In each of the cases examined so far the pendulum as well as the elliptical and circular orbits the

In this article we'll explore the relationship between gravity and energy, lowest possible airline tickets and consider some consequences for matters both large and small. This article lowest possible airline tickets uses animations and graphics to clarify its points, and some key equations are included and explained. Finally, we'll discuss a new theory about the universe how it might have come into being without either violating any laws of physics or requiring supernatural intervention.
If I lift a one-kilogram book from the floor and place it on a two-meter-high shelf, the book gains 19.6 joules of potential energy (enough to power a small flashlight lowest possible airline tickets for about one second) and 2.2 * 10 -16 kilograms of mass (about 1/3 that of a small bacterium).
Notice the minus sign in equation (4) above it means that gravitational potential energy is negative . Because this is an important property with cosmological lowest possible airline tickets significance, I would like to explain how it comes about.
The reader may recall my earlier remark lowest possible airline tickets than energy is the time integral of power, lowest possible airline tickets but this is just one example in mechanics, work (energy) can be expressed as the integral of force with respect to distance (x) rather than time:
Expressed in everyday language, work is equal to force times distance. Now we'll apply this to gravitation lowest possible airline tickets here is the force equation for gravitational lowest possible airline tickets attraction between two masses m 1 and m 2 , separated by a distance r , and under the influence of the gravitational constant lowest possible airline tickets term G :
One more thing under General Relativity , gravity is not a force, instead it arises as a result of spacetime curvature. But in ordinary circumstances the Newtonian conventions still apply, and energy is still a meaningful concept in orbital mechanics.
Remember that mass-energy lowest possible airline tickets cannot be created or destroyed, only changed in form. A more general way to say this is that the universe has a constant quantity Q of mass-energy, fixed at the moment of the Big Bang and unchanged since. We'll be discussing the quantity Q throughout this paper, and we'll eventually assign it a value.
As a mass moves in a gravitational field, it typically exchanges kinetic and potential energy. A swinging pendulum (Figure 2) has maximum kinetic energy at the lowest point in its swing, and zero kinetic energy at the highest. The pendulum's potential energy has the reverse lowest possible airline tickets relationship it increases (i.e. becomes less negative) with distance from the center of the earth, lowest possible airline tickets and in exchange, the kinetic energy must decrease. The important thing to understand about freely moving objects in a gravitational field is that their energy, the sum of kinetic and potential energy, is constant.
There is a well-known principle in mechanics called Newton's First Law which says that, unless acted on by an external force, an object will maintain a constant state of motion. There is, or should be, a corollary lowest possible airline tickets for freely moving objects in space:
This doesn't mean the object's velocity will remain the same, nor does it mean the object's kinetic and potential energy values will remain the same. It means the total energy, the sum of kinetic and potential energy, will remain constant.
The swinging pendulum in Figure 2 shows this even though there is a periodic exchange between kinetic and potential energy, the total energy ( e k + e p ) is constant. If our pendulum were located lowest possible airline tickets in a vacuum and had lossless bearings, it would continue to swing forever in the same way, perpetually conserving its energy. (In Figure 2, the height of the red/green bar at the left represents the sum of kinetic and potential energy. Because the pendulum's lowest possible airline tickets energy is constant, the bar's red and green sections always sum to the same height.)
For small-scale mechanical systems like the pendulum, it's convenient to establish an arbitrary zero point for potential energy. In this case, the zero point is set at the bottom of the swing, so potential energy is pictured as increasing from zero to positive values as the pendulum lowest possible airline tickets swings. This is a reasonable way to picture a physical system, but the absolute value of gravitational potential energy is typically a much larger value, and is always negative.
Pendulums don't usually get to swing in a vacuum with frictionless bearings, but an orbiting satellite is a better example of a frictionless system. Like the pendulum, as the satellite orbits it carries both kinetic and potential energy:
To be consistent with the Principle of Energy lowest possible airline tickets Conservation , for a freely orbiting body with no external forces acting on it, equation (10), the sum of kinetic and potential energies, always produces a constant.
Figure 3 shows a satellite lowest possible airline tickets in a very elliptical (oval-shaped) orbit around a central body. I chose this configuration to show that, even though there is an ongoing exchange between kinetic and potential energy, as with the pendulum the total energy remains lowest possible airline tickets constant. By the way, this orbital shape isn't hypothetical comets often have elongated orbits like this. Many comets dwell far beyond Pluto and only rarely descend into our neighborhood for a brief appearance.
