Ch5_SmithJ

=toc Homework Ch. 5=

Lesson 1 (a-e)

 * 1) Speed and Velocity
 * 2) The idea shown about linear speed and velocity can be applied to circular motion. Uniform circular motion is when an object is moving at constant speed, where average speed is calculated by distance over time but in a circle it is circumference/time. Although constant speed, not constant velocity. Velocity is a vector and is tangent to the circle; since tangent is facing different direction as object moves, the velocity is constantly changing.
 * 3) Acceleration
 * 4) Circular acceleration depends on the change of velocity and goes in the same direction as the velocity. It is calculated by the change of velocity over time. Constant circular speed accelerates toward the center of the circle. Accelerometer is used to measure acceleration and is made up of an object suspended in a fluid; it determines inertia of the object.
 * 5) The centripetal force requirement
 * 6) Centripetal force requirement is that an object moving in a circle must have an inward force acting upon it to cause the inward acceleration. Therefore, centripetal force is the force pushing or pulling object towards center of circle. Newton's first law states must be an unbalanced force to move in circular motion. Centripetal force changes direction of circle but not necessarily the speed. Force is perpendicular to tangential velocity, so doesn't change magnitude only direction.
 * 7) The forbidden f-word
 * 8) Centrifugal is a common misconception and is thought to be force moving away from the circle. The truth is that centripetal refers to the real motion of a circle.
 * 9) Mathematics of circular motion
 * 10) Average speed is calculated by circumference over time. Acceleration is calculated by 4pi2r/time. Net force is equal to mass times acceleration.

Lesson 2

 * 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. The law is often expressed in the form of the following two equations.

Applying the concept of a centripetal force requirement, we know that the net force acting upon the object is directed inwards.


 * 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 **. A clothoid is a section of a spiral in which the radius is constantly changing. The radius at the bottom of a clothoid loop is much larger than the radius at the top of the clothoid loop. 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.

An increase in height (and in turn an increase in potential energy) results in a decrease in kinetic energy and speed, and vice versa.

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 (** ac **) and attributes itself to the direction change; and there is a component that is directed tangent (** at **) 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.

As depicted in the free body diagram, the magnitude of Fnorm (or whatever centripetal force) is always greater at the bottom of the loop than it is at the top to overcome Fgrav. The normal force must always be of the appropriate size to combine with the Fgrav in such a way to produce the required inward or centripetal net force.

When at the top of the loop, a rider will __feel__ partially weightless if the normal forces become less than the person's weight. And at the bottom of the loop, a rider will feel very "weighty" due to the increased normal forces.

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 = v2 / 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.


 * Athletics **

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.

*I did this on a word document originally and the pictures were still there so the words include reference to the pictures

Lesson 3

 * Gravity is More Than a Name **

** Fgrav **is a force that exists between the Earth and the objects that are near it. ** The acceleration of gravity ** (** g **) is the acceleration experienced by an object when the only force acting upon it is the force of gravity.


 * The Apple, the Moon, and the Inverse Square Law **

German mathematician and astronomer Johannes 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. 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)

Kepler could only suggest that there was some sort of interaction between the sun and the planets that 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 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'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) that led him to his notion of ** universal gravitation **.

A survey of Newton's writings reveals an illustration similar to the one shown at the right. The illustration was accompanied by an extensive discussion of the motion of the moon as a projectile. Cannonball fired at speed such that the trajectory of the falling cannonball matched the curvature of the earth, then would fall around the earth instead of into it as an orbiting satellite (** path C **). And then at even greater launch speeds, a cannonball would once more orbit the earth, but in an elliptical path (** path D **). The motion of the cannonball orbiting to the earth under the influence of gravity is analogous to the motion of the moon orbiting the Earth and to falling apple. The same force that causes objects on Earth to fall to the earth also causes objects in the heavens to move along their circular and elliptical paths. The same law of mechanics can also be used.

Of course, Newton's dilemma was to provide reasonable evidence for the extension of the force of gravity from earth to the heavens. Newton knew that the force of gravity must somehow be "diluted" by distance. 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 **.

An increase in the separation distance causes a decrease in the force of gravity and a decrease in the separation distance causes an increase in the force of gravity. Furthermore, the factor by which the force of gravity is changed is the square of the factor by which the separation distance is changed. So if the separation distance is doubled (increased by a factor of 2), then the force of gravity is decreased by a factor of four (2 raised to the second power).
 * Using Equations as a Guide to Thinking **


 * Newton's Law of Universal Gravitation **

Distance is not the only variable affecting the magnitude of a gravitational force. Consider Newton's famous equation

** Fnet = m • a **

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 that separates the centers of the earth and the object.

But Newton's law of universal gravitation extends gravity beyond earth. **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. Newton's conclusion about the magnitude of gravitational forces is summarized symbolically as The proportionalities expressed by Newton's universal law of gravitation are represented graphically by the following illustration.

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.

The constant of proportionality (G) in the above equation is known as the ** universal gravitation constant **.

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.

