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 Chapter 5

12/13 Lesson 1 (Method 5)
OMG!! We already know how to measure speed and velocity of circular motion!!

The same concepts and principles of speed and velocity that we have learned about can also be used to explain the motion of objects moving in circles. The calculation for finding average speed is still equal to distance over time, which is also equal to circumference over time. Circumference is equal 2*pi*r.

More Acceleration!!

An object moving in a circle at constant speed is accelerating. It is accelerating because the direction of the velocity vector is changing. Average acceleration = change in velocity/time.

Quick Way to Know If There May Be a Centripetal Force

The centripetal force requirement is that for an object moving in a circle, there must be an inward force acting upon it in order to cause its inward acceleration. Centripetal means the center. There is a net force acting towrads the center which causes the object to seek the center. Centripetal force is not a new kind of force, it is just the direction of the forces we have already discussed.

The F Bomb!

Centrifugal means away from the center, or outward. This word often creates a misconception for students. An object moving in circular motion requires an inward net force, not an outward. If the outward force existed, the object would not be moving in a circle.

Circular Motion Involves Math!

When studying the motion of objects in circle, we will need to consider speed, acceleration, and force. Average speed equals distance/time or circumference/time. Acceleration equals v^2/R. Net force equals mass * acceleration.

Lesson 2(a-c)
** Newton's Second Law - Revisited ** Newton's second law states that the acceleration of an object is directly proportional to the net force acting upon the object and inversely proportional to the mass of the object. To illustrate how circular motion principles can be combined with Newton's second law to analyze a physical situation, consider a car moving in a horizontal circle on a level surface. Applying the concept of a centripetal force requirement, we know that the net force acting upon the object is directed inwards. Since the car is positioned on the left side of the circle, the net force is directed rightward. An analysis of the situation would reveal that there are three forces acting upon the object - the force of gravity (acting downwards), the normal force of the pavement (acting upwards), and the force of friction (acting inwards or rightwards). It is the friction force that supplies the centripetal force requirement for the car to move in a horizontal circle. Without friction, the car would turn its wheels but would not move in a circle (as is the case on an icy surface).


 * Roller Coasters and Amusement Park Physics **

The most obvious section on a roller coaster where centripetal acceleration occurs is within the so-called **clothoid loops**. Unlike a circular loop in which the radius is a constant value, the radius at the bottom of a clothoid loop is much larger than the radius at the top of the clothoid loop. A mere inspection of a clothoid reveals that the amount of curvature at the bottom of the loop is less than the amount of curvature at the top of the loop. To simplify our analysis of the physics of clothoid loops, we will approximate a clothoid loop as being a series of overlapping or adjoining circular sections. The radius of these circular sections is decreasing as one approaches the top of the loop. Furthermore, we will limit our analysis to two points on the clothoid loop - the top of the loop and the bottom of the loop. For this reason, our analysis will focus on the two circles that can be matched to the curvature of these two sections of the clothoid. As a roller coaster rider travels through a clothoid loop, she experiences an acceleration due to both a change in speed and a change in direction. As [|energy principles] would suggest, an increase in height (and in turn an increase in potential energy) results in a decrease in kinetic energy and speed. And conversely, a decrease in height (and in turn a decrease in potential energy) results in an increase in kinetic energy and speed. So the rider experiences the greatest speeds at the bottom of the loop - both upon entering and leaving the loop - and the lowest speeds at the top of the loop. This change in speed as the rider moves through the loop is the second aspect of the acceleration that a rider experiences. For a rider moving through a circular loop with a constant speed, the acceleration can be described as being centripetal or towards the center of the circle. In the case of a rider moving through a noncircular loop at non-constant speed, the acceleration of the rider has two components. There is a component that is directed towards the center of the circle (**a** ** c ** ) and attributes itself to the direction change; and there is a component that is directed tangent (**a** ** t ** ) to the track (either in the opposite or in the same direction as the car's direction of motion) and attributes itself to the car's change in speed. This tangential component would be directed opposite the direction of the car's motion as its speed decreases (on the ascent towards the top) and in the same direction as the car's motion as its speed increases (on the descent from the top). At the very top and the very bottom of the loop, the acceleration is primarily directed towards the center of the circle. At the top, this would be in the downward direction and at the bottom of the loop it would be in the upward direction.

