Ch5_RubensteinD

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**Chapter Five**

__**Homework 12/13/11 Lesson 1: Motion Characteristics for Circular Motion**__
Like we have learned before, speed and velocity are NOT the same thing. While speed is a scalar quality, which only requires a magnitude, velocity is a vector quality, which requires a magnitude in addition to a direction in which the object is going. Although we already knew this concept, we now can see how it is applied to circular motion. For example, the distance now has to do with the circumference of the circle and velocity is found by using the direction of the tangent line at every instant of the circle. Also, the velocity quality is always changing as the object moves in a circular motion.
 * a. Speed and Velocity, Scalar and Vector**

Acceleration, like we already learned, is the change in velocity over the change in time. However, while dealing with circular motion, it is much harder to understand where the different velocities are coming from. Since the velocity is always changing, one would have to draw two tangent lines, one being line a and one being line b, in order to determine two different velocities to use as the initial velocity and final velocity.
 * b. Acceleration: Same Term, Different Application**

The centripetal force is the force that is acting inwards towards the system in order to cause its inward acceleration. For object's in circular motion, there is a net force acting towards the center which causes the object to seek the center. As this was not learned before, one could see that the centripetal force is something that relates closely to circular motion.
 * c. Centripetal Force? Where Does That Come in!**

Centrifugal, which unfortunately always gets confused with centripetal, is a force going AWAY from the center of the circle. This mix up is the common misconception by most when it comes to learning about circular motion. This is a common mistake because people often think that this means that there is an outward force, however, THIS IS NOT TRUE! The important concept to remember is the centripetal force, and this can be remembered since the F in centrifugal makes it the forbidden F-word!
 * d. Centrifugal vs. Centripetal: What's the Difference?**

Just like any other application, circular motion as new equations that are used to solve the mathematical problems. Average speed equals 2(pi)(R)/t and acceleration equals 4(pi)^2R/t. Although these equations look different, they really come from the same place as the old equations, however, they just have to be different since circles deal with radii and the radius.
 * e. Less Concepts, More Math!**

__**Homework 12/11/11 Lesson 2: Applications of Circular Motion (Method 4)**__
- Just like previously learned, it still has to due with equations - However, the equations differ slightly than the old ones - a = (m^2) / R - Equations are used to solve circular motion problems - These principals along with circular motion concepts are used to analyze a variety of physical situations involving the motion of objects in circles or along curved paths
 * 1. What does Newton's Second Law have to do with circular motion and how can it be applied?**

- Circular motion has to due with sharp 180 degree banked turns and small dips and hills, and this obviously has to due with roller coasters - Centripetal acceleration is found on a roller coaster within the clothoid loops - A clothoid is a section of a spiral in which the radius is constantly changing - There is always a change in direction and this means there is always acceleration - There is a component directed to the center and a component directed to the tangent - Minimum and maximum speeds always need to be found
 * 2. How does physics and circular motion apply to amusement parks?**

- Circular motion is common to almost all sporting events - Everything is caused by an inward net force and characterized by an inward acceleration - Most common application is the turn; when people are running and turn - Because turning a corner involves the motion of an object that is momentarily moving along the path of a circle, concepts and the mathematics of circular motion can be applied to such motion - The 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 - Another example is an ice skater on a turn
 * 3. How does physics and circular motion apply to athletics?**

In this lesson I learned that Newton's second law can be applied to circular motion as well to help solve application problems. Also, I was able to see real examples of how this concept applies to life. For example, it explained the impact physics and circular motion has on amusement parks as well as athletics.

__Homework 1/3/12 Lesson 3: Universal Gravitation (Method 1)__
Gravity is the name associated with the reason for "what goes up, must come down." Gravity is the thing that causes objects to fall to Earth. Gravity must be understood in terms of its cause on the structure and the motion of the objects in the universe. We have become accustomed to calling gravity the force of gravity and have even represented it by the symbol F grav. In fact, many students have become accustomed to referring to the actual acceleration of such an object as the acceleration 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).
 * Part A:**

In the early 1600's, mathematician and astronomer Johannes Kepler 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 cause for how the planets moved as they did was never stated. 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. To Kepler, the planets were somehow "magnetically" driven by the sun to orbit in their elliptical trajectories. There was however no interaction between the planets themselves. 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 was 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.
 * Part B:**
 * 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)

Newton compared the acceleration of the moon to the acceleration of objects on earth. Believing that gravitational forces were responsible for each, Newton was able to draw an important conclusion about the dependence of gravity upon distance. 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. But Newton's law of universal gravitation extends gravity beyond earth. **ALL** objects attract each other with a force of gravitational attraction. This force of gravitational attraction is directly dependent upon the masses of both objects and inversely proportional to the square of the distance that separates their centers. 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. If the mass of one of the objects is doubled, then the force of gravity between them is doubled; 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. The constant of proportionality (G) is known as the universal gravitation constant. The value of G is found to be 6.673 x 10^-11. The units on G may seem rather odd; nonetheless they are sensible. When the units on G are substituted into the equation above and multiplied by m 1 • m 2 units and divided by d 2 units, the result will be Newton’s. Knowing the value of G allows us to calculate the force of gravitational attraction between any two objects of known mass and known separation distance.
 * Part C:**