The first conceptual comment is the inverse relationship between separation distance and the force of gravity (or in this case, the weight of the student). The student weighs less at the higher altitude. However, a mere change of 40 000 feet further from the center of the Earth is virtually negligible. The distance of separation becomes much more influential when a significant variation is made. The second conceptual comment to be made is that the use of Newton's universal gravitation equation to calculate the force of gravity (or weight) yields the same result as when calculating it using the equation presented: ** Fgrav = m•g = (70 kg)•(9.8 m/s2) = 686 N **

The constant of proportionality in this equation is ** G ** - the universal gravitation constant. The value of G was not experimentally determined until nearly a century later (1798) by Lord Henry Cavendish using a torsion balance.
 * Cavendish and the Value of G **

Cavendish's apparatus for experimentally determining the value of G involved a light, rigid rod about 2-feet long.

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 m1, m2, d and Fgrav, the value of G could be determined. Cavendish's measurements resulted in an experimentally determined value of 6.75 x 10-11 N m2/kg2. Today, the currently accepted value is 6.67259 x 10-11 N m2/kg2.

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.

In the first equation above, ** g ** is referred to as the acceleration of gravity. Its value is ** 9.8 m/s2 ** on Earth. 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 that 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
 * The Value of g **

To understand why the value of g is so location dependent, we will use the two equations above to derive an equation for the value of g. First, both expressions for the force of gravity are set equal to each other.

The above equation demonstrates that the acceleration of gravity is dependent upon the mass of the earth (approx. 5.98x1024 kg) and the distance (** d **) that an object is from the center of the earth.

The value of g varies inversely with the distance from the center of the earth. In fact, the variation in g with distance follows an [|inverse square law] where g is inversely proportional to the distance from earth's center.

The same equation used to determine the value of g on Earth' surface can also be used to determine the acceleration of gravity on the surface of other planets. The equation takes the following form:

The value of g is independent of the mass of the object and only dependent upon //location// - the planet the object is on and the distance from the center of that planet.

The Clockwork Universe

 * 1) Who were important figures in the discovery of the workings of the universe and what did they do?
 * Copernicus
 * Heliocentric universe where Earth revolved around the sun
 * Galileo
 * Supported Copernicus
 * Objects accelerate at same rate, regardless of mass and size
 * Basics of laws of motion
 * Newton
 * Expanded on Galileo's laws of motion
 * Inertia is resistance to change in velocity
 * Acceleration is caused by a force
 * Kepler
 * Modified Copernicus by saying planets orbit sun in elliptical motion
 * Wrote Astronomia Nova on his observations
 * 1) Who was Renee Descartes and what did he do?
 * Important mathematician
 * Discovered mathematical equations that linked algebra and geometry in coordinate system
 * 1) Expand upon Newton's discoveries.
 * Newton was able to connect mathematics, astronomy, and physics
 * He was able to combine his laws of motion with gravity to show elliptical pattern of planets around sun mathematically
 * His three key points:
 * There were deviations from standard motion
 * The cause of these deviations
 * Law of universal gravitation linked a force and this deviation
 * Used physics to show that gravitational attraction would cause planets to veer off from their elliptical track
 * Pierre Simon Lapace expanded on Newton's discoveries with the discovery of mechanics
 * Used determinism - once clockwork was set in motion

Lesson 4

 * Kepler's Three Laws**

Kepler's three laws of planetary motion can be described as follows:


 * 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)
 * 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)

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 (represented here by the tack locations) are known as the ** foci ** of the ellipse. 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.

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. A planet moves fastest when it is closest to the sun and slowest when it is furthest from the sun. Yet, if an imaginary line were drawn from the center of the planet to the center of the sun, that line would sweep out the same area in equal periods of time

Kepler's third law - sometimes referred to as the ** law of harmonies ** - compares the orbital period and radius of orbit of a planet to those of other planets. The third law makes a comparison between the motion characteristics of different planets. The ratio of the squares of the periods to the cubes of their average distances from the sun is the same for every one of the planets.

Amazingly, every planet has the same T2/R3 ratio.

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 T2/R3 ratio for the planets' orbits about the sun also accurately describes the T2/R3 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 T2/R3 ratio - something that must relate to basic fundamental principles of motion.


 * 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 **.

The fundamental principle to be understood concerning satellites is that a satellite is a [|projectile]. 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. Consider a projectile launched horizontally from the top of the legendary // Newton's Mountain // - at a location high above the influence of air drag. As the projectile moves horizontally in a direction tangent to the earth, the force of gravity would pull it downward.


 * Velocity, Acceleration and Force Vectors **

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.

This centripetal force is supplied by [|gravity - the force that universally] acts at a distance between any two objects that have mass. Were it not for this force, the satellite in motion would continue in motion at the same speed and in the same direction. It would follow its inertial, straight-line path. Like any projectile, gravity alone influences the satellite's trajectory such that it always falls below its [|straight-line, inertial path].


 * Elliptical Orbits of Satellites **

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. 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.

