Ch5_BehrensB

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 * Chapter 5 toc

= = = Circular Motion and Satellite Motion - Lesson 1; a, b, c, d, & e = December 13, 2011 Method 5

__Speed And Velocity Are //Not// the Same For Circular Motion!__ Although in previous chapters it was possible for objects in motion to have both constant speed and velocity, circular motion proves to be different. While an object traveling in uniform circular motion maintains a constant speed, the velocity vector continuously changes direction. This vector is tangential, meaning that its direction coincides with that of the tangent line. The magnitude of the velocity vector can be found by calculating the instantaneous speed, and the average speed can be calculated by using the formula: average speed = circumference / time.

__Acceleration is Everywhere!__ The misconception of no acceleration in circular motion is common, though false. When an object alters its direction as in uniform circular motion, acceleration is present and can be calculated by dividing the change in velocity by the total time. Devices called accelerometers can measure the acceleration of an object.

__Centripetal Forces Alter Motion!__ In the presence of acceleration, there is also a force acting upon it to cause the change in velocity. This is called the centripetal force requirement, in which there is a "center seeking" force compromising the motion of an object to a circular path; it can be justified by Newton's first law, the law of inertia. The idea of work, a force that acts on an object in motion to cause displacement, can help to answer the question of why an unbalanced force has the ability to change direction but not magnitude of a vector.

__Centrifugal Motion Does //Not// Cause Circular Motion!__ Centrifugal motion is commonly misperceived as an outwardly directed force while an object travels in a circular path, deducing the belief that this path is controlled by a force other than that of the centripetal. However, this is false because, for example, as a car passenger feels the sensation of an outward force, this is merely the tendency to follow the inertial path of a straight line upon a curve in the road.

__(title)__ Three prime factors to focus on during analysis of circular motion are speed, acceleration, and force. The equations that can be used to calculate these factors are: average speed = circumference / time, acceleration = v 2 / R , and ∑F = m•a.

= Circular Motion and Satellite Motion - Lesson 3; a, b, c, d, & e = January 3, 2012 Method 1

**Gravity is More Than a Name** 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. Approximately 9.8 m/s/s is the same acceleration value for all objects, regardless of their mass.

**The Apple, the Moon, and the Inverse Square Law** In the early 1600's, German mathematician and astronomer Johannes Kepler's three laws emerged from the analysis of data carefully collected by his Danish predecessor and teacher, Tycho Brahe. Kepler's three laws of planetary motion can be briefly described as follows: To Kepler, the planets were somehow "magnetically" driven by the sun to orbit in their elliptical trajectories.
 * 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)

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.

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**. The force of gravity is inversely related to 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.

**Newton's Law of Universal Gravitation** Newton's law of universal gravitation is about the **universality** of gravity. **ALL** objects attract each other with a force of gravitational attraction. 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. The proportionalities expressed by Newton's universal law of gravitation are represented graphically. Another means of representing the proportionalities is to express the relationships in the form of an equation using a constant of proportionality. **Universal gravitation constant (G) -** 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** Knowing that all objects exert gravitational influences on each other, the small perturbations in a planet's elliptical motion can be easily explained. As the planet Jupiter approaches the planet Saturn in its orbit, it tends to deviate from its otherwise smooth path; this deviation, or **perturbation**, is easily explained when considering the affect of the gravitational pull between Saturn and Jupiter.

**Cavendish and the Value of G** Newton's law of universal gravitation, in equation form, is often expressed as follows: 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'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. A diagram of the apparatus is shown below. 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 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** There are slight variations in the value of g about earth's surface. 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.

Then,

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 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 Clockwork Universe - Lesson 2; 1, 2, 3, & 4  January 5, 2012   Method 1 __Mechanics and Determinism Part 1__ In 1543, Nicolaus Copernicus launched a scientific revolution by rejecting the Earth-centred view of the Universe in favour of a **heliocentric** view in which the Earth moved round the Sun. By removing the Earth, and with it humankind, from the centre of creation, Copernicus had set the scene for a number of confrontations between the Catholic church and some of its more independently minded followers. The most famous of these must surely have been Galileo, who was summoned to appear before the Inquisition in 1633, on a charge of heresy for supporting Copernicus' ideas. As a result Galileo was 'shown the instruments of torture', and renounced his declared opinion that the Earth moves around the Sun. This he did, though at the end of his renunciation he muttered 'Eppur si muove' ('And yet it moves').