The magnitude of the force of gravity acting upon the passenger (or car) can easily be found using the equation F grav **m•g where g **


 * acceleration of gravity (9.8 m/s 2 ). The magnitude of the normal force depends on two factors - the speed of the car, the radius of the loop and the mass of the rider. **
 * Observe that the normal force is greater at the bottom of the loop than it is at the top of the loop. This becomes a reasonable fact when circular motion principles are considered. At all points along the loop - which we will refer to as circular in shape - there must be some inward component of net force. When at the top of the loop, the gravitational force is directed inwards (down) and so there is less of a need for a normal force in order to meet the net centripetal force requirement. When at the bottom of the loop, the gravitational force is directed outwards (down) and so now there is a need for a large upwards normal force in order to meet the centripetal force requirement. **


 * The magnitude of the normal forces along these various regions is dependent upon how sharply the track is curved along that region (the radius of the circle) and the speed of the car. These two variables affect the acceleration according to the equation a = v 2 / R and in turn affect the net force. As suggested by the equation, a large speed results in a large acceleration and thus increases the demand for a large net force. And a large radius (gradually curved) results in a small acceleration and thus lessens the demand for a large net force. The relationship between speed, radius, acceleration, mass and net force can be used to determine the magnitude of the //seat force// (i.e., normal force) upon a roller coaster rider at various sections of the track. **


 * Athletics **
 * Circular motion is common to almost all sporting events. Like any object moving in a circle, the motion of these objects that we view from the stadium bleachers or watch upon the television monitor are governed by Newton's laws of motion. Their circular motion - however brief or prolonged they may be - is characterized by an [|inward acceleration] and caused by an [|inward net force] . **
 * The most common example of the physics of circular motion in sports involves the turn. Whatever turning motion it happens to be, you can be sure that turning a corner involves circular motion principles. Now for certain not all turns involve a complete circle; nor do all turns have a perfectly circular shape. Some turns are only one-quarter of a turn - such as the fullback rounding the corner of the line in football. And some turns are hardly circular whatsoever. Nonetheless, any turn can be approximated as being a part of a larger circle or a part of several circles of varying size. A sharp turn can be considered part of a small circle. A more gradual turn is part of a larger circle. Some turns can begin sharply and gradually change in sharpness, or vice versa. In all cases, the motion around a turn can be approximated as part of a circle or a collection of circles. **

Because turning a corner involves the motion of an object that is momentarily moving along the path of a circle, both the concepts and the mathematics of circular motion can be applied to such a motion. Conceptually, such an object is moving with an inward acceleration - the inward direction being towards the center of whatever // circle // the object is moving along. There would also be a [|centripetal force requirement] for such a motion. That is, there must be some object supplying an inward force or inward [|component of force]. When a person makes a turn on a horizontal surface, the person often // leans into the turn //. By leaning, the surface pushes upward at an angle // to the vertical //. As such, there is both a horizontal and a vertical component resulting from contact with the surface below. This **contact force** supplies two roles - it balances the downward force of gravity and meets the centripetal force requirement for an object in uniform circular motion. The upward component of the contact force is sufficient to balance the downward force of gravity and the horizontal component of the contact force pushes the person towards the center of the circle. This contact force is depicted in the diagram below for a speed skater making a turn on ice.

1/3 Lesson 3
**Gravity is More Than a Name** <span style="font-family: Arial,Helvetica,sans-serif;">Gravity is the //thing// that causes objects to fall to Earth. Gravity is a force that exists between the Earth and the objects that are near it. As you stand upon the Earth, you experience this force. We have become accustomed to calling it the force of gravity and have even represented it by the symbol F grav. This same force of gravity acts upon our bodies as we jump upwards from the Earth. As we rise upwards after our jump, the force of gravity slows us down. And as we fall back to Earth after reaching the peak of our motion, the force of gravity speeds us up. In this sense, the force gravity causes an acceleration of our bodies during this brief trip away from the earth's surface and back. 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

<span style="font-family: Arial,Helvetica,sans-serif;">** The Apple, the Moon, and the Inverse Square Law ** <span style="font-family: Arial,Helvetica,sans-serif;">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 follows: The paths 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). 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 knew that there must be some sort of force that 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. Newton knew that the force of gravity must somehow be "diluted" by distance. The riddle is solved by a comparison of the distance from the apple to the center of the earth with the distance from the moon to the center of the earth. The force of gravity between the earth and any object is inversely proportional to the square of the distance that separates that object from the earth's center. The force of gravity follows an **inverse square law**. The relationship between the force of gravity (**F**** grav **) between the earth and any other object and the distance that separates their centers (**d**) can be expressed by the following relationship. Since the distance **d** is in the denominator of this relationship, it can be said that the force of gravity is inversely related to the distance. And since the distance is raised to the second power, it can be said that the force of gravity is inversely related to the square of the distance. This mathematical relationship is sometimes referred to as an inverse square law since one quantity depends inversely upon the square of the other quantity. The inverse square relation between the force of gravity and the distance of separation provided sufficient evidence for Newton's explanation of why gravity can be credited as the cause of both the falling apple's acceleration and the orbiting moon's acceleration. The inverse square law proposed by Newton suggests that the force of gravity acting between any two objects is inversely proportional to the square of the separation distance between the object's centers.

