Ch5_SellmanJ

=__**toc Lesson 1: Method 5**__= __Speed and Circular Motion__ Uniform circular motion means that an object moving in circular motion is at constant speed. Average speed can be calculated by distance over time, which is circumference over time. It is important to remember that the velocity is always directed in the direction that the object moves, while the speed has no direction. Direction is always tangent to the circle. __Common Misconceptions about Acceleration__ Most would think that because an object is moving at constant speed, there would be no acceleration, but this is untrue. The acceleration in this case is referring to the velocity, which has direction, and because it is changing direction, there is acceleration. Average acceleration is equal to the change in velocity over time. Acceleration of objects can be measured with an accelerometer. __All About Centripetal Forces__ Centripetal forces are forces which act upon objects and cause an inward acceleration.This force is either pulling or pushing the object towards the center of the circle. Work is equal to force times displacement times cosine of theta. Without the centripetal force, the object wouldn't change direction. __No Outward Force!__ Many students tend to think that there is an outward force on an object moving in circular motion, although it is really an inward force which is caused by inward acceleration. Sometimes people are misconstrued because it feels like there is an outward force, although this theory is untrue. If there was no inward force, then the object would keep moving in a straight line. __Three Main Equations!__ Average speed is equal to total distance over time, or the circumference over time. The acceleration of an object moving in a circle can be determined through acceleration equaling velocity squared over radius or acceleration equaling 4 times pi squared times radius divided by the period squared. The net force of an object in circular motion is directed inwards. The vector sum should add up to the net force. Net force is related to acceleration through the following three equations:

=**Lesson 2** = Part 1 Applying the concept of a centripetal force requirement, we know that the net force acting upon the object is directed inwards. Since the car is positioned on the left side of the circle, the net force is directed rightward. An analysis of the situation would reveal that there are three forces acting upon the object - the force of gravity (acting downwards), the normal force of the pavement (acting upwards), and the force of friction (acting inwards or rightwards). It is the friction force that supplies the centripetal force requirement for the car to move in a horizontal circle. Without friction, the car would turn its wheels but would not move in a circle (as is the case on an icy surface). Part 2 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. 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. The normal force provides a //feel// for a person's weight. The more you weigh, the more normal force that you will experience when at rest in your seat. When at the top of the loop, a rider will __feel__ partially weightless if the normal forces become less than the person's weight. And at the bottom of the loop, a rider will feel very "weighty" due to the increased normal forces. It is important to realize that the force of gravity and the weight of your body are not changing. Only the magnitude of the supporting normal force is changing. Part 3 Circular motion is characterized by an [|inward acceleration] and caused by an [|inward net force] .Nonetheless, any turn can be approximated as being a part of a larger circle or a part of several circles of varying size. A sharp turn can be considered part of a small circle. A more gradual turn is part of a larger circle. There must be some object supplying an inward force or inward [|component of force]. When a person makes a turn on a horizontal surface, the person often //leans into the turn//. By leaning, the surface pushes upward at an angle //to the vertical//. As such, there is both a horizontal and a vertical component resulting from contact with the surface below. This ** contact force ** supplies two roles - it balances the downward force of gravity and meets the centripetal force requirement for an object in uniform circular motion. The upward component of the contact force is sufficient to balance the downward force of gravity and the horizontal component of the contact force pushes the person towards the center of the circle.

=**Lesson 3** = **Gravity is More Than a Name** Certainly gravity is a force that exists between the Earth and the objects that are near it. The force gravity causes an acceleration of our bodies during this brief trip away from the earth's surface and back. The acceleration of gravity ( **g** ) is the acceleration experienced by an object when the only force acting upon it is the force of gravity. On and near Earth's surface, the value for the acceleration of gravity is approximately 9.8 m/s/s. It is the same acceleration value for all objects, regardless of their mass (and assuming that the only significant force is gravity). **The Apple, the Moon, and the Inverse Square Law** Kepler's three laws: To Kepler, the planets were somehow "magnetically" driven by the sun to orbit in their elliptical trajectories. Newton's ability to relate the cause for heavenly motion (the orbit of the moon about the earth) to the cause for Earthly motion (the falling of an apple to the Earth) that led him to his notion of **universal gravitation**. 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.
 * <span style="font-family: Arial,Helvetica,sans-serif;">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: Arial,Helvetica,sans-serif;">An imaginary line drawn from the center of the sun to the center of the planet will sweep out equal areas in equal intervals of time. (The Law of Equal Areas)
 * <span style="font-family: Arial,Helvetica,sans-serif;">The ratio of the squares of the periods of any two planets is equal to the ratio of the cubes of their average distances from the sun. (The Law of Harmonies)