In the early seventeenth century, German-born astronomer Johannes Kepler (1571-1630) devised a modified form of Copernicanism that was in good agreement with the best observational data available at the time. According to Kepler, the planets //did// move around the Sun, but their orbital paths were ellipses rather than collections of circles. Kepler had no real reason to //expect// that the planets would move in ellipses, though he did speculate that they might be impelled by some kind of magnetic influence emanating from the Sun. __Mechanics and Determinism Part 2__ Kepler's ideas were underpinned by new discoveries in mathematics. Chief among these was the realization, by René Descartes, that problems in geometry can be recast as problems in algebra. Like most revolutionary ideas, the concept is disarmingly simple. Imagine a giant grid extending over the whole of space. This is the beginning of a branch of mathematics, called //coordinate geometry//, which represents geometrical shapes by equations, and which establishes geometrical truths by combining and rearranging those equations. Sometimes, what is difficult to show using traditional geometry is easy to establish using algebra, so this 'mapping' of geometry into algebra gave scientists new ways of tackling geometrical problems, allowing them to go further than the greatest mathematicians of ancient Greece. __Mechanics and Determinism Part 3__ At the core of Newton's world-view is the belief that all the motion we see around us can be explained in terms of a single set of laws. We cannot give the details of these laws now, but it is appropriate to mention three key points: **3.** Finally Newton produced a quantitative link between force and deviation from steady motion and, at least in the case of gravity, quantified the force by proposing his famous law of universal gravitation. __Mechanics and Determinism Part 4__In the hands of Newton's successors, notably the French scientist Pierre Simon Laplace (1749-1827), Newtonís discoveries became the basis for a detailed and comprehensive study of **mechanics** (the study of force and motion). The detailed character of the Newtonian laws was such that once this majestic clockwork had been set in motion, its future development was, in principle, entirely predictable. This property of Newtonian mechanics is called **determinism**. It had an enormously important implication. Given an accurate description of the character, position and velocity of every particle in the Universe at some particular moment (i.e. the //initial condition// of the Universe), and an understanding of the forces that operated between those particles, the subsequent development of the Universe could be predicted with as much accuracy as desired. Needless to say, obtaining a completely detailed description of the entire Universe at any one time was not a realistic undertaking, nor was solving all the equations required to predict its future course. But that wasn't the point. It was enough that the future was ordained. If you accepted the proposition that humans were entirely physical systems, composed of particles of matter obeying physical laws of motion, then in principle, every future human action would be already determined by the past. For some this was the ultimate indication of God: where there was a design there must be a Designer, where there was a clock there must have been a Clockmaker. For others it was just the opposite, a denial of the doctrine of **free will** which asserts that human beings are free to determine their own actions. Even for those without religious convictions, the notion that our every thought and action was pre-determined in principle, even if unpredictable in practice, made the Newtonian Universe seem strangely discordant with our everyday experience of the vagaries of human life.
 * 1.** Newton concentrated not so much on motion, as on //deviation from steady motion// - deviation that occurs, for example, when an object speeds up, or slows down, or veers off in a new direction.
 * 2.** Wherever deviation from steady motion occurred, Newton looked for a cause. Slowing down, for example, might be caused by braking. He described such a cause as a force. We are all familiar with the idea of applying a force, whenever we use our muscles to push or pull anything.

= Circular Motion and Satellite Motion - Lesson 4; a, b, & c = January 6, 2012 Method 1

**Kepler's Three Laws** In the 1600s, Kepler proposed three laws of planetary motion that summarize the 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:
 * 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. 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. 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. Unlike Kepler's first and second laws, the third law makes a comparison between the motion characteristics of different planets. The comparison being made is that 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. 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.

**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 moon, the planets and comets are examples of natural 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. So what launch speed does a satellite need in order to orbit the earth? For a projectile to orbit the earth, it must travel horizontally a distance of 8000 meters for every5 meters of vertical fall. If shot with a speed greater than 8000 m/s, it would orbit the earth in an elliptical path.

** Velocity, Acceleration and Force Vectors ** 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.

**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. 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.

**Mathematics of Satellite Motion** The motion of objects is governed by Newton's laws. Consider a satellite with mass Msat orbiting a central body with a mass of mass MCentral. 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 **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. 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. 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. When air resistance is negligible and only gravity is present, the mass of the moving object becomes a non-factor. Such is the case of orbiting satellites.

= Circular Motion and Satellite Motion - Lesson 4; d & e = January 9, 2012  Method 1 ** Weightlessness in Orbit ** ** Contact versus Non-Contact 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 does not require that the two interacting objects (your body and the Earth) make physical contact; it can act over a distance through space. Without the contact force (the normal force), there is no means of feeling the non-contact force (the force of gravity).

**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. These sensations are common to any situation in which you are momentarily (or perpetually) in a state of free fall. Weightlessness is only a sensation; it is not a reality corresponding to an individual who has lost weight. Weightlessness has very little to do with weight and mostly to do with the presence or absence of contact forces.
 * Meaning and Cause of Weightlessness **

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. Now consider Otis L. Evaderz who is conducting one of his famous elevator experiments. He stands on a bathroom scale and rides an elevator up and down. As he is accelerating upward and downward, the scale reading is different than when he is at rest and traveling at constant speed. When he is accelerating, the upward and downward forces are not equal. But when he is at rest or moving at constant speed, the opposing forces balance each other. Knowing that the scale reading is a measure of the upward normal force of the scale upon his body, its value could be predicted for various stages of motion. For instance, the value of the normal force (Fnorm) on Otis's 80-kg body could be predicted if the acceleration is known. **Energy Relationships for Satellites**  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. 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.
 * Scale Readings and Weight **

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** ** Energy Analysis of Circular Orbits ** When in circular motion, a satellite remains the same distance above the surface of the earth; that is, its radius of orbit is fixed. Furthermore, its speed remains constant. Since kinetic energy is dependent upon the speed of an object, the amount of kinetic energy will be constant throughout the satellite's motion. And since potential energy is dependent upon the height of an object, the amount of potential energy will be constant throughout the satellite's motion. So if the KE and the PE remain constant, it is quite reasonable to believe that the TME remains constant.

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. As mentioned above, the force of gravity does work upon a satellite to slow it down as it moves away from the earth and to speed it up as it moves towards the earth. So if the speed is changing, the kinetic energy will also be changing. The elliptical trajectory of a satellite is shown below.
 * Energy Analysis of Elliptical Orbits **