This section is optional and covers the physics and mathematical details of the two-body problem and three-body problem. The two-body problem Consider two point masses, m₁ and m₂, moving within a plane (that is, the 2-dimensional case). Let the position of m₁ be denoted by (x₁, y₁) and m₂ by (x₂, y₂). Because the masses are moving with respect to time, their positional coordinates (x, y) are actually functions of time t, that is (x(t), y(t)). Figure 1 shows such a situation for a particular value of t (a moment frozen in time): Figure 1 In figure 1, we see that m₁ and m₂ are a distance r apart and are mutually attracted to one another through the force of gravity F₁ and F₂ (whose magnitudes are the same). This force of gravity (F) is given by Newton's law of universal gravitation: The force vector F₁ acting on the first point mass m₁ has the following x- and y-components (Fₓ and Fy, respectively), as can be seen here: Figure 2 Using trigonometry, we see that: And, using the larger triangle, shows that: If we focus on Fₓ and substitute for F and cos θ, we have: Where F(t) and (t) are both vector functions of time. If, for now, we consider just the x-direction of m₁, we see that: Where, in the last expression, we drop the implied function of time variable t. We now have two expressions for Fₓ, the first from Newton’s second law and the second from Newton’s law of universal gravitation: By using the Pythagorean theorem (see figure 2), the distance r between m₁ and m₂ is given by: Therefore (for m₁ in the x-direction), we have: Now given the initial positions (x₁, y₁) and (x₂, y₂) at t = 0, we can calculate the acceleration of m₁ due to m₂ in the x-direction using equation 14. We can use the same reasoning as above to show that the y-component of m₁’s acceleration (due to m₂) is given by: From m₂’s perspective, we see that the acceleration equations are identical except for a change in sign (indicating opposite force directions) as provided by the swap in positional terms in the numerator: Thus, we have the following set of equations, where the numeric subscripts indicate the mass m₁ or m₂: Where: (when α = 0, an infinite acceleration is imparted when the two masses exactly collide, which is physically impossible). Now we have acceleration as a function of each mass’s current position but how do we approximate its velocity and position at some small amount of time h (that is, Δt) later? This is where numerical integration comes in. In particular, we’ll use leapfrog integration (the position Verlet algorithm) in that it’s relatively simple but reasonably accurate and stable. Consider N celestial masses. Let i indicate one of the masses (i = 1, …, N) and h a small time interval. For the position Verlet algorithm, the ith mass’s next position and velocity values are calculated as follows: Where, for a given directional component, xⁱ is the position, vⁱ the velocity, and aⁱ the acceleration of the ith mass. In order to improve accuracy, on first use, equation 22 is replaced with the following: This information leads to the following algorithm: Choose a small value for h (Δt). Choose initial (component-wise) position and velocity values for all masses. Using equations 18 through 21, calculate initial acceleration values. Using the values from the prior step, use equation 25 to calculate the initial values. Using the previously calculated values, use equations 18 through 21 to calculate the values. Using the values from the prior step, calculate the values using equation 23 Using the values from the prior step, calculate the values using equation 24. Using the values from the prior step, use equation 22 to calculate the values. Go to step 5 until the user chooses to stop. The three-body problem When there are three masses, any one mass has two forces acting on it from the other two masses. For example, m₁ has the following forces (F₂ and F₃) acting on it: Figure 3 To start, we note that the net force F₁ acting on m₁ is the sum of F₂ and F₃. That is, F₁ = m₁₁ = F₂ + F₃ Now applying the same trigonometry used in figure 2 to figure 3, the magnitude of the net force F₁ on m₁ can be broken into its x- and y-components as follows: Using the red and green triangles in figure 3, we see that: From Newton’s law of universal gravitation, F₂ and F₃ can be expressed as: Substituting equation 28 through 33 into equations 26 and 27 results in: Simplifying 34 and 35 yields: Replacing r₂ and r₃ with their Pythagorean formulae yields the following x- and y-component acceleration equation for m₁:

]]>For irregular, nonspherical mass distributions in three dimensions, Newton’s original vector equation (4) is inefficient, though theoretically it could be used for finding the resulting gravitational field. The main progress in classical gravitational theory after Newton was the development of potential theory, which provides the mathematical representation of gravitational fields. It allows... ...to determine the large-scale structure of the entire universe. Gravity is a fundamental quantity whether it is an essentially geometric parameter, as in general relativity, or the strength of a field, as in one aspect of a more-general field of unified forces. The fact that, so far as is known, gravitation depends on no other physical factors makes it likely that the value of ... Earth Earth’s gravitational field is manifested as the attractive force acting on a free body at rest, causing it to accelerate in the general direction of the centre of the planet. Departures from the spherical shape and the effect of Earth’s rotation cause gravity to vary with latitude over the terrestrial surface. The average gravitational... passage of electromagnetic rays ...of a few of the more common terms. Around every particle, whether it be at rest or in motion, whether it be charged or uncharged, there are potential fields of various kinds. As one example, a gravitational field exists around the Earth and indeed around every particle of mass that moves with it. At every point in space, the field has direction in respect to the particle. The strength of... Saturn Information about the interior structure of Saturn is obtained from studying its gravitational field, which is not spherically symmetrical. The rapid rotation and low mean density that lead to distortion of the planet’s physical shape also distort the shape of its gravitational field. The shape of the field can be measured precisely from its effects on the motion of spacecraft in the vicinity...

]]>Unit 4 Test - Part 1 - Earth Science 1110 with Sirvatka at College

]]>On the last day of my residence at the artists' colony Yaddo, I shared with my co-residents an excerpt from my book, Dark Matter and the Dinosaurs . I read from the first chapter, in which I liken dark matter - matter present throughout the universe that is invisible to us because it doesn't emit or absorb light - to other entities that remain unnoticed but influence the workings of the world, from the bacterial cells in our bodies, which outnumber human cells by a factor of 10, to the myriad Internet communities and subcultures that thrive outside our awareness. The goal was to illuminate the gap between our limited observations and the many barely perceived phenomena that permeate our reality. I was gratified to observe the audience's increased comfort with dark matter and its unseen but important influences. But the most surprising and rewarding response came the following day, when Jefferson Pinder, a young African-American artist, stopped me as I was leaving and asked, "I know this might sound like a crazy question, but were you really talking about race?" The crazy thing is that I was. People's attitude toward dark matter is bedeviled by the same instincts that influence their responses to different races, castes or classes whom they might not truly see but who are nonetheless essential to society. Jefferson understood that the real issue I was addressing was the transparency - both metaphorical and literal - of people, phenomena, particles, and forces that we don't necessarily appreciate but that are important to our shared reality. The metaphor returned in a different guise in a seminar I teach at Harvard, which I begin with a discussion of how science advances, and how the optimal scientific description can depend on the frame of reference. The students relished the science, but the classroom discussion also took a surprising turn - into questions of empathy. Race and class differences call for empathy largely because of our difficulties in understanding what we can't experience or see, including the often hidden cultural forces that animate other people and their communities. Such blindspots challenge us in the scientific realm too - but in ways that are usually more obvious and readily acknowledged. The world looks entirely different at the scale of the atom - or the Higgs boson - than it does when viewed from your chair or from space. This is why the rules of quantum mechanics can appear unintuitive or illogical. Their unfamiliarity makes them difficult to comprehend. People relate best to scales they encounter in their daily lives, perhaps a millimeter to a kilometer in size - the scale our brain's visual system readily processes via optical wavelengths. As science and technology advance, sophisticated measuring instruments allow us to explore distances increasingly removed from our immediate experience. But our human blinders endure - the greater that distance, the more irrelevant the phenomena tend to seem.

]]>We've all heard the story. A young Isaac Newton is sitting beneath an apple tree contemplating the mysterious universe. Suddenly - boink! -an apple hits him on the head. "Aha!" he shouts, or perhaps, "Eureka!" In a flash he understands that the very same force that brought the apple crashing toward the ground also keeps the moon falling toward the Earth and the Earth falling toward the sun: gravity. Or something like that. The apocryphal story is one of the most famous in the history of science and now you can see for yourself what Newton actually said. Squirreled away in the archives of London's Royal Society was a manuscript containing the truth about the apple. It is the manuscript for what would become a biography of Newton entitled Memoirs of Sir Isaac Newton's Life written by William Stukeley, an archaeologist and one of Newton's first biographers, and published in 1752. Newton told the apple story to Stukeley, who relayed it as such: "After dinner, the weather being warm, we went into the garden and drank thea, under the shade of some apple trees...he told me, he was just in the same situation, as when formerly, the notion of gravitation came into his mind. It was occasion'd by the fall of an apple, as he sat in contemplative mood. Why should that apple always descend perpendicularly to the ground, thought he to himself..." The Royal Society has made the manuscript available today for the first time in a fully interactive digital form on their website at The digital release is occurring on the same day as the publication of Seeing Further (HarperPress, £25), an illustrated history of the Royal Society edited by Bill Bryson, which marks the Royal Society's 350th anniversary this year. So it turns out the apple story is true - for the most part. The apple may not have hit Newton in the head, but I'll still picture it that way. Meanwhile, three and a half centuries and an Albert Einstein later, physicists still don't really understand gravity. We're gonna need a bigger apple.

