Universal Gravitation

Gravity is a force which exists between the Earth and the objects which are near it.. We have become accustomed to calling it the force of gravity and have even represented it by the symbol Fgrav.
Not to be confused with the force of gravity (Fgrav), the acceleration of gravity (g) is the acceleration experienced by an object when the only force acting upon it is the force of gravity. On and near Earth's surface, the value for the acceleration of gravity is approximately 9.8 m/s/s. It is the same acceleration value for all objects, regardless of their mass (and assuming that the only significant force is gravity).
The Apple, the Moon, and the Inverse Square Law
In the early 1600's, German mathematician and astronomer Johannes Kepler mathematically analyzed known astronomical data in order to develop three laws to describe the motion of planets about the sun. Kepler's three laws emerged from the analysis of data carefully collected over a span of several years by his Danish predecessor and teacher, Tycho Brahe. Kepler's three laws of planetary motion can be briefly described as foflows:
The path of the planets about the sun are elliptical in shape, with the center of the sun being located at one focus. (The Law of Ellipses)
An imaginary line drawn from the center of the sun to the center of the planet will sweep out equal areas in equal intervals of time. (The Law of Equal Areas)
The ratio of the squares of the periods of any two planets is equal to the ratio of the cubes of their average distances from the sun. (The Law of Harmonies)
While Kepler's laws provided a suitable framework for describing the motion and paths of planets about the sun, there was no accepted explanation for why such paths existed. The cause for how the planets moved as they did was never stated. Kepler could only suggest that there was some sort of interaction between the sun and the planets which provided the driving force for the planet's motion. To Kepler, the planets were somehow "magnetically" driven by the sun to orbit in their elliptical trajectories. There was however no interaction between the planets themselves.
Newton was troubled by the lack of explanation for the planet's orbits. To Newton, there must be some cause for such elliptical motion. Even more troubling was the ircular motion of the moon about the earth. Newton knew that there must be some sort of force which governed the heavens; for the motion of the moon in a circular path. and of the planets in an elliptical path required that there be an inward component of force. Circular and elliptical motion were clearly departures from the inertial paths (straight-line) of objects. And as such, these celestial motions required a cause in the form of an unbalanced force. The nature of such a force - its cause and its origin - bothered Newton for some time and was the fuel for much mental pondering. And according to legend, a breakthrough came at age 24 in an apple orchard in England. Newton never wrote of such an event, yet it is often claimed that the notion of gravity as the cause of all heavenly motion was instigated when he was struck in the head by an apple while lying under a tree in an orchard in England. Whether it is a myth or a reality, the fact is certain that it was Newton's ability to relate the cause for heavenly motion (the orbit of the moon about the earth) to the cause for Earthly motion (the falling of an apple to the Earth) which led him to his notion of universal gravitation
Newton's Law of Universal Gravitation
Isaac Newton compared the acceleration of the moon to the acceleration of objects on earth. Believing that gravitational forces were responsible for each, Newton was able to draw an important conclusion about the dependence of gravity upon distance. This comparison led him to conclude that the force of gravitational attraction between the Earth and other objects is inversely proportional to the distance separting the earth's center from the object's center. But distance is not the only variable affecting the magnitude of gravitational force. Consider Newton's famous equation
Fnet = m • a
Newton knew that the force which caused the apple's acceleration (gravity) must be dependent upon the mass of the apple. And since the force acting to cause the apple's downward acceleration also causes the earth's upward acceleration (Newton's third law), that force must also depend upon the mass of the earth. So for Newton, the force of gravity acting between the earth and any other object is directly proportional to the mass of the earth, directly proportional to the mass of the object, and inversely proportional to the square of the distance which separates the centers of the earth and the object.
But Newton's law of universal gravitation extends gravity beyond earth. Newton's law of universal gravition is about the universality of gravity. Newton's place in the Gravity Hall of Fame is not due to his discovery of gravity, but rather due to his discovery that gravitation is universal. All objects attract each other with a force of gravitational attraction. Gravity is universal. This force of gravitational attraction is directly dependent upon the masses of both objects and inversely proportional to the square of the distance which separates their centers. Newton's conclusion about the magnitude of gravitational forces is summarized symbolically as
Fgrav ~ m1 * m2 / d2
when Fgrav represents the force of gravity between two objects
~ means*proportional to*
m1 represents the mass of object 1
m2 represents the mass of object 2
d represents the distance separating the objects centers
Since the gravitational force is directly proportional to the mass of both interacting objects, more massive objects will attract each other with a greater gravitational force. So as the mass of either object increases, the force of gravitational attraction between them also increases. If the mass of one of the objects is doubled, then the force of gravity between them is doubled. If the mass of one of the objects is tropled, then the force of gravity between them is tripled. If the mass of both of the objects is doubled, then the force of gravity between them is quadrupled; and so on.