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 14pt;">**Newton's Law of Universal Gravitation** <span style="font-family: Arial,Helvetica,sans-serif;">The force of gravitational attraction between the Earth and other objects is inversely proportional to the distance separating the earth's center from the object's center. But distance is not the only variable affecting the magnitude of a gravitational force. Consider Newton's famous equation Fnet = m • a. 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 that separates the centers of the earth and the object. Newton's law of universal gravitation is about the universality of gravity. 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 that separates their 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. 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. The force of gravity is directly proportional to the product of the two masses and inversely proportional to the square of the distance of separation. Another means of representing the proportionalities is to express the relationships in the form of an equation using a constant of proportionality. 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. When the units on G are substituted into the equation above and multiplied by m 1 • m 2 units and divided by d 2 units, the result will be Newtons - the unit of force. Knowing the value of G allows us to calculate the force of gravitational attraction between any two objects of known mass and known separation distance. Gravitational interactions do not simply exist between the earth and other objects; and not simply between the sun and other planets. Gravitational interactions exist between all objects with an intensity that is directly proportional to the product of their masses. Gravitational forces are only recognizable as the masses of objects become large.

<span style="color: #000000; font-family: Arial,Helvetica,sans-serif; font-size: 14pt;">**Cavendish and the Value of G**

<span style="font-family: Arial,Helvetica,sans-serif;">Cavendish's apparatus for experimentally determining the value of G involved a light, rigid rod about 2-feet long. Two small lead spheres were attached to the ends of the rod and the rod was suspended by a thin wire. When the rod becomes twisted, the torsion of the wire begins to exert a torsional force that is proportional to the angle of rotation of the rod. The more twist of the wire, the more the system pushes //backwards// to restore itself towards the original position. Cavendish had calibrated his instrument to determine the relationship between the angle of rotation and the amount of torsional force. Cavendish then brought two large lead spheres near the smaller spheres attached to the rod. Since all masses attract, the large spheres exerted a gravitational force upon the smaller spheres and twisted the rod a measurable amount. Once the torsional force balanced the gravitational force, the rod and spheres came to rest and Cavendish was able to determine the gravitational force of attraction between the masses. By measuring m 1, m 2 , d and F grav , the value of G could be determined. Cavendish's measurements resulted in an experimentally determined value of 6.75 x 10 -11 N m 2 /kg 2. Today, the currently accepted value is 6.67259 x 10 -11 N m 2 /kg 2. The value of G is an extremely small numerical value. Its smallness accounts for the fact that the force of gravitational attraction is only appreciable for objects with large mass. While two students will indeed exert gravitational forces upon each other, these forces are too small to be noticeable. Yet if one of the students is replaced with a planet, then the gravitational force between the other student and the planet becomes noticeable.

<span style="color: #000000; font-family: Arial,Helvetica,sans-serif; font-size: 14pt;">**The Value of g** <span style="font-family: Arial,Helvetica,sans-serif;">In the first equation above, **g** is referred to as the acceleration of gravity. Its value is ** 9.8 m/s **** 2 ** on Earth. That is to say, the acceleration of gravity on the surface of the earth at sea level is 9.8 m/s 2. When discussing the acceleration of gravity, it was mentioned that the value of g is dependent upon location. There are slight variations in the value of g about earth's surface. These variations result from the varying density of the geologic structures below each specific surface location. They also result from the fact that the earth is not truly spherical; the earth's surface is further from its center at the equator than it is at the poles. This would result in larger g values at the poles. As one proceeds further from earth's surface - say into a location of orbit about the earth - the value of g changes still.

1/5 The Clockwork Universe
An Incredible Progress in Thought!

First View: The Earth-centered view of the ancient Greeks and of the Catholic church in the sixteenth century. Second View: The Copernican system, in which the planets move in collections of circles around the sun. Third View: The Keplerian system in which a planet follows an elliptical orbit, with the Sun at one focus of the ellipse.

Galileo dies, Newton is born!

Galileo measured that all bodies accelerate at the same rate regardless of their size or mass. One of the most dangerous implications of Descartes' mechanical universe is that it raises sensitive questions about God's relationship to nature. Newton - physics emerges Newton's idea = Clockwork Universe model; a concept that states that the total moment the Universe is conserved, interactions redistribute the momentum, but the total never changes.

