Ch5_malleys

=Circular Motion and Satellite Motion= toc

Speed and Velocity
Headline: Vectors, Vectors... Article: The motion of an object in a circle with a constant/uniform speed is called uniform circular motion. It is only one of several types of circular motion. Because the speed and circumference are uniform, you can calculate the average speed of an object in a circle by dividing circumference (2πr) by time. It is very important to keep in mind the direction of the vector. The direction of the vector at any given time is directed towards a tangent line at the object's location; thus the name tangential.

Acceleration
Headline: And More Vectors! Article: An accelerometer is used to measure objects moving in circles at a constant speed; they accelerate towards the center of the circle.

The Centripetal Force Requirement
Headline: The Word is Centripetal Article: Objects in uniform circular motion experience an inward net force, sometimes called the centripetal force. (Centripetal is describing the direction of the object). Without this, the object could not change directions. The force has the ability to change the direction of the object's velocity (without changing its magnitude) because it is directed perpendicularly to the tangential velocity.

The Forbidden F-Word
Headline: The forbidden f-word is not what you'd think! Article: Centrifugal refers to an object moving away from a circle, which is not the kind of object we want to work with. We use centripetal forces for now. To make circular motion, there necessarily must be a net force that is directed toward the center of the circle - otherwise, it would continue in its tangential path. This force would be the centripetal force.

Mathematics of Circular Motion
Headline: So Much Math, So Little Time Article: The equation for average speed is 2πr/t. The equation for acceleration is 4π 2 r/t 2. By expanding upon the net force equation that we already know, we can figure out that we know that (assuming a constant mass and a constant radius) F net = m*(4π 2 r/t 2 ) which equals F net m*v 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. If any given physical situation is analyzed in terms of the individual forces that are acting upon an object, then those individual forces must add up as vectors to the net force. Furthermore, the net force must be equal to the mass times the acceleration. Subsequently, the acceleration of an object can be found if the mass of the object and the magnitudes and directions of each individual force are known. And the magnitude of any individual force can be determined if the mass of the object, the acceleration of the object, and the magnitude of the other individual forces are known. The most obvious section on a roller coaster where centripetal acceleration occurs is within the so-called **clothoid loops**. Roller coaster loops assume a tear-dropped shape that is geometrically referred to as a clothoid. A clothoid is a section of a spiral in which the radius is constantly changing. 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. 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. A change in direction is one characteristic of an accelerating object. In addition to changing directions, the rider also changes speed. As the rider begins to ascend (climb upward) the loop, she begins to slow down. 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 ( **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. 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 **Fgrav = m•g** where g = acceleration of gravity (9.8 m/s2). 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. The most common example of the physics of circular motion in sports involves the turn. 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 cicle. 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. ** C ** **ontact 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.

Gravity is More Than a Name
Certainly 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 **Fgrav**. Many students of physics have become accustomed to referring to the actual acceleration of such an object as the **acceleration of gravity**. It is the same acceleration value for all objects, regardless of their mass (and assuming that the only significant force is gravity).

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

While Kepler's laws provided a suitable framework for describing the motion and paths of planets about the sun, there was no accepted explanation for why such paths existed. Newton was troubled by the lack of explanation for the planet's orbits. 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. It was Newton's ability to relate the cause for heavenly motion (the orbit of the moon about the earth) to the cause for Earthly motion that led him to his notion of **universal gravitation**. Quite amazingly, the laws of mechanics that govern the motions of objects on Earth also govern the movement of objects in the heavens. 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 follows an **inverse square law**. The relationship between the force of gravity ( **Fgrav** ) 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. **Using Equations as a Guide to Thinking** 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. Altering the separation distance (d) results in an alteration in the force of gravity acting between the objects. Since the two quantities are inversely proportional, an increase in one quantity results in a decrease in the value of the other quantity. That is, 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). And if the separation distance is tripled (increased by a factor of 3), then the force of gravity is decreased by a factor of nine (3 raised to the second power).