Since gravitational force is inversely proportional to the separation distance between the two interacting objects, more separation distance will result in weaker gravitational forces. So as two objects are separated from each other, the force of gravitational attraction between them also decreases.
Another means of representing the proportionalities is to express the relationships in the form of an equation using a constant of proportionality. This equation is shown below.
Fgrav = G + m1 * m2 / d2
where G represents the universal gravitation constant
(G = 6.67 * 10-11 N.m2 / kg2)
The constant of proportionality (G) in the above equation is known as the universal gravitation constant. The precise value of G was determined experimentally by Henry Cavendish in the century after Newton's death. The value of G is found to be
G = 6.673 x 10-11 N m2 / kg2
The units on G may seem rather odd; nonetheless they are sensible. When the units on G are substituted into the equation above and multiplied by m1•m2 units and divided by d2 units, the result will be Newtons - the unit of force.
Planetary and Satellite Motion
Kepler's Three Laws
In the early 1600s, Johannes Kepler proposed three laws of planetary motion. Kepler was able to summarize the carefully collected data of his mentor - Tycho Brahe - with three statements which described the motion of planets in a sun - centered solar system. Kepler's efforts to explain the underlying reasons for such motions are no longer accepted; nonetheless, the actual laws themselves are still considered an accurate description of the motion of any ptanet and any satellite.
Kepler's three laws of planetary motion can be described as follows:
The path of the planets about the sun are ellitical in shape, with the center of the sun being located at one focus. (The Law of Ellipses)
An imaginary line drawn form the center of the sun to the center of the planet will sweep out equal areas in equal intervals of time. (The Law of Equal Areas)
The ratio of the squares of the periods of any two planets is equal to the ratio of the cubes of their average distances form the sun. (The Law of Harmonies)
Circular Motion Principles for Satellites
A satellite is any object which is orbiting the earth, sun or other massive body. Satellites can be categorized as natural satellites or man-made satellites. The moon, the planets and comets are examples of natural satellites. Accompanying the orbit of natural satellites are a host of satellites launched form earth for purposes of communication, scientific research, weather forecasting, intelligence, etc. Whether a moon, a planet, or some man-made satellite, every satellite's motion is governed by the same physics principles and described by the same mathematical equations.
The fundamental principle to be understood concerning satellites is that a satellite is a projectile. That is to say, a satellite is an object upon which the only force is gravity. Once launched into orbit, the only force governing the motion of a satellite is the force of gravity. Newton was the first to theorize that a projectile launched with sufficient speed would actually orbit the earth
So what launch speed do;s a satellite need in order to orbit the earth? The answer emerges from a basic fact about the curvature of the earth. For every 8000 meters measured along the horizon of the earth, the earth's surface curves downward by approximately 5 meters. So if you were to look out horizontally along the horizon of the Earth for 8000 meters, you would observe that the Earth curves downwards below this straight-line path a distance of 5 meters. For a projectile to orbit the earth, it must travel horizontally a distance of 8000 meters for every 5 meters of vertical fall. It so happens that the vertical distance which a horizontally launched projectile would fall in its first second is approximately 5 meters (0.5*g*t2). For this reason, a projectile launched horizontally with a speed of about 8000 m/s will be capable of orbiting the earth in a circular path. This assumes that it is launched
above the surface of the earth and encounters negligible atmospheric drag. As the projectile travels tangentially a distance of 8000 meters in I second, it will drop approximately 5 meters towards the earth. Yet, the projectile will remain the same distance above the earth due to the fact that the earth curves at the same rate that the projectile falls. If shot with a speed greater than 8000 mis, it would orbit the earth in an elliptical path.
Elliptical Orbits of Satellites
Occasionally satellites will orbit in paths which can be described as ellipses. In such cases, the central body is located at one of the foci of the ellipse. Similar motion characteristics apply for satellites moving in elliptical paths. The velocity of the satellite is directed tangent to the ellipse. The acceleration of the satellite is directed towards the focus of the ellipse. And in accord with Newton's second law of motion, the net force acting upon the satellite is directed in the same direction as the acceleration - towards the focus of the ellipse. Once more, this net force is supplied by the force of gravitational attraction between the central body and the orbiting satellite. In the case of elliptical paths, there is a component of force in the same direction as (or opposite direction as) the motion of the object. such a component of force can cause the satellite to either speed up or slow down in addition to changing. directions. So unlike uniform circular motion, the elliptical motion of satellites is not characterized by a constant speed.