<span style="font-family: Arial,Helvetica,sans-serif;">1/6 Lesson 4(a-c)
<span style="font-family: Arial,Helvetica,sans-serif;">** Kepler's Three Laws **

<span style="font-family: Arial,Helvetica,sans-serif;">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 that described the motion of planets in a sun-centered solar system. Kepler's three laws of planetary motion can be described as follows: <span style="font-family: Arial,Helvetica,sans-serif;">Kepler's first law - sometimes referred to as the law of ellipses - explains that planets are orbiting the sun in a path described as an ellipse. An ellipse is a special curve in which the sum of the distances from every point on the curve to two other points is a constant. The two other points are known as the **foci** of the ellipse. The closer together that these points are, the more closely that the ellipse resembles the shape of a circle. Kepler's first law is rather simple - all planets orbit the sun in a path that resembles an ellipse, with the sun being located at one of the foci of that ellipse. <span style="font-family: Arial,Helvetica,sans-serif;">Kepler's second law - sometimes referred to as the law of equal areas - describes the speed at which any given planet will move while orbiting the sun. The speed at which any planet moves through space is constantly changing. A planet moves fastest when it is closest to the sun and slowest when it is furthest from the sun. <span style="font-family: Arial,Helvetica,sans-serif;">Kepler's third law provides an accurate description of the period and distance for a planet's orbits about the sun. Additionally, the same law that describes the T 2 /R 3 ratio for the planets' orbits about the sun also accurately describes the T 2 /R 3 ratio for any satellite (whether a moon or a man-made satellite) about any planet. There is something much deeper to be found in this T 2 /R 3 ratio - something that must relate to basic fundamental principles of motion.
 * <span style="font-family: Arial,Helvetica,sans-serif;">The path of the planets about the sun is elliptical in shape, with the center of the sun being located at one focus. (The Law of Ellipses)
 * <span style="font-family: Arial,Helvetica,sans-serif;">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)
 * <span style="font-family: Arial,Helvetica,sans-serif;">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)


 * Circular Motion Principles for Satellites **

A satellite is any object that is orbiting the earth, sun or other massive body. Satellites can be categorized as **natural satellites** or **man-made satellites**. Accompanying the orbit of natural satellites are a host of satellites launched from earth for purposes of communication, scientific research, weather forecasting, intelligence, etc. 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. For every 8000 meters measured along the horizon of the earth, the earth's surface curves downward by approximately 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 that a horizontally launched projectile would fall in its first second is approximately 5 meters (0.5*g*t 2 ). The motion of an orbiting satellite can be described by the same motion characteristics as any object in circular motion. The [|velocity] of the satellite would be directed tangent to the circle at every point along its path. The [|acceleration] of the satellite would be directed towards the center of the circle - towards the central body that it is orbiting. And this acceleration is caused by a [|net force] that is directed inwards in the same direction as the acceleration. Occasionally satellites will orbit in paths that 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.
 * Velocity, Acceleration and Force Vectors **
 * Elliptical Orbits of Satellites **


 * Mathematics of Satellite Motion **

The motion of objects is governed by Newton's laws. The same simple laws that govern the motion of objects on earth also extend to the //heavens// to govern the motion of planets, moons, and other satellites. Consider a satellite with mass M sat orbiting a central body with a mass of mass M Central. The central body could be a planet, the sun or some other large mass capable of causing sufficient acceleration on a less massive nearby object. If the satellite moves in circular motion, then the [|net centripetal force] acting upon this orbiting satellite is given by the relationship ** F **** net **** = ( M **** sat **** • v **** 2 **** ) / R **

**G** is 6.673 x 10 -11 N•m 2 /kg 2, **M** ** central ** is the mass of the central body about which the satellite orbits, and **R** is the radius of orbit for the satellite. The acceleration value of a satellite is equal to the acceleration of gravity of the satellite at whatever location that it is orbiting. The equation is: **g = (G • Mcentral)/R2** The final equation that is useful in describing the motion of satellites is Newton's form of Kepler's third law. The equation is: There is an important concept evident in all three of these equations - the period, speed and the acceleration of an orbiting satellite are not dependent upon the mass of the satellite.

