Ch5_LeibowitzG

=**__Lesson 1-__** **//Motion Characteristics for Circular Motion//** = **(Method 5)** toc
 * **Speed and Velocity**
 * **Acceleration**
 * **The Centripetal Force Requirement**
 * **The Forbidden F-Word**
 * **Mathematics of Circular Motion**

**Speed And Velocity Come Into Play Once Again** We have been dealing with the concepts of speed and velocity since unit one. It’s no surprise that in this unit, we see these same concepts and principles and use them to describe the motion of objects moving in circles. This can then be applied to every day situations. For example, the motion of a car can be described as experiencing uniform circular motion, or circular motion at a constant speed. Furthermore, when moving in a circle, the car crosses a distance around the perimeter of the circle. In other words, if a car is moving with a constant speed of 5m/s, it would travel 5 meters along the perimeter, or circumference, in each second of time. But, just because objects moving in uniform circular motion will have a constant speed doesn’t mean it will have a constant velocity. As we know, velocity, a vector quantity, and speed, a scalar quantity, are two different concepts. Since a vector quantity includes direction, and an object is moving in a circle, the direction is continuously changing. This is described as being tangential. This is because the direction of the velocity vector at any instant is in the direction of a tangent line drawn to the circle at the object’s location.

**Acceleration To The Next Level** It is a common misconception that there is no acceleration for an object moving in uniform circular motion since the speed is constant. However, this is not true. The truth is that an accelerating object is an object that is changing its //velocity//, in this case, a change in its direction. To find acceleration in these cases, one must use the equation final velocity – initial velocity / time. To find final velocity, one must draw two tangent lines to the circle in order to determine two different velocities that will serve as the initial and final. Once this is achieved, plugging the values into the equation will complete the puzzle!

**Centripetal Forces Are More Simple Than They Sound** Although they sound complicated, centripetal forces are pretty basic. A centripetal force is the force that is acting inwards towards the system in order to cause its inward acceleration. The law of inertia comes into play here, stating that objects in motion tend to stay in motion with the same speed and direction unless acted upon by an unbalanced force. Therefore, the presence of an unbalanced force is definitely necessary for objects to move in circles.

**Don’t Get Caught Using The F-Word** When learning about circular motion, do not get confused between the terms centrifugal and centripetal! While these two terms sound very similar, they are not. Centrifugal forces move away from the center or outwards. However, objects in circular motion are NOT experiencing an outward force. There is an INWARD-directed acceleration that demands an INWARD force. An easy way to remember to avoid centriFugal forces is to remember to stay away from the forbidden F-word!

**The Math Behind It All** <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Just like every other aspect in physics, circular motion requires mathematic understanding as well. A key equation to know is that average speed equals 2(pi)(R) all over 2. R represents the radius and T represents the period. Another important equation is that acceleration equals 4(pi^2)(R) all over T^2. Finally, since net force is related to the acceleration of the object, in order to find net force you simply take the above equation and multiply it by the mass. All these equations are derived from the previous equations we have worked with in the past.

=<span style="font-family: 'Times New Roman',Times,serif; font-size: 22px;">**__Lesson 2-__** **//Applications of Circular Motion//** = <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">**(Method 1)**
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">**Newton's Second Law- Revisited**
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">**Amusement Park Physics**
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">**Athletics**

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">__**Newton's Second Law- Revisited**__ <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Circular motion principles can be combined with Newton's second law to analyze a physical situation <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">* Ex: consider a car moving in a horizontal circle on a level surface

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">- The net force acting upon the object is directed inwards <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">- Since the car is positioned on the left side of the circle, the net force is directed rightward <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">- 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) <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">- The friction force that supplies the centripetal force requirement for the car to move in a horizontal circle

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">__**Amusement Park Physics**__ <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">- The most obvious section on a roller coaster where centripetal acceleration occurs is within the so-called clothoid loops - a section of a spiral in which the radius is constantly changing <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">- 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 <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;"> - There is a continuous change in the direction of the rider as she moves through the clothoid loop <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">- An increase in height results in a decrease in kinetic energy and speed and vice versa <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">- Knowing this, the magnitude of the force of gravity acting upon the passenger (or car) can then easily be found using the equation Fgrav = m•g

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">__**Athletics**__ <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;"> Circular motion is common to almost all sporting events <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">- Circular motion in sports is characterized by an inward acceleration and caused by an inward net force <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">- Ex: A downhill skier makes her turn by leaning into the snow. The snow pushes back in both an inward and an upward direction - balancing the force of gravity and supplying the centripetal force

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">- Such an object is moving with an inward acceleration - the inward direction is towards the center of whatever //circle// the object is moving along <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">- There would also be centripetal force requirement for such a motion <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">- There is both a horizontal and a vertical component resulting from contact with the surface below <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">- 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

=<span style="font-family: 'Times New Roman',Times,serif; font-size: 22px;">**__Lesson 3-__** **//Universal Gravitation//** = <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">**(Method 2a)**
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">**Gravity is More than a Name**
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">**The Apple, the Moon, and the Inverse Square Law**
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">**Newton's Law of Universal Gravitation**
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">**Cavendish and the Value of G**
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">**The Value of g**