Meaning and Cause of Weightlessness
Weightlessness is simply a sensation experienced by an individual when there are no external objects touching one's body and exerting a push or pull upon it. Weightless sensations exist when all contact forces are removed. These sensations are common to any situation in which you are momentarily (or perpetually) in a state of free fall. When in free fall, the only force acting upon your body is the force of gravity - a non-contact force. Since the force of gravity cannot be felt without any other opposing forces, you would have no sensation of it. You would feel weightless when in a state of gree fall.
These feelings of weightlessness are common at amusement parks for riders of roller coasters and other rides in which riders are momentarily airborne and lifted out of their seats. Suppose that you were lifted in your chair to the top of a very high tower and then your chair was suddenly dropped. As you and your chair fall towards the ground, you both accelerate at the same rate - g. Since the chair is unstable, falling at the same rate as you, it is unable to push upon you. Normal forces only result from contact with stable, supporting surfaces. The force of gravity is the only force acting upon your body. There are no external objects touching your body and exerting a force. As such, you would experience a weightless sensation. You would weigh as much as you always do (or as little), yet you would not have any sensation ofthis weight.
Weightlessness is only a sensation; it is not a reality corresponding to an individual who has lost weight. As you are free-falling on a roller coaster ride (or other amusement park ride), you have not momentarily lost your weight. Weightlessness has very little to do with weight and mostly to do with the presence or absence of contact forces. If by "weight" we are referring to the force of gravitational attraction to the Earth, a free- falling person has not "lost their weight;" they are still experiencing the Earth's gravitational attraction. Unfortunately, the confusion of a person's actual weight with one's feeling of weight is the source of many misconceptions.
Weightlessness in Orbit
Earth-orbiting astronauts are weightless for the same reasons that riders of a frcffalling amusement park ride or a free-falling elevator are weightless. They are weightless because there is no external contact force pushing or pulling upon their body. In each case, gravity is the only force acting upon their body. Being an action-at-a-distance force, it cannot be felt and therefore would not provide any sensation of their weight. But for certain, the orbiting astronauts weigh something; that is, there is a force of gravity acting upon their body. In fact, if it were not for the force of gravity, the astronauts would not be orbiting in circular motion. It is the force of gravity which supplies the centripetal force requirement to allow the inward acceleration which is characteristic of circular motion. The force of gravity is the only force acting upon their body. The astronauts are in free-fall. Like the falling amusement park rider and the falling elevator rider, the astronauts and their surroundings are falling towards the Earth under the sole influence of gravity. The astronauts and all their surroundings - the space station with its contents - are falling towards the Earth without colliding into it. Their tangential velocity allows them to remain in orbital motion while the force of gravity pulls them inward.
Many students believe that orbiting astronauts are weightless because they do not experiencea force of gravity. So to presume that the absence of gravity is the cause of the weightlessness experienced by orbiting astronauts would be in violation of circular motion principles. If a person believes that the absence of gavity is the cause of their weightlessness, then that person is hard-pressed to come up with a reason for why the astronauts are orbiting in the first place. The fact is that there must be a force of gravity in order for there to be an orbit.
One might respond to this discussion by adhering to a second misconception: the astronauts are weightless because the force of gravity is reduced in space. The reasoning goes as follows: "with less gravity, there would be less weight and thus they would feel less than their normal weight." While this is partly true, it does not explain their sense of weightlessness. The force of gravity acting upon an astronaut on the space station is certainly less than on Earth's surface. But how much less? Is it small enough to account for a significant reduction in weight? Absolutely not! If the space station orbits at an altitude of approximately 400 km above the Earth's surface, then the value of g at that location will be reduced from 9.8 m/s/s (at Earth's surface) to approximately 8.7 m/s/s. This would cause an astronaut weighing 1000 N at Earth's surface to be reduced in weight to approximately 890 N when in orbit. While this is certainly a reduction in weight, it does not account for the absolutely weightless sensations which astronauts experience. Their absolutely weightless sensations are the result of having "the floor pulled out from under them" (so to speak) as they are free-falling towards the Earth.
Still other students believe that weightlessness is due to the absence of air in space. Their misconception lies in the idea that there is no force of gravity when their is no air. According to them, gravity does not exist in a vacuum. But this is not the case. Gravity is a force which acts between the Earth's mass and the mass of other objects which surround it. The force of gravity can act across large distances and its affect can even penetrate across and into the vacuum of outer space. Perhaps students who own this misconception are confusing the force of gravity with air pressure. Air pressure is the result of surrounding air particles pressing upon the surface of an object in equal amounts from all directions. The force of gravity is not affected by air pressure. While air pressure reduces to zero in a location void of air (such as space), the force of gravity does not become 0 N. Indeed the presence of a vacuum results in the absence of air resistance; but this would not account for the weightless sensations. Astronauts merely feel weightless because there is no external contact force pushing or pulling upon their body. They are in a state of free fall.