1/9 Lesson 4 (d-e)

 * Weightlessness in Orbit **

Before understanding weightlessness, we will have to [|review two categories of forces] - **contact forces** and **action-at-a-distance forces**. [|Contact forces] can only result from the actual touching of the two interacting objects. The force of gravity acting upon your body is not a contact force; it is often categorized as an [|action-at-a-distance force]. The force of gravity is the result of your center of mass and the Earth's center of mass exerting a mutual pull on each other; this force would even exist if you were not in contact with the Earth. The force of gravity does not require that the two interacting objects (your body and the Earth) make physical contact; it can act over a distance through space. Since the force of gravity is not a contact force, it cannot be felt through contact. Without the contact force (the normal force), there is no means of feeling the non-contact force (the force of gravity). Technically speaking, a scale does not measure one's weight. While we use a scale to measure one's weight, the scale reading is actually a measure of the upward force applied by the scale to balance the downward force of gravity acting upon an object. When an object is in a state of equilibrium (either at rest or in motion at constant speed), these two forces are balanced. The upward force of the scale upon the person equals the downward pull of gravity (also known as weight). And in this instance, the scale reading equals the weight of the person. However, if you stand on the scale and bounce up and down, the scale reading undergoes a rapid change. As you undergo this bouncing motion, your body is accelerating. During the acceleration periods, the upward force of the scale is changing. And as such, the scale reading is changing. The scale is only measuring the external contact force that is being applied to your body. It is the force of gravity that supplies the [|centripetal force requirement] to allow the [|inward acceleration] that 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.
 * Contact versus Non-Contact Forces **
 * 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 free fall. Weightlessness is only a sensation; it is not a reality corresponding to an individual who has lost weight. 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.
 * Scale Readings and Weight **
 * Weightlessness in Orbit **

The orbits of satellites about a central massive body can be described as either circular or elliptical. At all instances during its trajectory, the force of gravity acts in a direction perpendicular to the direction that the satellite is moving. Since [|perpendicular components of motion are independent] of each other, the inward force cannot affect the magnitude of the tangential velocity. For this reason, there is no acceleration in the tangential direction and the satellite remains in circular motion at a constant speed. A satellite orbiting the earth in elliptical motion will experience a component of force in the same or the opposite direction as its motion. This force is capable of doing [|work] upon the satellite. Thus, the force is capable of slowing down and speeding up the satellite. When the satellite moves away from the earth, there is a component of force in the opposite direction as its motion. During this portion of the satellite's trajectory, the force does negative work upon the satellite and slows it down. When the satellite moves towards the earth, there is a component of force in the same direction as its motion. During this portion of the satellite's trajectory, the force does positive work upon the satellite and speeds it up. Subsequently, the speed of a satellite in elliptical motion is constantly changing - increasing as it moves closer to the earth and decreasing as it moves further from the earth. The governing principle that directed our analysis of motion was the **work-energy theorem**. Simply put, the theorem states that the initial amount of total mechanical energy (TME i ) of a system plus the work done by external forces (W ext ) on that system is equal to the final amount of total mechanical energy (TME f ) of the system. The mechanical energy can be either in the form of potential energy (energy of position - usually vertical height) <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">or kinetic energy (energy of motion). The work-energy theorem is expressed in equation form as <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">KEi + PEi + Wext**KEf + PEf, The Wext term in this equation is representative of the amount of work done by [|external forces]. For satellites, the only force is gravity. Since gravity is considered an [|internal (conservative) force], the Wext term is zero. The equation can then be simplified to the following form. KEi + PEi KEf + PEf. In such a situation as this, we often say that the total mechanical energy of the system is conserved. That is, the sum of kinetic and potential energies is unchanging. While energy can be transformed from kinetic energy into potential energy, the total amount remains the same - mechanical energy is //conserved//. As a satellite orbits earth, its total mechanical energy remains the same. Whether in circular or elliptical motion, there are no external forces capable of altering its total energy.**
 * Energy Relationships for Satellites **

Energy Analysis of Circular Orbits
 * One means of representing the amount and the type of energy possessed by an object is a [|work-energy bar chart] . A work-energy bar chart represents the energy of an object by means of a vertical bar. The length of the bar is representative of the amount of energy present - a longer bar representing a greater amount of energy. In a work-energy bar chart, a bar is constructed for each form of energy. **

Energy Analysis of Elliptical Orbits Like the case of circular motion, the total amount of mechanical energy of a satellite in elliptical motion also remains constant. Since the only force doing work upon the satellite is an [|internal (conservative) force], the W ext term is zero and mechanical energy is conserved. Unlike the case of circular motion, the energy of a satellite in elliptical motion will change forms. An energy analysis of satellite motion yields the same conclusions as any analysis guided by Newton's laws of motion. A satellite orbiting in circular motion maintains a constant radius of orbit and therefore a constant speed and a constant height above the earth. A satellite orbiting in elliptical motion will speed up as its height (or distance from the earth) is decreasing and slow down as its height (or distance from the earth) is increasing. The same principles of motion that apply to objects on earth - Newton's laws and the work-energy theorem - also govern the motion of satellites in the heavens.