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">**1. What specifically did you read that you already understood well from our class discussion? Describe at least 2 items fully.** <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">In class we already discussed the concept of gravity. Therefore, reading this material I was already aware of the fact gravity is a force that exists between the Earth and the objects that are near it. This force of gravity acts upon our bodies as we jump upwards from the Earth. As we rise upwards, the force of gravity slows us down versus when we fall back to Earth, the force of gravity speeds us up. Therefore, the force of gravity causes an acceleration of our bodies. In addition to this, we also discussed in class the meaning of "g." The acceleration of gravity, or g, is acceleration experienced by an object when the only force acting upon it is the force of gravity. We previously learned that on/near Earth;s surface, the value for "g" is approximately 9.8 m/s/s. This is FOR ALL OBJECTS.

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">**2. What specifically did you read that you were a little confused/unclear/shaky about from class, but the reading helped to clarify? Describe the misconception you were having as well as your new understanding.** <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Today we did a problem in class which had no numbers and merely dealt with variables. It became somewhat overwhelming and confusing. However, in this reading, it shows not only a step by step problem with the variables, but then substitutes numbers in their place which clarifies things for me tremendously.

<span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">**3. What specifically did you read that you still don't understand? Please word these in the form of a question.** <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">Everything was clear and I have no questions.

<span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">**4. What specifically did you read that was not gone over during class today?** <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">In class we did not yet discuss the inverse square law. The inverse square law explains that the relationship between the force of gravity between the earth and any other object and the distance that separates their centers can be expressed by the following relationship: Fgrav ~ 1/d^2. Since the distance is in the denominator, it is said that the force of gravity is inversely related to the distance. And since the distance is raised to the second power, it is said that the force of gravity is inversely related to the square of the distance.

=<span style="font-family: 'Times New Roman',Times,serif; font-size: 22px;">**__Lesson 2-__** **//The Clockwork Universe//** =

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;"> **(Method 4)**

<span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">**1. What is meant by the term "heliocentric"?** <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Copernicus launched a scientific revolution by rejecting the previous Earth-centeredview of the universe in favor of a heliocentric view. This claimed that the Earth moved around the Sun. This set the scene for a number of confrontations between the Catholic church and some of its more independently minded followers. This is because the heliocentric view removed Earth, and therefore, humankind, from the "center of creation."

<span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">**2. What are the three views of planetary motion?** <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">1. The Earth-centered view of the ancient Greeks and of the Catholic church in the sixteenth century <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">2. The Copernican system, in which the planets move in collections of circles around the Sun <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">3. The Keplerian system in which a planet follows an elliptical orbit, with the Sun at one focus of the ellipse

<span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">**3. How does this concept involve mathematics?** <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Kepler's ideas led to new discoveries in mathematics, mainly by Descartes, who realized that problems in gemoetry can be recast as problems in algebra. He developed a two-dimensional coordinate system that was used to locate the position of any point in terms of its x and y coordinates. This coordinate system could also be used to represent lines and other geometrical shapes by equations. This is characterized by an equation stating the square root of x squared plus y squared is equal to 2cm. This began the branch of mathematics known as coordinate geometry.

<span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">**4. Explain Newton's beliefs and findings.** <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Newton's central belief was that all motion can be explained in terms of a single set of laws. He concentrated no as much on motion as he did on a concept known as deviation from steady motion, which occurs when an object speeds up, or slows down, or veers off in a new direction. He tried finding the cause of this, which he described as being the presence of a force. Finally, he produced a quantitative link between force and deviation from stead motion. This led to his proposition of his well-known law of universal gravitation.

<span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">**5. What is meant by the terms "mechanics" and "determinism"?** <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">Newton's discoveries became the basis for the study of mechanics, or the study of force and motion. These Newtonian laws were characterized by the fact that once this "clockwork" had been set in motion, its future development was entirely predictable, a property known as determinism.

<span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">* In summary, this reading goes into detail about the history behind the all the laws we studied this chapter in physics. It goes chronologically through the original, and inaccurate, beliefs of how our solar system functioned to the current and well-known theorems. Knowing this information definitely aids a physics student in their studying because it gives one insight into the thinking process that went behind determining these laws. It also gives one perspective of the meaning behind these laws, preventing students from merely memorizing the equations.

=<span style="font-family: 'Times New Roman',Times,serif; font-size: 22px;">**__Lesson 4-__** **//Planetary and Satellite Motion (a-c)//** = <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">**(Method 2a)**
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px; line-height: 24px;">**Kepler's Three Laws**
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">**Circular Motion Principles for Satellites**
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">**Mathematics of Satellite Motion**

<span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">**1. What specifically did you read that you already understood well from our class discussion? Describe at least 2 items fully.** <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">In class, we already discussed Kepler's three laws of planetary motion, in addition to the history behind it. Kepler was able to summarize the collected data of his mentor, Brahe, with three statements that described the motion of planets in a sun centered solar system. The first one explains that the path of planets about the sun is elliptical in shape, with the center of the sun being located at one focus. The second law states 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. Finally, the last law states 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. Together, these laws are known as the Law of Ellipses, the Law of Equal Areas, and the Law of Harmonies. In addition, in class we also discussed the equation for the velocity of a satellite moving about a central body in circular motion. The equation for this, as both mentioned in class and in the reading, is v is equal to the square root of G times M(central) all over R.

