Ch5+KloorfainN

=Chapter 5 = toc

12/13 Lesson 1 Motion Characteristics for Circular Motion
Circular Speed and Velocity Uniform Circular Motion is the motion of an object in a circle with a constant or uniform speed and constant radius. To find the average speed of this motion is the circumference divided by time. The circumference of a circle is 2(pi)(R). The __magnitude is constant while the direction is changing.__ The direction can be described the tangential because it is always pointing in the new direction.

What is acceleration? Acceleration is the changing of velocity but it can either have a __changing magnitude or direction__. Objects moving in circles at a constant speed accelerate towards the center of the circle. The lighter the objects experience greater acceleration and will lean in the direction of the accleration.

Inward Force Facts Centripetal force requirement refers to an object moving in a circle that //must have inward force acting up on// it in order to cause its inward acceleration. Inward forces are the centripetal forces. The centripetal force is directed perpdenicular to the tangential velocity. **Objects will keep moving with the same speed in the same direction as long as there is an unbalanced force.** __Unbalanced force causes the object to turn.__ Inertia is the tendency to resist acceleration. In cars, the direction of the passenger leaning is the opposite direction of the acceleration. There is a physical force pushing or pulling the object towards the center of the circle. Work is a force acting upon an object to cause a displacement. Work = force (displacement) (cosTHETA)

What is this Forbidden F Word? Is it Appropriate for Physics Class? Centrifugal means away from the center or **outward**. Without the inward force, circular motion would be unachievable. //An outward force does not exist.// If there was one, you would **//__not__//** be moving in a circle. An object in motion stays in the same motion and direction.

The Epic Trio of Circles Motion and Math! The average speed is circumference divided by time. The acceleration is 2 (pi squared) (R) all divided by time. The inward force is larger than the outward force. The unbalanced force is in the direction of the center of the circle. The acceleration is directly proportional to the net force. Acceleration is inversely proportional to mass.

1/3/12 Lesson 3 Universal Gravitation
Gravity is More than a Name Gravity is the name associated with the reason for "what goes up, must come down. We all know of the word gravity - it is the //thing// that causes objects to fall to Earth. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">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** . As we rise upwards after our jump, the force of gravity slows us down. In this sense, the force gravity __causes an acceleration__ of our bodies during this brief trip away from the earth's surface and back. In fact, many students of physics have become accustomed to referring to the actual acceleration of such an object as the **acceleration of gravity**. 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. The acceleration of gravity is approximately 9.8 m/s/s. It is the same acceleration value for all objects, regardless of their mass (and assuming that the only significant force is gravity).

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">The Apple, The Moon, the Inverse Square Law <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">German mathematician and astronomer Johannes Kepler developed 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: <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">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. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">To Newton, there must be some cause for such elliptical motion/orbit. 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. These celestial motions required a cause in the form of an unbalanced force. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">
 * <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">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)
 * <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">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: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">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)

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Suppose a cannonball is fired horizontally from a very high mountain in a region devoid of air resistance. In the absence of gravity, the cannonball would travel in a straight-line, tangential path. Yet in the presence of gravity, the cannonball would drop below this straight-line path and eventually fall to Earth (as in **path A** ). The cannonball is fired horizontally again, yet with a greater speed. The cannonball would still fall below its straight-line tangential path and eventually drop to earth. Only this time, the cannonball would travel further before striking the ground (as in **path B** ). There is a speed at which the cannonball could be fired such that the trajectory of the falling cannonball matched the curvature of the earth. If such a speed could be obtained, then the cannonball would fall around the earth instead of into it. The cannonball would fall towards the Earth without ever colliding into it and subsequently become a satellite orbiting in circular motion (as in **path C** ). And then at even greater launch speeds, a cannonball would once more orbit the earth, but in an elliptical path (as in **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. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">The key to this extension demanded that he be able to show how the affect of gravity is diluted with distance. It was known at the time, that the force of gravity causes earthbound objects (such as falling apples) to accelerate towards the earth at a rate of 9.8 m/s2. And it was also known that the moon accelerated towards the earth at a rate of 0.00272 m/s2. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">The moon in its orbit about the earth is approximately 60 times further from the earth's center than the apple is. 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 moon, being 60 times further away than the apple, experiences a force of gravity that is 1/(60)2 times that of the apple. The force of gravity follows an **inverse square law**.



<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">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.

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Newton's Law of Universal Gravitation <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">This comparison led him to conclude that 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** <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Newton knew that the force that caused the apple's acceleration (gravity) must be dependent upon the mass of the apple. And since the force acting to cause the apple's downward acceleration also causes the earth's upward acceleration (Newton's third law), that force must also depend upon the mass of the earth. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">**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. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">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. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">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.

