Ch5_WangS


 * Chapter 5: Circular Motion and Satellite Motion**

Lesson 1: Motion Characteristics for Circular Motion
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a) Speed and Velocity

 * Headline: Find out //now// how speed and velocity are different.
 * When an object is traveling in a circle at a constant speed, it is called uniformed circular motion. It is important to note that even thought the speed remains constant, the acceleration changes due to the changing directions. Also, its average speed can be calculated with the formula (2*pi*r)/t. This can be simplified into circumference/time. Also, the direction of the velocity at each moment in a direction is tangent to the circle.

b) Acceleration

 * Headline: We can't forget about acceleration!
 * Because objects moving in uniform circular paths are constantly changing direction, they have an acceleration. The average acceleration can be found by subtracting the initial velocity from the final velocity and dividing this quantity by the time. Also, the object always accelerates towards the center of the circle. Smaller masses experience greater acceleration because they have less inertia. This makes them lean into the direction of acceleration.

c) The Centripetal Force Requirement

 * Headline: And now to add something new-- centripetal force
 * All objects that move in a uniform circular path have a "centripetal force requirement." This basically means that in order for an object to accelerate inward, there must be a force present. This force can come in different forms, such as tension, weight, or friction. Because of Newton's First Law of Motion, only an unbalanced force can move an object in a circle. This force is always perpendicular to the tangential velocity, which lets the direction of the object's velocity vector change directions without changing magnitude.

d) The Forbidden F-Word

 * Headline: Want to know about a word that physicists both fear and hate?
 * Centrifugal means outward and away from the center. This force does not exist for objects moving in a circular path, since they all lean towards the middle and inwards. So, this is the forbidden f-word. However, there is an inward-directed acceleration that requires an inward force. This is centripetal force. Without this inward force, an objects would keep traveling in a straight line tangent to the perimeter of the circle. In this scenario, there would be no circular motion. Although at first it may look like an object is being pushed outward, that is only relative and not real.

e) Mathematics of Circular Motion

 * Headline: Here are some formulas for circular motion
 * You can calculate the speed that an object is moving (in a circular motion) by using the equation (2*pi*r)/t. You can calculate the acceleration by using the equation (4*pi^2*radius)/t^2. The net force acting on an object, directed inwards, can be found using the equation mass*(4*pi^2*radius)/t^2. As the acceleration of the object increases, the net force also increases. As the mass increases, the acceleration decreases. Also, if the mass and radius are constant, then the net force is proportional to the speed squared.

a) Newton's Second Law - Revisited

 * Headline: Newton's Laws are everywhere, and they affect circular motion too!
 * Newton's Second Law applies to circular motion too! Here's what you do: draw a free body diagram. Then, find the direction of centripetal force and use the equation fc = mac. Since we already know what centripetal acceleration is, the equation simplifies to fc=mv^2/R. Use fx = max and fy = may if necessary to obtain other information (such as friction).

b) Amusement Park Physics

 * Headline: Attention! Roller coasters use circular motion
 * There are two different sections of a roller coaster. There are hills/dips and there are loops. When a roller coaster is in a loop, both the speed and direction change, since it is slowest at the top. This means that the centripetal force also changes. This explains why someone riding a roller coaster feels heaviest at the bottom --because there is the most normal force. There is the same concept behind hills and dips. The amount of normal force also depends on the radius of the circle. The larger the radius, the smaller the acceleration, and the smaller the normal force.

c) Athletics

 * Headline: What does skating have to do with circular motion? Find out here!
 * These types of problems are relevant to sports. Sports such as skating, race car driving, and skiing that involve turns also involve a centripetal force. This usually comes in the form of friction. If there is a leaning towards the ground, then components will be involved.

a) Gravity is More Than a Name

 * Headline: Why don't we fall up?
 * Gravity, Fgrav, pulls us towards the center of the Earth. It allows us to fall down, towards the Earth, and not float up. On Earth, the acceleration of gravity is 9.8 m/s/s.