]]>Yes, Sir Isaac Newton is best known for his work on gravity. He also worked on and discovered many other scientific wonders during his lifetime (1642-1727). His work in physics was so advanced that he was the first scientist to be knighted, which is a great honor in England and the reason "Sir" preceeds his name. So what else did Sir Isaac Newton discover? Besides his work on universal gravitation (gravity), Newton developed the three laws of motion which form the basic principles of modern physics. His discovery of calculus led the way to more powerful methods of solving mathematical problems. His work in optics included the study of white light and the discovery of the color spectrum. It was his experiments with light that first made him famous. How did he discover the color spectrum? Newton performed an experiment using a glass prism. For the experiment he placed a glass prism in front of a beam of light projected through a tiny hole in a window shade. You can perform a virtual version of the experiment below. You just need to place the mouse cursor over the prism to darken the room and reveal the color spectrum.

]]>Ali Pretty | Walking With Newton – Gravity Fields Festival

]]>In my draft article, Five Lessons from the Health Care Cases, I queried what is the significance of the fact that in NFIB v. Sebelius, the cahllengers did not primarily advance an originalist argument, though the outcome of the case moved the Constitution closer to its original meaning. I had trouble reconciling the two concepts. Though, after attending the Federalist Society convention this weekend, and chatting at length with Randy Barnett, I’ll offer a very cosmic constitutional theory (my apologies to Judge Wilkinson). A brief astronomy lesson. In our solar system, 9 planets orbit around the Sun. The gravitational pull of our nearest star star keeps the planet in orbit. But, gravity goes both ways (grossly oversimplifying). Our planet exerts a pull, however small, on our star. To the extent that the planet exerts a pull on the star, the star will wobble a bit towards the planet. This principle of physics has enabled astronomers to locate planets outside of our solar system (extrasolar planets). Astronomers are only able to detect extrasolar planets–which are too small to be visible even with advanced telescopes–by measuring shifts in the movement of stars. If a star “wobbles, ” that is a sign that a planet’s gravitational forces is pulling on it. In our jurisprudential solar system, think of a star as our Constitution. Various planets that orbit the star represent different constitutional theories (now I’m just having fun with Judge Wilkinson). The strength of the theory can be viewed as a function of the gravitational pull the planet places on the star. If a theory has some pull on the star, even if the theory is not that close to the star itself, it still has some influence. Let me explain in the context of originalism. As Randy Barnett noted, there are three views of federalism (and I would add, relatedly, federalism’s structural protection of individual liberty, see Bond ). First, there is the pre-1937 originalist view, where the Court, unbound by modern precedents, rules in accordance with the original public meaning of the Constitution. Second, there is the New Deal-era view of federalism, wherein Congress has a plenary police power to do whatever it deems necessary, and any law that fills within the New Deal’s ambit will be upheld. Third, there is the New Federalism of the Rehnquist Court. This third strand can be best characterized as “this far, but no farther.” In other words, the New Federalism did not repudiate the New Deal view of federalism, nor did it effect a return to the pre-1937 view of Federalism. Rather, it asserted that if the federal government seeks to assert a power that goes beyond what had already been upheld, it must justify that extension of an unprecedented assertion of power. Now, the Court would not adjudge the constitutionality of the new law purely based on originalism, but instead based on what Chief Justice Rehnquist referred to as “first principles” in Lopez. It is noteworthy that Justice Thomas’s originalist opinion in Lopez was not joined by Justice Scalia (same for Morrison). This tripartite taxonomy helps to explain why originalism has, and has not been successful in recent cases. Perhaps the best examples in the first category are Heller in McDonald. In these cases, the Court was largely writing on a blank slate. The Court was in no way bound by any sort of New Deal compromise, as the precedential slate is clear. Thus, the Court was free to receive, and did apply originalist arguments. In fact, both the majority and dissent in Heller and McDonald advanced originalist arguments. For decades, until Lopez and Morrison and other Rehnquist era precedents, the Supreme Court was steadfastly locked in the second zone of the New Deal-era view of Federalism. In the words of Larry Solum, the gestalt had crystalized. However, what helped to break federalism free from its Roosevelt-imposed chrysalis? I would suggest originalism. Originalist scholarship began to emerge in the 1970s and 1980s that showed that the Court had departed from the original understanding of the Constitution in the New Deal cases, particularly with respect to federalism and structural protections of individual liberty. These cases exerted a pull on the Court’s jurisprudence ever so subtle at first, but soon enough the law, like a star being attracted to a planet, began to wobble. Progressives observed this wobble, worried, and hoped that the Constitution would remain in the sole-pull of the New Deal. Cases like New York v. United States, United States v. Lopez, Printz v. United States, United States v. Morrison, Seminole Tribe v. Florida and others are collectively dubbed part of the “New Federalism.” None of these cases were argued in terms of restoring the original meaning of the Constitution. The advocates didn’t need to. It was sufficient for the Justices to know that errors were made, those errors would not be fixed (in Justice Scalia’s words, they were “water over the dam”), but that the court should go no further from the Constitution’s original meaning without a sufficient justification from the government.