Newton's Law of Universal Gravitation
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. 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. Newton's conclusion about the magnitude of gravitational forces is summarized symbolically as Since the gravitational force is directly proportional to the mass of both interacting objects, more massive objects will attract each other with a greater gravitational force. So as the mass of either object increases, the force of gravitational attraction between them also increases. If the mass of one of the objects is doubled, then the force of gravity between them is doubled. If the mass of one of the objects is tripled, then the force of gravity between them is tripled. If the mass of both of the objects is doubled, then the force of gravity between them is quadrupled; and so on. Since gravitational force is inversely proportional to the separation distance between the two interacting objects, more separation distance will result in weaker gravitational forces. So as two objects are separated from each other, the force of gravitational attraction between them also decreases. If the separation distance between two objects is doubled (increased by a factor of 2), then the force of gravitational attraction is decreased by a factor of 4 (2 raised to the second power). If the separation distance between any two objects is tripled (increased by a factor of 3), then the force of gravitational attraction is decreased by a factor of 9 (3 raised to the second power). 

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 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 **m1• m2** units and divided by **d2** units, the result will be Newtons - the unit of force.

Cavendish and the Value of G
Isaac Newton's law of universal gravitation proposed that the gravitational attraction between any two objects is directly proportional to the product of their masses and inversely proportional to the distance between their centers. In equation form, this 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. 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 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.

The Value of g
A second equation has been introduced for calculating the force of gravity with which an object is attracted to the earth. where **d** represents the distance from the center of the object to the center of the 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. 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. 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. Now observe that the mass of the object - **m** - is present on both sides of the equal sign. Thus, m can be canceled from the equation. This leaves us with an equation for the acceleration of gravity. 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. This inverse square relationship means that as the distance is doubled, the value of g decreases by a factor of 4. As the distance is tripled, the value of g decreases by a factor of 9. And so on. This inverse square relationship is depicted in the graphic at the right.

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 value of g on any other planet can be calculated from the mass of the planet and the radius of the planet. The equation takes the following form: The acceleration of gravity of an object is a measurable quantity. 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
It is probably fair to say that no single individual has had a greater influence on the scientific view of the world than Isaac Newton.

In 1543, Nicolaus Copernicus launched a scientific revolution by rejecting the prevailing Earth-centred view of the Universe in favour of a **heliocentric** view in which the Earth moved round the Sun. 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. 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'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. 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.  Newton's great achievement was to provide a synthesis of scientific knowledge. He did not claim to have all the answers, but he discovered a convincing quantitative framework that seemed to underlie everything else.

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.

**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. **3.** Newton produced a quantitative link between force and deviation from steady motion and quantified the force by proposing his famous law of universal gravitation.

Newton proposed just one law for gravity - a law that worked for every scrap of matter in the Universe. Newton was able to demonstrate mathematically that a single planet would move around the Sun in an elliptical orbit. Newtonian physics was able to predict that gravitational attractions between the planets would cause small departures from the purely elliptical motion that Kepler had described.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 entirely predictable. This property of Newtonian mechanics is called **determinism**.

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 second law 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. Kepler's third law 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. It 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.

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.  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 acceleratio. 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]. 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.

Mathematics of Satellite Motion
Consider a satellite with mass Msat orbiting a central body with a mass of mass MCentral. 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

The acceleration of a satellite in circular motion about some central body is given by the following equation The period of a satellite ( **T** ) and the mean distance from the central body ( **R** ) are related by the following equation:

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. 

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]. Theforce 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. **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.  Weightlessness is only a sensation. Weightlessness has very little to do with weight and mostly to do with the presence or absence of contact forces. **Scale Readings and 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. 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. If it were not for the force of gravity, the astronauts would not be orbiting in circular motion. The astronauts and all their surroundings 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. The fact is that there must be a force of gravity in order for there to be an orbit.

Energy Relationships for Satellites
The orbits of satellites about a central massive body can be described as either circular or elliptical. 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 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//. **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. The amount of kinetic energy will be constant throughout the satellite's motion. 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. 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** The total amount of mechanical energy of a satellite in elliptical motion remains constant. The Wext term is zero and mechanical energy is conserved. The energy of a satellite in elliptical motion will change forms. So if the speed is changing, the kinetic energy will also be changing. The speed of this satellite is greatest when the satellite is closest to the earth and least when the satellite is furthest from the earth. Throughout the entire elliptical trajectory, the total mechanical energy of the satellite remains constant. 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.