b) The Apple, the Moon, and the Inverse Square Law

 * Headline: Hey, apples fall down too!
 * Planets move in elliptical paths. Kepler explained this along with the inverse square law. However, people didn't know why planets moved in elliptical paths. Newton explained that it was because of gravity. He first came up with this theory when an apple fell on his head. He developed the universal laws of gravitation. Newton said that ratio of acceleration between the moon and the acceleration of the apple is 1:3600. This is because the moon and center of the Earth are located 60 times farther than the apple and the center of the Earth. This makes sense because it follows the inverse square law, which states that t he force of gravity equals 1 over the square of the distance.

c) Newton's Law of Universal Gravitation

 * Headline: Planets are affected by laws, too!
 * The force of gravity is equal to a mass times a second mass divided by their distance squared. This means that more mass equals more gravity, and more distance equals less gravity. The universal gravitation constant applies this to all objects. This is G, aka, 6.673 x 10-11 N m2/kg2, which gets put on the numerator.

d) Cavendish and the Value of G

 * Headline: Ancient news: Man discovers universal gravitation constant
 * The man we are referring to is Lord Cavendish. He discovered the universal gravitation constant by using simple tools. He attached 2 small lead balls to the ends of a 2- ft rod. Then, he put bigger lead balls on the smaller lead balls. The large spheres exerted a gravitational force upon the smaller spheres and twisted the rod. When the rod and spheres came to a stop, that meant the torsional forces were balanced. Using this method, Cavendish was able to discover G.

e)The Value of g

 * Headline: What's this? 9.8 isn't always the acceleration of gravity on Earth?
 * 9.8 is only the acceleration of gravity on Earth when we are at sea level. The closer we are to the center of the Earth, the bigger this value becomes, and vice versa. This is because of the formula g=(G*Mearth**)/**d^2. G is a constant, and the mass of the Earth doesn't change, so distance is the only factor that determines acceleration. This is an inverse relationship. This means that the farther you are from the Earth's center, the lower the acceleration of gravity. The acceleration of other planets can be found with the formula g=(G*Mplanet)/(Rplanet^2). Basically, the larger the planet is, the larger g is.

The Clockwork Universe (using method 3)
Theme: Many scientists contributed to the development of ideas about the Earth, and the universe. Copernicus and Kepler were two of the first to claim that the Earth revolved around the sun, and Newton discovered gravity. These two discoveries were the basis for many for scientific discoveries regarding the world around us.
 * 1) **Why was the statement that Copernicus made so revolutionary?** Before Copernicus, people thought the Earth was the center of the universe. Copernicus stated that the Earth moved around the sun, and proposed a heliocentric model of the solar system. Many people were religious and disagreed with him. They said that everything revolved around the Earth because God created the Earth as the center of the Universe.
 * 2) **Why wasn't Copernicus shunned when he also said the Earth revolves around the sun?** Kepler lived in England, which was less religious since it wasn't controlled by the Pope. However, he didn't have the same thoughts as Copernicus. He said that the Earth moves in ellipses, not in circles as Copernicus had thought.
 * 3) **Who is Descartes?** He realized that algebra can help solve geometry problems. He made a coordinate plane system and rearranged equations to fit them on a graph. His research and ideas helped Kepler figure out that the Earth moves in an elliptical path.
 * 4) **Who is Newton?** Newton used calculus to show that the planets move in an elliptical path around the sun. He discovered gravity, and made the law of gravitation. He also said that gravity pulls the planets towards the sun.
 * 5) **What is determinism?** It is an idea that Newton first came up with. He said that all the actions of the universe, and all of our actions were predetermined in the past. This makes all future motions and developments entirely predictable.

a) Kepler's Three Laws (method 1)
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. Kepler's three laws of planetary motion can be described as follows:
 * 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)

Theme: Kepler's Three Laws are the Law of Ellipses, The Law of Equal Areas, and The Law of Harmonies.

b) Circular Motion Principles for Satellites (method 1)
Satellites are projectiles that orbit around a central massive body instead of falling into it. They are acted upon by the force of gravity - a universal force that acts over even large distances between any two masses. Newton's Laws of Motion affect satellites. 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.