Interestingly, Johannes Kepler lowest possible airline tickets computed the properties of orbits and wrote what we know as Kepler's Laws of Planetary Motion , but without lowest possible airline tickets understanding the reason orbits behave as they do. The secret to understanding orbits is to recognize that their motion conserves energy, and if they behaved at all differently, nature would need different laws.
Where m 1 is the satellite's mass, m 2 is the central body's mass, r is the orbital radius and G is the universal gravitational constant. If the satellite is much less massive than the central body, this approximate equation is suitable:
Where M is the central mass, r is the orbital radius and G is the universal gravitational constant. It turns out that, for circular orbits, the relationship between kinetic and potential energy is fixed, regardless of the orbit's other properties the negative gravitational potential energy is always twice the magnitude of the positive kinetic energy. Another way to say this is that, for a circular orbit, 2/3 of the energy is negative potential and 1/3 is positive kinetic.
In each of the cases examined so far the pendulum as well as the elliptical and circular orbits the sum of energies has been negative, dominated lowest possible airline tickets by negative gravitational potential energy. Obviously lowest possible airline tickets we might select a very high velocity and produce a positive result for equation (10) above, the total orbital energy. But is there an orbital velocity that exactly balances the two kinds of energy and produces zero? Yes, there is it's called escape velocity . Here is its value, using the terminology from the previous section:
Escape lowest possible airline tickets velocity has some interesting properties. If an object is propelled away from an airless planet with an initial impulse of escape velocity (sort of like Alan Shepard's famous golf shot on the moon, but a much higher velocity), that object will continue to move away, at gradually decreasing speed, but it will never stop and return. In fact, at an infinite distance, an escape-velocity object will achieve zero velocity. Here are the properties of an object initially given escape velocity:
There are two canonical orbital velocities one is the circular velocity v o provided lowest possible airline tickets by equations (11) or (12), the other is escape velocity ( v e ) provided by equation (13). Velocities greater than escape velocity v e or less than circular velocity v o produce interesting effects, like the large family of elliptical orbits resulting from initial velocities in the range 0 v v o and shown in Figure 3 above. But because escape velocity has properties of cosmological significance, it merits a closer look.
Because of its importance to what follows, we should prove that escape velocity results in zero net energy (i.e. e k + e p = 0). First, lowest possible airline tickets let's simplify equation (10) let's normalize the mass of the orbiting body m 1 to 1. Here is the result:
Remember that equation (14) provides lowest possible airline tickets the total energy of an orbiting body, the sum of positive kinetic and negative potential energy. At this point, those sufficiently adept at mathematics will compare equations (13) (escape velocity) and (14) (total energy) and a light bulb will go off. For the rest of us, here's a step-by-step proof:
When Georges Lemaître first proposed the Big Bang theory, there were a number of objections at the time there was no evidence in support of the idea, it seemed counterintuitive, and it appeared to violate basic physical principles. How could the universe arise out of nothing?
Astronomer Edwin Hubble detected a systematic redshift in the spectra of distant galaxies (more distant galaxies have proportionally more redshift), and this was eventually taken to mean those galaxies were moving away from each other and us.
Physicist George Gamow conjectured that, if the Big Bang was real, there would be a residual radiation left over from the exceedingly high temperatures lowest possible airline tickets of the explosion, but very much redshifted, all the way into the microwave region of the spectrum and with a characteristic lowest possible airline tickets temperature of about 5 Kelvins.
Radio astronomers Arno Penzias and Robert Wilson inadvertently discovered this microwave signal, now known as the cosmic microwave background (CMB) radiation , and this signal has become the subject of intense study.
All this evidence has given the Big Bang the status of a scientific theory, that is to say, a theory supported by evidence and falsifiable in principle. But one objection to the Big Bang remains, and it is serious there is no physical law so well-established as the conservation of mass-energy, and the Big Bang seems to violate it. By creating an entire universe of mass-energy out of nothing, the Big Bang seems to break the most basic rule of physics: no free lunch .
But this final objection is answered by the idea expressed in this article if the universe began with an exact balance between positive mass-energy and negative gravitational potential energy, the law of mass-energy conservation is honored.
lowest possible airline tickets For this condition to be met, the Big Bang would have to