Ch5_LuoR

toc =Chapter 5: Circular Motion=

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A.) "An E'motion'al New Discovery!" Leading researchers have recently discovered a way to describe an object moving in a circular path and derived an equation for the average speed of the object. Objects moving in a circular path can be described using kinematics, and the researchers call an object moving in a circular path at constant speed to be moving at uniform circular motion. These researchers told Honors Physics 4 News (HPP4N) that to find the average speed, you take the circumference of the circular path and divide it by the amount of time the object takes to travel the path and you will have the average speed, as seen in the equation below. However, these esteemed physics pioneers have admonished HPP4N that although the object is moving at a constant speed, it does not mean the velocity remains constant, too. They said that because the path is circular, the path will continuously change directions, and thus, continue to change velocity (shown in the picture next to the equation).
 * Lesson 1: Motion Characteristics for Circular Motion**



B.) "Who Needs Speed When You Can Change Direction" HPP4N has uncovered how velocity changes in an uniform circular path. Unlike linear changes in velocity, the changes in velocity in an uniform circular path are due to changes in direction and not speed. This is because velocity is always directed to the center of the circle created by the circular path, so at each point of the path, the direction of the velocity changes. Because acceleration of an object is in the direction of the velocity change vector, acceleration is directed to the center of the of this circle.

C.) "I Need YOU!" After being enlightened by the what the researchers found in parts A and B, HPP4N has taken the liberty to experiment on some of these findings to see if we could learn new things about centripetal force. What we discovered was shocking. Our experiment involved driving a car around a traffic circle to observe the movement of the car and use these observations to calculate what forces are acting on the car. Our findings showed that the car, when going around a turn, has friction force pulling the car towards the center of the circle, and as a result, the car moves in a circular path. There is always a force acting on an object moving in a circular path that draws the object towards the center of the circle, even in space, where the object is usually affected by gravity.

D.) "A F or A P?" While HPP4N continues to delve further into the mystery of uniform circular paths, an ancient miscue has arisen. From the inceptions of the two words, centri//**fugal**// and centri//**peta****l**// have been commonly mistaken for one another. Centrifugal refers to an object moving away from the center of the circle from the circular path, while centripetal means moving towards the center. This concept, according to the ancient scriptures, is essential to circular motion, but people seem to always confuse the two words. This confusion may be caused by the notion that objects in circular motion are experiencing an outward force. HPP4N interviewed a student currently enrolled in a physics class named R. Luo said, "I can recall vividly the sensation of being thrown outward away from the center of the circle on that roller coaster ride. Therefore, circular motion must be characterized by an outward force."

E.) "Where There's A Concept, There's An Equation!" In the final segment of "Motion Characteristics of Circular Motion", HPP4N uses all of the information it has gathered and has come up with equations to calculate the main variables of uniform circular motion.

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A.) Newton's Second Law-Revisited Q. What does Newton's Second Law have to do with circular motion? R. The concept of calculating the net force of an object in circular motion and linear motion is similar. Summary: The equation for calculating net force is the same for linear motion and circular motion in that it is ∑F=m*a. However, in circular motion, acceleration equals the velocity squared divided by the radius of the circle.
 * Lesson 2: Application of Circular Motion**

B.) Roller Coasters and Amusement Park Physics Q. 1. How does circular motion apply to roller coasters? 2. What is a clothoid loop? R. 1. Roller coasters are created based on the principles of circular motion. 2. A clothoid loop is a teardrop-shaped area where centripetal acceleration occurs. Summary: Humans have taken advantage of the principles of circular motion and used it to create fun in the form of roller coasters. The thrill and rush that people feel when going down a roller coaster are due to centripetal acceleration that occurs in the clothoid loop.

C.) Athletics Q. 1. What is the most common example of physics with athletics? R. 1. The most common example is a turn. A turn can be anything from a baseball player rounding the bases or a NASCAR driver speeding around a turn at high speeds.

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A.) Gravity is More Than a Name Q. What is the difference between acceleration of gravity and force of gravity? R. Acceleration of gravity is the acceleration experienced by an object when the only force acting upon it is the force of gravity. Summary: Gravity is something that is present in everyday life. However, people can become confused between the acceleration of gravity and the force of gravity. The acceleration of gravity is the acceleration experienced by an object when the only force acting upon it is the force of gravity, while the force of gravity is the force that slows us down when we jump up and speeds us up when returning to the ground from the jump.
 * Lesson 3: Universal Gravitation**

B.) The Apple, the Moon, and the Inverse Square Law "The Birth of Newton's Notion of Universal Gravitation" Through his observations of planetary motion, 17th century German mathematician and astronomer Johannes Kepler developed 3 laws that can be summarized as such: While Kepler made these hypotheses, there was no explanation to these planetary movements during his time period. Along came Isaac Newton, who was troubled by the fact that there were no explanations for the motion of the planet's orbit. Newton knew that there was a relationship between the Moon and the Earth, and because of an apparent myth of an apple falling on his head, he was able to relate these events to create his notion of universal gravitation. From this, Newton used the resources and knowledge of the time to realize the force of gravity follows an inverse square law. C.) Newton's Law of Universal Gravitation "Not Just One, But ALL" Newton's Law of Universal Gravitation does not just apply to the Earth, but to ALL objects in the universe, as 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 Also, the force of gravity is directly proportional to the product of the two masses and inversely proportional to the square of the distance of separation. D.) Cavendish and the Value of G "The G-Man" Isaac Newton's equation did not explain what the value of G is, and it was not until a hundred years later before this value was determined. In 1798, Lord Henry Cavendish, using a torsion balance, calculated each of the values that can be measured on Earth. 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 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)