Theme: Satellites, projectiles that orbit around planets and other large bodies, are acted upon by gravity. Problem solving with satellites is similar to that of circular motion.

c) Mathematics of Satellite Motion (using method 3)

 * How can you calculate the net force of satellites?
 * You use Newton's Second Law equation, F=ma. m is the mass of the satellite, and the distance between the center of the planet and the satellite is R. Then, you use the formula Fnet = ( Msat • v2 ) / R. This is derived from F = ma.
 * How can you determine out the gravity of satellites?
 * You use Newton's law of gravitation equation, which is Fg=(G*M1*M2)/R^2.
 * How can you calculate the velocity of satellites?
 * You use the equation [[image:jitskyhonorsphysics/Screen_shot_2012-01-05_at_2.17.01_PM.png caption="Screen_shot_2012-01-05_at_2.17.01_PM.png"]]
 * How can you calculate the acceleration due to gravity of satellites?
 * You can use the Equation [[image:jitskyhonorsphysics/Screen_shot_2012-01-05_at_2.17.27_PM.png caption="Screen_shot_2012-01-05_at_2.17.27_PM.png"]]


 * How can you calculate the period of satellites?
 * You use the equation[[image:jitskyhonorsphysics/Screen_shot_2012-01-05_at_2.18.02_PM.png caption="Screen_shot_2012-01-05_at_2.18.02_PM.png"]] which also helps verify Kepler's Law of Harmonies.
 * Theme: Using different formulas, you can calculate different aspects of satellite motion including net force, gravity, acceleration, and period.

d) Weightlessness in Orbit (using method 3)

 * What is weightlessness?
 * It is a sensation you feel when you are in freefall. You feel like you are weightless, even though you still have a weight. It is caused by the sensation of having nothing pushing or pulling you.
 * Contact and non-contact force: What's the difference?
 * A contact force are when two surfaces have to touch for a force. An example is normal force. Gravity is a non-contact force, since it two objects don't have to touch for there to be gravity. For example, the moon and Earth exert gravity on one another, but don't touch.
 * When we measure our weights with a scale, why is it different in outer space?
 * Scales measure the normal force (the upward force on our body) at the point of contact with the scale. This force changes if there is an acceleration, but your weight remains the same. If there is an upwards acceleration, the scale would display a higher number for weight. If there is a downwards acceleration, the scale would display a lower number for weight.
 * Where else do you feel weightless?
 * Some examples would be a free falling elevator and a free falling amusement park ride.
 * Why do you feel weightless in outer space?
 * Astronauts have no external forces acting on them, just like in freefall. Though the gravity from the planets affect the astronaut, there are no contact forces, so astronauts feel weightless.
 * Theme: Weightlessness is a sensation that a person experiences when they are in freefall. That is because the only force acting on them then is gravity, which is an at-a-distance or non-contact force.

e) Energy Relationships for Satellites (using method 3)

 * What is the work-energy theorem?
 * It says that the initial amount of the total mechanical energy, TMEi, of a system, plus the external forces,(Wext), on that system is equal to the total mechanical energy, TMEf, of the system. This mechanical energy can be in the form of potential energy or kinetic energy. The equation for this theorem is KEi + PEi + Wext = KEf + PEf
 * What is the Wext term for satellites?
 * Since this term represents the amount of work being down by external forces, it is only gravity for satellites. Also, since gravity is not an external force, but an internal (conservative) force, the Wext term is zero. This means the equation can be simplified into KEi + PEi = KEf + PEf
 * What are some characteristics of the energy analysis of elliptical orbits of satellites?
 * TME is constant
 * Wext is zero
 * The satellite moves fastest near Earth and slows down farther away from Earth
 * Kinetic energy changes
 * When it is going faster, the satellite's distance from the Earth decreases. Therefor, there is a gain of kinetic energy and a loss of potential energy
 * Mechanical energy is conserved
 * Total mechanical energy of the satellite remains constant
 * What are some characteristics of the energy analysis of circular orbits of satellites?
 * Speed is constant
 * Height above Earth is constant
 * Kinetic energy is constant
 * Potential energy is constant
 * If KE and PE are constant, TME is also constant.
 * What is a work-energy bar chart?
 * It is a way of representing the quantity and type of energy possessed by an object using a vertical bar. The length of the bar shows how much energy is 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. Observe that the bar chart depicts that the potential and kinetic energy of the satellite are the same at all four labeled positions of its trajectory (the diagram above shows the trajectory).
 * Theme: The orbit of satellites can be either elliptical or circular, with different laws guarding each one.