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">The constant of proportionality (G) in the above equation is known as the **universal gravitation constant**. <span style="color: #ff0000; font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">**G = 6.673 x 10-11 N m2/kg2**

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;"> 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. Distance of separation becomes much more influential when a significant variation is made. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;"> <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">The second conceptual comment to be made about the above sample calculations 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 in Unit 2: <span style="color: #ff0000; font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">**Fgrav = m•g = (70 kg)•(9.8 m/s2) = 686 N**

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Newton's revolutionary idea was that gravity is universal - ALL objects attract in proportion to the product of their masses. Gravity is universal. Most gravitational forces are so minimal to be noticed. Gravitational forces are only recognizable as the masses of objects become large.

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">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.

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Cavendish and the Value of G <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">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.

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">The Value of G <span style="color: #ff0000; font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">**Fgrav = m*g** <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Calculating the force of gravity with which an object is attracted to the earth. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">where **d** represents the distance from the center of the object to the center of the earth. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;"> 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. Earth is not truly spherical; the earth's surface is further from its center at the equator than it is at the poles. One proceeds further from earth's surface - say into a location of orbit about the earth - the value of g changes still. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">To understand why the value of g is so location dependent <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">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. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;"> Equation for another planet: <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">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.

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">1/5 Lesson 2 The Clockwork Universe
<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Mechanism and Determinism <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">In 1543, a century before Newton's birth, 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. The most famous of these must surely have been Galileo for supporting Copernicus' ideas. As a result Galileo was 'shown the instruments of torture', and invited to renounce his declared opinion that the Earth moves around the Sun.

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">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.

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Kepler's ideas were underpinned by new discoveries in mathematics. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">The grid is calibrated (in centimetres) so the position of any point can be specified by giving its //x-// and //y-// coordinates on the grid. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">This idea becomes more powerful when we consider lines and geometrical shapes. The straight line shown in Figure 1.5 is characterized by the fact that, at each point along the line, the //y-//coordinate is half the -coordinate. Thus, the //x-// and y- coordinates of each point on the line obey the equation //y// = 0.5//x//, and this is said to be the equation of the line. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;"> <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">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. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;"> Newton's great achievement was to provide a synthesis of scientific knowledge.
 * <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">[[image:http://physicalworld.org/restless_universe/figs/fig_1_5.gif width="175" height="137" align="left" caption="figure 1.5, a 2-D coordinate system can represent lines and other geometrical shapes by equationsrep"]] ||  ||

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">**'There is only one Universe... It can happen to only one man in the world's history to be the interpreter of its laws.'**

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">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:

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">**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. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">**2.** Wherever deviation from steady motion occurred, Newton looked for a cause. 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. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">**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.

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">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, just as Kepler claimed each of the planets did. 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.

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Newtonís discoveries became the basis for a detailed and comprehensive study of **mechanics** (the study off force and motion). The upshot of all this was a mechanical world-view that regarded the Universe as something that unfolded according to mathematical laws with all the precision and inevitability of a well-made clock. 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**.

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">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.Every future human action would be already determined by the past. 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.

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">1/6 Lesson 4 A-C
<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Kepler's Three Laws

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">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.

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Kepler's three laws of planetary motion can be described as follows: <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">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. The closer together that these points are, the more closely that the ellipse resembles the shape of a circle. In fact, a circle is the special case of an ellipse in which the two foci are at the same location. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">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. The line through the center would sweep out the same area in equal periods of time. The areas formed when the earth is closest to the sun can be approximated as a wide but short triangle; whereas the areas formed when the earth is farthest from the sun can be approximated as a narrow but long triangle. These areas are the same size. Since the //base// of these triangles are shortest when the earth is farthest from the sun, the earth would have to be moving more slowly in order for this imaginary area to be the same size as when the earth is closest to the sun.
 * <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">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: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">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: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">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)



<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">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.

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Observe that the T2/R3 ratio is the same for Earth as it is for mars. IAmazingly, every planet has the same T2/R3 ratio. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">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.

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Circular Motion Principles for Satelites <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">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**. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">The fundamental principle to be understood concerning satellites is that a satellite is a __ projectile __. 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. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">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. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">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 __. Observe that the inward net force pushes (or pulls) the satellite (denoted by blue circle) inwards relative to its straight-line path tangent to the circle.



<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">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.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. The elliptical motion of satellites is not characterized by a constant speed. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;"> <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Mathematics of Satellite Motion

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">The motion of objects is governed by Newton's laws.

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">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. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Acceleration for Gravity <span style="color: #ff0000; font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">**g = (G • Mcentral)/R2** <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">The final equation that is useful in describing the motion of satellites is Newton's form of Kepler's third law. The period of a satellite ( **T** ) and the mean distance from the central body ( **R** ) are related by the following equation: <span style="color: #000000; font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">The period, speed and the acceleration of an orbiting satellite are not dependent upon the mass of the satellite. <span style="color: #000000; font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">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.

