propulsion systems are used infrequently; spacecraft operate for the most part in free drift mode. Having knowledge of the acceleration due to gravity is essential in predicting where a spacecraft will go. Since this acceleration cannot be measured, a spacecraft's inertial navigation system must resort to computing this acceleration. Spacecraft going to the Moon do not pass anywhere close to your "neutral point". Doing so would require consumption of vast quantities of fuel. Spacecraft that go to the Moon either pass well in front of the Moon or well behind the Moon so that the Moon itself can do most of the work needed to turn the vehicle's trajectory. NASA doesn't publish the locations of the neutral point because they don't use the location of the neutral point in any calculations. The location of the neutral point is completely irrelevant when it comes to plotting a course to the Moon.

What is so unbelievable about a legend lasting 40,000+ years? I can think of one legend/myth that has lasted far longer than that: religion. From http://en.wikipedia.org/wiki/Paleolithic_Religion There are suggestions for the first appearance of religious or spiritual experience in the Lower Paleolithic (significantly earlier than 300,000 years ago, pre-Homo sapiens), but these are controversial and have limited support.

An data structure is just that: A bunch of data grouped together for some reason. All one can do with a data structure is access/modify the data within the structure. An abstract data type is a collection of data plus the operations that can be performed on the data. In the extreme, the data that comprise the ADT are not directly accessible from the outside. When implemented in this manner, the only way to get at the data (or modify it) is through the defined operations on the ADT. An IEEE double precision floating point numbers (e.g., the type doubles in C/C++) is an ADT. Most people don't care that such numbers comprise a sign bit, an 11 bit base 2 exponent, and a 52-bit mantissa. Most people don't care that the exponent is expressed in excess-1023 notation, that an exponent of zero has very special meaning, that the binary point is right before the first bit in the mantissa, or that there is an implied one to the left of the binary point except when there isn't. All most people care about is that the operators +, -, *, /, and = work on these numbers.

Yes! You are definitely on to something here, but you have only scratched the surface. Look what happened to the poor Apollo 1 astronauts. They died because the fire in the capsule spread far too fast because of the 100% oxygen atmosphere. We must add oxygen to the list of banned substances. And food, of course.

In other words, it's yet another namby-pamby, feel-good law that does nothing but makes the people who passed the law think they solved all of the world's problems. Stop with the logical fallacies, young man, or I will take you to the woodshed and spank you! Skeptic, the law did work in a very real sense. It made those who passed the law think they had solved all of the world's problems. Isn't that the point of all legislation?

This is not how science works. Even if it was, you do not have a theory. For that matter, you don't even have a speculation, let alone a conjecture. Let me elaborate. You do not have a theory. You are using the lay meaning for theory here. Imagine a man discussing some vexing issue with his wife: "Hey honey! I have a theory that might help us solve the problem!" In this context, the word "theory" means little more than "wild-ass guess". That is not the meaning of the word "theory" where science is concerned. In science, only the very best of ideas are designated as scientific theories. Scientific ideas sometimes do indeed start their lives as speculations. After trimming off some of the rough edges and adding some mathematical underpinnings the idea becomes something a bit better than speculation -- it becomes a hypothesis or conjecture. This is not how science works. The way that science does work is quite simple: The burden of proof in science always falls upon the claimant of a new model. Several things need to happen for a scientific model of some process or event to be designated as a scientific theory. The mathematical underpinnings of the model need to be made very strong throughout. The basis of the model must be shown to be consistent with the reality. The proponent must identify some outcome of the new model that distinguishes it from existing theory. Finally, that predicted outcome must be shown to exist -- and of course be in favor of the new model. Reality is the ultimate judge of a scientific model. You have several things wrong here. First off, E=mc2 is not a scientific law. It is a derived result. As an aside, do you even know what this statement means? In classical mechanics, gravitation is a conservative force, meaning that it does indeed conserve energy. I suggest we stick with classical physics for now for the simple reason that modern physics is built on top of classical physics. A pair of bodies transforms gravitational potential energy into kinetic energy as the bodies get closer to one another, and kinetic energy into potential energy as they move away from one another. Actually, they do. However, the radiated energy is typically very, very small because gravitation is the weakest of the four forces. For example, the power radiated by the Earth and Sun as they two orbit their common center of mass is a paltry 313 watts.