<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">**Newton's Law of Universal Gravitation** <span style="font-family: Arial,Helvetica,sans-serif;">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. Gravity is universal. Newton's conclusion about the magnitude of gravitational forces is summarized symbolically as <span style="font-family: Arial,Helvetica,sans-serif;"> <span style="font-family: Arial,Helvetica,sans-serif;">As the mass of either object increases, the force of gravitational attraction between them also increases. So as two objects are separated from each other, the force of gravitational attraction between them also decreases. <span style="font-family: Arial,Helvetica,sans-serif;">

<span style="font-family: Arial,Helvetica,sans-serif;">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. Value of G is found to be **G = 6.673 x 10-11 N m2/kg2** Value of g is equivalent to the ratio of (G•Mearth)/(Rearth)2. ALL objects attract in proportion to the product of their masses. Gravity is universal. <span style="color: #000000; font-family: Arial,Helvetica,sans-serif; font-size: 14px;">**Cavendish and the Value of G** <span style="font-family: Arial,Helvetica,sans-serif;">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. <span style="color: #000000; font-family: Arial,Helvetica,sans-serif; font-size: 14px;">**The Value of g** <span style="font-family: Arial,Helvetica,sans-serif;">A second equation has been introduced for calculating the force of gravity with which an object is attracted to the earth. <span style="font-family: Arial,Helvetica,sans-serif;"> <span style="font-family: Arial,Helvetica,sans-serif;">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 <span style="font-family: Arial,Helvetica,sans-serif;">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. If the value 6.38x106 m (a typical earth radius value) is used for the distance from Earth's center, then g will be calculated to be 9.8 m/s2. And of course, the value of g will change as an object is moved further from Earth's center <span style="font-family: Arial,Helvetica,sans-serif;">The value of g varies inversely with the distance 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 value of g on any other planet can be calculated from the mass of the planet and the radius of the planet. <span style="font-family: Arial,Helvetica,sans-serif;">

=**Lesson 2 Part 1-4: Method 5**= Part 1: What were some original theories about the universe? Copernicus thought that the universe was heliocentric and the earth moved around the sun. Originally, there had been the Earth-centered view of the ancient Greeks and the Catholic church in the 16th century. Then the Copernican System, in which the planets move in collections of circles around the Sun. Then the Keplerian system in which a planet follows an elliptical orbit, with the Sun at one focus of the ellipse. According to Kepler, the planets did move around the sun but their paths were ellipses not circles. Part 2: What was discovered next? There was a 2-d coordinate system created by Rene Descartes. This could show lines and 2-d shapes. The equation of the early circle was then formed. This was a branch of coordinate geometry. Part 3: What was Newton's deal? Lagrange remarked "There is only one universe...it can happen to only one man in the world's history to be the interpretor of its laws." Newton focused on deviation from steady motion. He looked for causes for deviation in motion, especially in pushes and pulls. He then proposed his laws of gravitation. Part 4: How did Newton take it further? Newton was able to show mathematically that a planet could move around the sun in an elliptical orbit. He was able to show that Kepler's theories were correct, but he was able to prove even more. Mechanics was the study of force and motion. Determinism is that once clockwork was set in motion, the rest was predictable. According to Newtonian Mechanics, it is possible to predict the entire future behavior of the universe provided initial positions and velocities of all the particles in it are known.