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Mearth = 5.98x1024 kg <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">G = 6.673 x 10-11 N m2/k ||<  || <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">
 * < <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;"> **Given/Known:** R = Rearth + height = 6.47 x 106 m

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">**1/9 Lesson 4 D&E**
<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Weightlessness in Orbit <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Sitting in a chair <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">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 __. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">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. If there were no upward normal force acting upon your body, you would not have any sensation of your weight. Without the contact force (the normal force), there is no means of feeling the non-contact force (the force of gravity).

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;"> **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. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">As you and your chair fall towards the ground, you both accelerate at the same rate - **g**. Normal forces only result from contact with stable, supporting surfaces. The force of gravity is the only force acting upon your body. There are no external objects touching your body and exerting a force. Weightlessness has very little to do with weight and mostly to do with the presence or absence of contact forces. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">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 scale is only measuring the external contact force that is being applied to your body. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">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. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">The interaction of the two forces - the upward normal force and the downward force of gravity - can be thought of as a tug-of-war. The net force acting upon the person indicates who wins the tug-of-war (the up force or the down force) and by how much. The gravitational force acting upon the rider is found using the equation **Fgrav = m*g**. ||< <span style="color: #ff0000; font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">**Stage B** ||< <span style="color: #ff0000; font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">**Stage C** ||< <span style="color: #ff0000; font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">**Stage D** || <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">The normal force is greater than the force of gravity when there is an upward acceleration (B), less than the force of gravity when there is a downward acceleration (C and D), and equal to the force of gravity when there is no acceleration (A). Since it is the normal force that provides a sensation of one's weight, the elevator rider would feel his normal weight in case A, more than his normal weight in case B, and less than his normal weight in case C. In case D, the elevator rider would feel absolutely weightless; without an external contact force, he would have no sensation of his weight. In all four cases, the elevator rider weighs the same amount - 784 N. Yet the rider's sensation of his weight is fluctuating throughout the elevator ride.
 * < <span style="color: #ff0000; font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">**Stage A**
 * < <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Fnet = m*aFnet = 0 N ||< <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Fnet = m*aFnet = 400 N, up ||< <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Fnet = m*aFnet = 400 N, down ||< <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Fnet = m*aFnet = 784 N, down ||
 * < <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Fnorm equals Fgrav **Fnorm = 784 N** ||< <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Fnorm > Fgrav by 400 N **Fnorm = 1184 N**  ||< <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Fnorm < Fgrav by 400 N **Fnorm = 384 N**  ||< <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Fnorm < Fgrav by 784 N **Fnorm = 0 N**  ||

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">They are weightless because there is no external contact force pushing or pulling upon their body. In each case, gravity is the only force acting upon their body. Being an action-at-a-distance force, it cannot be felt and therefore would not provide any sensation of their weight. In fact, if it were not for the force of gravity, the astronauts would not be orbiting in circular motion. It is the force of gravity 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 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. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Their absolutely weightless sensations are the result of having "the floor pulled out from under them" (so to speak) as they are free falling towards the Earth.

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Energy Relationships for Satellites

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">The orbits of satellites about a central massive body can be described as either circular or elliptical. It accomplishes this feat by moving with a tangential velocity that allows it to fall at the same rate at which the earth curves. 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. There is no acceleration in the tangential direction and the satellite remains in circular motion at a constant speed. 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. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;"> <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">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 <span style="color: #ff0000; font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">**KEi + PEi + Wext = KEf + PEf** <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;"> The Wext term in this equation is representative of the amount of work done by __ external forces __. Since gravity is considered an __ internal (conservative) force __, the Wext term is zero. <span style="color: #ff0000; font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">**KEi + PEi = KEf + PEf** <span style="color: #000000; font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">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.

<span style="color: #000000; font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;"> The speed at positions A, B, C and D are the same. The heights above the earth's surface at A, B, C and D are also the same. 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. <span style="color: #000000; font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;"> 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. A work-energy bar chart is presented below for a satellite in uniform circular motion about the earth. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">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. So if the speed is changing, the kinetic energy will also be changing. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">The speed of this satellite is greatest at location A (when the satellite is closest to the earth) and least at location C (when the satellite is furthest from the earth). So as the satellite moves from A to B to C, it loses kinetic energy and gains potential energy. The gain of potential energy as it moves from A to B to C is consistent with the fact that the satellite moves further from the surface of the earth. As the satellite moves from C to D to E and back to A, it gains speed and loses height; subsequently there is a gain of kinetic energy and a loss of potential energy. Yet throughout the entire elliptical trajectory, the total mechanical energy of the satellite remains constant. The work-energy bar chart below depicts these very principles.

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">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.