]]>You probably have an idea of what gravity is, but did you know that you, right now, are actually pulling on every other object in the universe? Find out more about the gravitational force and learn an equation to calculate its pull on other objects. Click "next lesson" whenever you finish a lesson and quiz. Got It You now have full access to our lessons and courses. Watch the lesson now or keep exploring. Got It You're 25% of the way through this course! Keep going at this rate, and you'll be done before you know it. Way to go! If you watch at least 30 minutes of lessons each day you'll master your goals before you know it. Go to Next Lesson Take Quiz Congratulations on earning a badge for watching 10 videos but you've only scratched the surface. Keep it up! Go to Next Lesson Take Quiz You've just earned a badge for watching 50 different lessons. Keep it up, you're making great progress! Go to Next Lesson Take Quiz You have earned a badge for watching 20 minutes of lessons. You have earned a badge for watching 50 minutes of lessons. You have earned a badge for watching 100 minutes of lessons. You have earned a badge for watching 250 minutes of lessons. You have earned a badge for watching 500 minutes of lessons.

]]>The most familiar examples of material particles are the electron, the proton and the neutron. Combinations of these particles form atoms. There are more than 100 different kinds of atoms, each kind constituting a unique chemical element. A combination of atoms forms a molecule. Atoms and/or molecules can join together to form a compound. Matter can exist in several states, also called phases. The three most common states are known as solid, liquid and gas. A single element or compound of matter might exist in more than one of the three states, depending on the temperature and pressure. Less familiar states of matter include plasma, foam and Bose-Einstein condensate. These states occur under special conditions. Different kinds of matter can combine to form substances that may not resemble any of the original ingredients. For example, hydrogen (a gaseous element) and oxygen (another gaseous element) combine to form water (a liquid compound at room temperature). The process of such combination is called a chemical reaction. A chemical reaction involves interactions between the electrons of the atoms, but does not affect the nuclei of the atoms. In some situations, matter is converted into energy by atomic reactions, also known as nuclear reactions. This type of reaction is fundamentally different from the chemical reaction because it involves changes in the nuclei of atoms. The most common example of an atomic reaction is the hydrogen fusion that occurs inside the sun. The immense pressure inside the sun, and inside other stars, forces atoms of hydrogen together to form atoms of helium. In this process, some of the mass is converted to energy according to the formula = mc 2 where is the energy in joules, is the mass in kilograms, and is the speed of light, which is approximately 2.99792 x 10 8 meters per second in a vacuum. In recent years, scientists have confirmed the existence of a substance called antimatter. The electron has an antiparticle twin called a positron, with equal mass but opposite electric charge. Similarly, the proton has an antimatter twin called an antiproton, and the neutron has an antimatter twin called an antineutron. If a particle of matter encounters its antiparticle, both are converted entirely to energy according to the above formula, where is the combined mass of the particle and the antiparticle. Small amounts of antimatter have been isolated in laboratory conditions, but no one has yet succeeded in creating a controlled a matter/antimatter reaction, or even an uncontrolled reaction of significant size.

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