E.) The Value of G "This is Not Your Father's Big G" Little "g" refers to the force of gravity being exerted on an object, and it is not to be mixed with big "G", which represents the universal gravitation constant. The value of "g" is 9.8 m/s2 at sea level on Earth, but this value changes the farther away from the object in question is from the Earth, if you are comparing the object to Earth. The equation to find this is.

__1/4/12 (physicalworld.org)__
Part 1.) Mechanics and Determinism "Heliocentric vs. Geocentric" In a time period when almost everybody was in favorite of the Roman Catholic Church's geocentric view of the world, where the Earth is the center of the universe, Copernicus released a heliocentric theory, where the Sun is the center of the solar system, not the Earth. Galileo, another famous astronomer, supported the heliocentric views of Copernicus, and was tried for heresy by the Church. Johannes Kepler built on Copernicus's theory, but said that instead of circles, planets moved in ellipses. These men all went against the geocentric view of the Church, and supported heliocentric views.
 * 2 The Clockwork Universe**

Part 2.) Mechanics and Determinism "A New Addition to the Math Family" While Kepler made his observations, his discoveries were undermined by the fact that numerous new discoveries in math. The biggest discovery is René Descartes's realization that problems in geometry can be recast as problems in algebra. Also, this time period marked the beginning of coordinate geometry, which represents geometrical shapes in equations.

Part 3.) Mechanics and Determinism "Isaac Newton: The Right Man in the Right Place of Time" Newton's fortune of being active in the field of physics around the time where Kepler's discoveries were unexplained and new concepts of geometry were uncovered allowed Newton to devise equations to explain Kepler's theories. Using the resources available to him, Newton explained Kepler's laws in three key point: **1.** Newton concentrated not so much on motion, as on //deviation from steady motion// - deviation that occurs, for example, when an object speeds up, or slows down, or veers off in a new direction. **2.** Wherever deviation from steady motion occurred, Newton looked for a cause. Slowing down, for example, might be caused by braking. He described such a cause as a force. We are all familiar with the idea of applying a force, whenever we use our muscles to push or pull anything. **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.

Part 4.) Mechanics and Determinism "Mechanics and Determinism" Newton proposed a single law for gravity that worked for all matter in the Universe. From Newton's discoveries, a detailed and comprehensive study called mechanics was created. The detailed characters of Newtonian Laws made them predictable, and this characteristic was called determinism.

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A.) Kepler's Three Laws "The Beginning of Something More" After the death of his mentor, Tycho Brahe, Johannes Kepler summarized the findings of Brahe into three laws. These became known as Kepler's Three Laws of Planetary Motion. The first law states that the path of the planets about the sun is elliptical in shape, with the center of the sun being located at one focus. This law is also called the law of ellipses because of the statement about the elliptical orbits of the planets. The second of Kepler's three laws, also known as the law of equal areas, says that 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. Last but not least, Kepler's third law, or the law of harmonies, tells us that 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. Although Kepler's explanations to why his laws are true may no longer be accepted today, the laws were actually accurate depictions of the motions of any planet or satellite.
 * Lesson 4: Planetary and Satellite Motion**

B.) Circular Motion Principles for Satellites "Saturday with Satellites" There are two types of satellites in outer space: natural satellites and man-made satellites. A natural satellite can be anything produced by nature that orbits another object, such as the Moon orbits the Earth, while man-made satellites are those hulking pieces of metal humans create and send into outer space. The key to understanding the motion of satellites is that a satellite is a projectile, because we can assume that the only force acting on these satellites orbiting whatever they are orbiting is gravity. If the object being launched is not at high enough speeds to break the gravitational pull of the planet, the object will begin to orbit the planet it was launched from. The higher the speed, the more elliptical the orbit will become.

C.) Mathematics of Satellite Motion "Behind the Scenes With the Force of Gravity" The force of gravity can be represented through this formula, Fgrav = ( G • Msat • MCentral ) / R2, in which "G" represents the universal gravitational constant, "M" represents the masses of each object, and R represents the distance between the two objects. The acceleration and velocity of the object orbiting the other object can be found using this equation, and the derived form of acceleration, velocity, and the relationship between the radius of the planet and the period becomes.

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D.) Weightlessness in Orbit "A Common Misconception" Weightlessness is something numerous people feel they understand but are actually confusing some things. Many people believe that weightlessness is caused by there being no weight force, but in reality, there is actually no normal force.
 * Lesson 4: Planetary and Satellite Motion**

E.) Energy Relationships for Satellites "Satellites and Energy" 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. 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 doingwork upon the satellite. The governing principle of analyzing motion is the work-energy theorem. This 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 can be written as the equation on the bottom. **KEi + PEi + Wext = KEf + PEf**