=**Lesson 4**= Part 1: Kepler's Three Laws Kepler's three laws of planetary motion can be described as follows: 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. Newton's comparison of the acceleration of the moon to the acceleration of objects on earth allowed him to establish that the __ [|moon is held in a circular orbit by the force of gravity] __ - a force that is inversely dependent upon the distance between the two objects' centers. **T2 / R3 = (4 * pi2) / (G * MSun ).** The right side of the above equation will be the same value for every planet regardless of the planet's mass. Subsequently, it is reasonable that the **T2/R3** ratio would be the same value for all planets if the force that holds the planets in their orbits is the force of gravity. Circular Motion Principles for Satellites A satellite is any object that is orbiting the earth, sun or other massive body. The fundamental principle to be understood concerning satellites is that a satellite is a __ [|projectile] __. That is to say, a satellite is an object upon which the only force is gravity. Part 2: **Velocity, Acceleration and Force Vectors** The motion of an orbiting satellite can be described by the same motion characteristics as any object in circular motion. The __ [|velocity] __ of the satellite would be directed tangent to the circle at every point along its path. The __ [|acceleration] __ of the satellite would be directed towards the center of the circle - towards the central body that it is orbiting. And this acceleration is caused by a __ [|net force] __ that is directed inwards in the same direction as the acceleration. Like any projectile, gravity alone influences the satellite's trajectory such that it always falls below its __ [|straight-line, inertial path] __. This is depicted in the diagram below. 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.
 * 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)

 Satellites are projectiles that orbit around a central massive body instead of falling into it. Being projectiles, they are acted upon by the force of gravity - a universal force that acts over even large distances between any two masses. The motion of satellites, like any projectile, is governed by Newton's laws of motion. For this reason, the mathematics of these satellites emerges from an application of Newton's universal law of gravitation to the mathematics of circular motion. Part 3: 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.

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.  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. Part 4: <span style="color: #000000; font-family: Arial,Helvetica,sans-serif; font-size: 12px;">Contact forces can only result from the actual touching of the two interacting objects - in this case, the chair and you. The force of gravity acting upon your body is not a contact force; it is often categorized as an __action-at-a-distance force__. The force of gravity is the result of your center of mass and the Earth's center of mass exerting a mutual pull on each other; this force would even exist if you were not in contact with the Earth. The force of gravity does not require that the two interacting objects (your body and the Earth) make physical contact; it can act over a distance through space. **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. <span style="font-family: Arial,Helvetica,sans-serif;">Weightlessness is only a sensation; it is not a reality corresponding to an individual who has lost weight.The scale is only measuring the external contact force that is being applied to your body. Now consider Otis L. Evaderz who is conducting one of his famous elevator experiments. 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. <span style="font-family: Arial,Helvetica,sans-serif;">

Part 5: The orbits of satellites about a central massive body can be described as either circular or elliptical. Since __ [|perpendicular components of motion are independent] __ of each other, the inward force cannot affect the magnitude of the tangential velocity. For this reason, there is no acceleration in the tangential direction and the satellite remains in circular motion at a constant speed. A satellite orbiting the earth in elliptical motion will experience a component of force in the same or the opposite direction as its motion. This force is capable of doing __ [|work] __ upon the satellite. 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** Let's consider the circular motion of a satellite first. 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. Like the case of circular motion, the total amount of mechanical energy of a satellite in elliptical motion also remains constant.

Lesson 1 Part a When a force acts upon an object to cause a displacement of the object, it is said that **work** was done upon the object. In order for a force to qualify as having done//work// on an object, there must be a displacement and the force must //cause// the displacement. Mathematically, work can be expressed by the following equation.**where **F** is the force, **d** is the displacement, and the angle ( **theta** ) is defined as the angle between the force and the displacement vector.** A vertical force can never cause a horizontal displacement; thus, a vertical force does not do work on a horizontally displaced object!! The equation for work lists three variables - each variable is associated with one of the three key words mentioned in the __ [|definition of work] __ (force, displacement, and cause). The angle theta in the equation is associated with the amount of force that causes a displacement. When determining the measure of the angle in the work equation, it is important to recognize that the angle has a precise definition - it is the angle between the force and the displacement vector. 1 Joule=1 Newton* 1 meter
 * Scenario A: A force acts rightward upon an object as it is displaced rightward. In such an instance, the force vector and the displacement vector are in the same direction. Thus, the angle between F and d is 0 degrees.
 * Scenario B: A force acts leftward upon an object that is displaced rightward. In such an instance, the force vector and the displacement vector are in the opposite direction. Thus, the angle between F and d is 180 degrees.
 * Scenario C: A force acts upward on an object as it is displaced rightward. In such an instance, the force vector and the displacement vector are at right angles to each other. Thus, the angle between F and d is 90 degrees.