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. 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. Cavendish had calibrated his instrument to determine the relationship between the angle of rotation and the amount of torsional force. 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 m 1, m 2 , d and F grav , the value of G could be determined. Cavendish's measurements resulted in an experimentally determined value of 6.75 x 10 -11 N m 2 /kg 2. Today, the currently accepted value is 6.67259 x 10 -11 N m 2 /kg 2. 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.
 * Part D:**

In the original equation, g is referred to as the acceleration of gravity. Its value is 9.8 m/s 2 on Earth. 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.
 * Part E:**

__Homework 1/5/12 The Clockwork Universe__
- Isaac Newton had great contributions, however, he was not involved in the initial discovery - Nicolaus Copernicus launched a scientific revolution by rejecting the prevailing Earth-centred view of the Universe in favor of a heliocentric view - Copernicus set the scene for a number of confrontations between the Catholic church and some of its more independently minded followers - Most famous was Galileo who was summoned to appear before the Inquisition for following Copernicus - Johannes Kepler devised a modified form of Copernicanism that was in good agreement with the best observational data 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 lead to new discoveries in mathematics - The equation of the circle was found, which launched a beginning of a branch of mathematics called coordinate geometry - This gave scientists new ways of tackling geometrical problems, allowing them to go further than the greatest mathematicians of ancient Greece - The new astronomy called for a new physics which Newton had the ability and the opportunity to devise - Newton's great achievement was to provide a synthesis of scientific knowledge - He discovered a convincing quantitative framework that seemed to underlie everything else - Before Newton, few could have imagined that such a world-view would be possible - Newton concentrated not so much on motion, as on deviation from steady motion - Wherever deviation from steady motion occurred, Newton looked for a cause - Newton produced a quantitative link between force and deviation from steady motion - Newton proposed just one law of gravity - a law that worked for every scrap of matter in the Universe - By combining this law with his general laws of motion, 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 described
 * 1. Who were the main people involved in discovering the Universe**
 * 2. How was mathematics involved?**
 * 3. How did Newton contribute to what was already known?**
 * 4. What was Newton's single law of gravity and how is that still in use today?**

After completing this homework, I talked to other people in Honors Physics to see if they learned the same things I did and if there was anything they learned that I missed. After, I talked to my family about what I learned and asked if they knew anything about it as well.

__Homework 1/6/12 Lesson 4: Planetary and Satellite Motion a-c__
- Kepler was known for creating three different laws - They were added onto by Newton later - The Law of Ellipses: the path of the planets about the sun is in elliptical shape, with the center of the sun being located at one focus. - The Law of Equal Areas: 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 Harmonies: the ratio of the squares of of the periods of any two planets is equal to the ratio of the cubes of their average distances from the sun - A satellite is any object that is orbiting the earth, sun, or any other massive body - It can be either natural or man made - They move in an orbit - They act similarly to projectiles as gravity is the single force acting upon it - The motion of satellites can be described by its' acceleration and velocity - These two components are constantly changing - Satellites also move in an elliptical motion - The final equation for velocity of a satellite moving about a central body in circular motion is:
 * 1. What are Kepler's three laws?**
 * 2. What are circular motion principles for satellites?**
 * 3. Where does mathematics come into satellite motion?**

After taking notes on this lesson, I explained what I learned to a friend from another town who is learning the same thing but had trouble understanding. We mainly talked about how mathematics came into play and how it could be applied to real life situations.

__Homework 1/9/12 Lesson 4: Planetary and Satellite Motion d-e__
**1. What is weightlessness and how does it apply to the situation?** - Sensations are the same for both astronauts and regular people on a roller-coaster - Additionally, the causes for these sensations are the same as well - Weightlessness is simply a sensation experienced by an individual when there are no external objects touching one's body and exerting a push or a pull on it - Weightlessness is ONLY a sensation - It has very little to do with weight and mostly to do with the presence or absence of contact forces - In addition, a scale does not actually measure one's weight - Earth-orbiting astronauts are weightless because there is no external contact force pushing or pulling on their body - There are many misconceptions concerning this concept - The orbits of satellites about a central massive body can be described as either circular or elliptical - There is no acceleration in the tangential direction and the satellite remains in circular motion 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 speeding up or slowing down the satellite - Work-energy theorem- the initial amount of total mechanical energy of a system plus the work done by external forces on that system is equal to the final amount of total mechanical energy of the system - The mechanical energy can be either in the form of potential energy or kinetic energy - A work energy bar chart represents the energy of an object by means of a vertical bar - A satellite orbiting in elliptical motion will speed up as its height or distance of from the earth is decreasing and will slow down as they both are decreasing
 * 2. What are energy relationships for satellites? **

From taking notes on this lesson I learned things about both weightlessness and the energy relationships for satellites. Weightlessness is the same for all people, whether they are orbiting in space or just sitting on a roller coaster. Also, weightlessness is ONLY a sensation and is not an actual physics. Finally, weightlessness as little to do with weight and more to do with the presence or absence of contact forces. Next, I learned that orbits of satellites could be both circular or elliptical. Additionally, work energy bar charts can be used to represent the energy of an object. Finally, a satellite orbiting in elliptical motion will speed up as the height and distance decrease and will slow down when they increase.