In summary, satellites are projectiles that orbit around a central massive body instead of falling into it. Being projectiles, they are acted upon by the force of gravity - a universal force that acts over even large distances between any two masses. The motion of satellites, like any projectile, is governed by Newton's laws of motion. For this reason, the mathematics of these satellites emerges from an application of Newton's universal law of gravitation to the mathematics of circular motion.


 * Mathematics of Satellite Motion**

** Fnet = ( Msat • v2 ) / R **

This net centripetal force is the result of the [|gravitational force] that attracts the satellite towards the central body and can be represented as

** Fgrav = ( G • Msat • MCentral ) / R2 **

Since Fgrav = Fnet, the above expressions for centripetal force and gravitational force can be set equal to each other. Thus,

** (Msat • v2) / R = (G • Msat • MCentral ) / R2 **

Observe that the mass of the satellite is present on both sides of the equation; thus it can be canceled by dividing through by ** Msat **. Then both sides of the equation can be multiplied by ** R **, leaving the following equation.

** v2 = (G • MCentral ) / R **

Taking the square root of each side, leaves the following equation for the velocity of a satellite moving about a central body in circular motion

where ** G ** is 6.673 x 10-11 N•m2/kg2, ** Mcentral ** is the mass of the central body about which the satellite orbits, and ** R ** is the radius of orbit for the satellite.

Similar reasoning can be used to determine an equation for the acceleration of our satellite that is expressed in terms of masses and radius of orbit. The acceleration value of a satellite is equal to the acceleration of gravity of the satellite at whatever location that it is orbiting. In [|Lesson 3], the equation for the acceleration of gravity was given as

** g = (G • Mcentral)/R2 **

Thus, the acceleration of a satellite in circular motion about some central body is given by the following equation

where ** G ** is 6.673 x 10-11 N•m2/kg2, ** Mcentral ** is the mass of the central body about which the satellite orbits, and ** R ** is the average radius of orbit for the satellite.

The final equation that is useful in describing the motion of satellites is Newton's form of Kepler's third law. Since the logic behind the development of the equation has been presented [|elsewhere], only the equation will be presented here. The period of a satellite (** T **) and the mean distance from the central body (** R **) are related by the following equation:

where ** T ** is the period of the satellite, ** R ** is the average radius of orbit for the satellite (distance from center of central planet), and ** G ** is 6.673 x 10-11 N•m2/kg2.

the period, speed and the acceleration of an orbiting satellite are not dependent upon the mass of the satellite.

None of these three equations has the variable ** Msatellite ** in them. The period, speed and acceleration of a satellite are only dependent upon the radius of orbit and the mass of the central body that the satellite is orbiting. Just as in the case of the motion of projectiles on earth, the mass of the projectile has no affect upon the acceleration towards the earth and the speed at any instant.

TYPED UP EQUATIONS FOR WIKI:

We start with this equation Fg= (G*m1*m2)/d2

Velocity= sqrt((G*mcentral)/R)

Acceleration= (G*Mcentral)/R2

T2/R3 = 4π2/G*Mcentral


 * Weightlessness in Orbit**

The cause of weightlessness is quite simple to understand. However, the stubbornness of one's preconceptions on the topic often stand in the way of one's ability to understand. Consider the following multiple choice question about weightlessness as a test of your preconceived notions on the topic:


 * Contact versus Non-Contact Forces **

Before understanding weightlessness, we will have to [|review two categories of forces] - ** contact forces ** and ** action-at-a-distance forces **. As you sit in a chair, you experience two forces - the force of the Earth's gravitational field pulling you downward toward the Earth and the force of the chair pushing you upward. The upward chair force is sometimes referred to as a normal force and results from the contact between the chair top and your bottom end. This normal force is categorized as a contact force. [|Contact forces] can only result from the actual touching of the two interacting objects - in this case, the chair and you. 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. You can never feel the force of gravity pulling upon your body in the same way that you would feel a contact force.


 * 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. 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.


 * Scale Readings and Weight **

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). 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. Is your weight changing? Absolutely not! You weigh as much (or as little) as you always do. The scale reading is changing, but remember: the SCALE DOES NOT MEASURE YOUR WEIGHT. The scale is only measuring the external contact force that is being applied to your body.


 * Weightlessness in Orbit **

Earth-orbiting astronauts are weightless for the same reasons that riders of a free-falling 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. 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.

Many students believe that orbiting astronauts are weightless because they do not experience a 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 gravity 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.

Still other physics 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 there is no air. According to them, gravity does not exist in a vacuum. But this is not the case. Gravity is a force that acts between the Earth's mass and the mass of other objects that 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.


 * Energy Relationships for Satellites**

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.

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 (TMEi) of a system plus the work done by external forces (Wext) on that system is equal to the final amount of total mechanical energy (TMEf) of the system. The mechanical energy can be either in the form of potential energy (energy of position - usually vertical height) or kinetic energy (energy of motion). The work-energy theorem is expressed in equation form as

** 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 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.


 * 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 Wext 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.