<span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">**2. What specifically did you read that you were a little confused/unclear/shaky about from class, but the reading helped to clarify? Describe the misconception you were having as well as your new understanding.** <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">Today in class, we discussed the deviation for how you reach the equation for force of gravity. I was a little shaky about how the equation came about, but after doing this reading, and seeing the step-by-step process of how they reached the final equation, this concept is much more clear. I now understand how the equation F(net) = (Msat * v^2) / R is basically, the same as v^2 = (G*Mcentral) / R. This will make solving equations but easier, for I will be able to logically rearrange the equation as necessary, opposed to merely memorizing formulas.

<span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">**3. What specifically did you read that you still don't understand? Please word these in the form of a question.** <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">Everything was clear and I have no questions.

<span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">**4. What specifically did you read that was not gone over during class today?** <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">In class we did not yet discuss the inclusion of force vectors into our problem solving. The reading expands upon what we learned in class, especially in the area of 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 o the circle at every point along its path. In addition, the acceleration of the satellite would be directed towards the center of the circle. In other words, it would be directed towards the central body that it is orbiting. This is caused by a net force, directed inwards in the same direction as acceleration.

=<span style="font-family: 'Times New Roman',Times,serif; font-size: 22px;">**__Lesson 4-__** **//Planetary and Satellite Motion (d-e)//** = <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">**(Method 4)**
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%; line-height: 24px;">**Weightlessness in Orbit**
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 120%; line-height: 24px;">**Energy Relationships for Satellites**

<span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">**1. What is the difference between contact and non-contact forces?** <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">As one sits in a chair, they experience two forces, the force of teh Earth;s gravitational field pulling downward toward the Earth and the force of the chair pushing one upward. The upward chair force is referred to as a normal force which is further categorized as a contact force. Contact forces result from the actual touching of the two interacting objects ONLY. The force of gravity acting upon one's body is not a contact force, but instead, an action-at-a-distance force. This force would even exist if one were not in contact with the Earth. Contact forces include friction, tension, and normal forces. The force of gravity is an example of an action-at-a-distance force.

<span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">**2. What is the meaning of weightlessness and how is it caused?** <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">Weightlessness is a sensation experienced by an individual when there are no external objects touching one's body and exerting a push/pull upon it. This sensation exists when all contact forces are removed. One would feel weightless when in a state of free fall. A common example of feeling weightlessness is on the top of a very high roller coaster. The force of gravity is the only force acting upon one's body in this scenario because normal forces only result from contact with stable, supporting surfaces.

<span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">**3. What is the misconception about scale readings?** <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">A scale does not actually measure ones weight. While people use scales to measure one's weight, the reading is actually a measure of the upward force applied by the scale to balance the downward force of gravity acting upon it. When an object is in a state of equilibrium, these two forces are balanced. Therefore, the upward force of the scale upon the person equals the downward pull of gravity. So, in this instance, the scale reading equals the weight of the person, if they are standing perfectly still on the scale. This clearly changes if the person were to move when on the scale.

<span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">**4. Why does there exist weightlessness during an orbit?** <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">An astronaut orbiting the Earth would be weightless. There is no external contact force pushing or pulling upon their body. Gravity is the only force acting upon their body, and since it is n action-at-a-distance force, it cannot be felt and therefore, would not provide any sensation of their weight.

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">Motion can be looked at from an energy perspective. This revolves around the work-energy theorem, stating that 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 equation is: KEi +PEi + Wext = KEf + PEf. Wext represents the amount of work done by external forces. For satellites, the only force is gravity, and the Wext term is zero. Therefore, the equation is simplified to KEi + PE i = KEf + PEf.
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 16px;">5. Describe the relationships that exist for satellites. **

<span style="font-family: 'Times New Roman',Times,serif; font-size: 120%;">* In summary, this reading first focuses on the concept of weightlessness. Before doing this, the reading briefly introduces previously learned concepts of contact and action-at-a-distance (or non-contact) forces. Weightlessness is basically a feeling in which there is no objects touching one's body, and therefore, no push or pull upon it. The reading gives the example of a roller coaster. When at the top of a very high roller coaster, this feeling of weightlessness is present, in which the normal force is equal to zero. After establishing the definition of weightlessness, the reading goes into detail about WHY weightlessness exists during an orbit. This is for the same reason as the roller coaster, in which gravity is the only force acting on, an astronaut, for example, and therefore, no other force to exert a force. The next reading focusses on satellites, and centers upon the relationships that exist between the variables in the work-energy theorem equation provided.