In what way? If the different materials have different masses, yes, then different materials affect gravity. If the different materials are of the same mass, then no, they do not affect gravity as far as we know, and we know this extremely well. Sheesh. Did you read any of my prior posts? Everything you wrote after that indicates that you completely misunderstood what I wrote. Axioms in physics not only can be questioned, questioning them is an essential part of physics. Physical axioms must be tested rigorously. A couple of examples: [math]F=ma[/math] is a 300+ year old axiom in classical physics. Physicists to this day come up with novel tests of this axiom. The speed of light is the same to all observers is a 100+ year old axiom that sits at the heart of special relativity. Scientists test this axiom to this day as well. Only to a limited extent. Read the rules. Too much speculating without offering any proof is a recipe for being banned. That is not how science works. It is up to the people making the extraordinary claims to prove their claims are right.

Whoa there! I suggest you read the speculation policy. "Whatever the case is, this forum is not a home for just any science-related idea you have. It has a few rules. Speculations must be backed up by evidence or some sort of proof." Questioning the status quo is the lifeblood of scientists. Almost all good scientists have a bit of the crackpot in them. Almost all good scientists also learn to temper their crackpot tendencies because the status quo in science tends to have a strong theoretical underpinning and immense amount of confirming evidence. It is usually a good idea to learn those theoretical underpinnings and examine the experimental evidence that supports the status quo. You still don't get the point. I guess I need to be extremely explicit. I was not talking about you making assumptions. I was talking about physicists doing so. The equivalence principle is an assumption. All physical theories have some set of axioms on which the theories are based. Axiom is just a high-falutin' term for assumption. Mathematicians are free to invent new mathematics by coming up with a new set of axioms. Physicists are not free to do so. Their axioms must conform with reality. A physical theory whose assumptions do not conform with reality is faulty from the onset. Physicists do not particularly like unwarranted assumptions. Physically invalid assumptions are far, far worse than unwarranted assumptions. Experimentalists poke at a physical theory in two places: They test whether the assumptions are valid and they test whether distinguishing results predicted by the theory do indeed arise. The article you posted and the results that I posted are an example of experimental physicists doing their job. One thing experimentalists love doing more than almost anything else is ripping a hole in the high-falutin' theories of those hoity-toity theoretical physicists. Failing that, they don't mind finding new evidence that indicates that their theoretical brethren are indeed correct.

So does this mean you are relinquishing your claim to the Moon have a vastly different surface gravity than stated by NASA (and every other space agency in the world)? Righto. I am not giving you my email address. You can, however, send it to me here at ScienceForums via private message. A link: http://www.scienceforums.net/forum/member.php?u=8197. You can put it down in equations right here. You have not done so. If you haven't noticed, we have the standard mechanism used by scientists worldwide to represent mathematics in papers -- LaTeX. If that is the case, one of a very few things is going on: You have "cooked the books" by fudging your numbers. Intellectual dishonesty is even worse than cracking pots. One thing that makes me think this might be the case is your stated incorporation of eight asteroids into your results. Asteroids have very small mass and thus will follow Kepler's Laws extremely well. You also have "cooked the books" by only modeling planets and asteroids. The ratio of the square of the period to the cube of the semi-major axis for satellites of planets is markedly different from the ratio for objects that orbit the SUn. You failed to test for outliers, specifically Jupiter and Saturn. Jupiter and Saturn (Jupiter particularly so) do not follow Kepler's Laws as well do the other planets because Jupiter and Saturn are quite massive. You have recreated Newton's law of gravitation (you have mentioned perturbations, and Kepler's Laws don't accommodate perturbations). If you have merely recreated Newton's law of gravitation, what would motivate choosing your concept over Newton's? The above begs the question: So what? Four nines of accuracy is lousy in the case of modeling planetary orbits. Think about it this way: We know by astronomical observations the perihelion precession of Mercury to within 0.1 arc seconds per century. Think what that means in terms of the number of significant digits to which we know the period of Mercury.