Chapter+5

=Physics Classroom Readings= toc

Lesson 1: Circular Motion
__Speed and Velocity of Circular Motion__ The distance traveled when in a circular motion is twice pi times the radius, or the circumference. The average speed, then, is distance over time, or circumference over time. While the speed may remain constant, the velocity is constantly changing due to the different direction. The direction of the given point is the line tangent to the circle at that point.

__Acceleration Changes in Circular Motion__ While many may have the misconception that there is no acceleration because speed is constant, this is not the case because of the changing direction. The average acceleration is found by the final velocity minus initial velocity, all over time. Since velocity is a vector there must be vector subtraction involved.

__Centripetal Forces Are Required__ A centripetal force is a force that goes in the direction towards the center of the circle, which changes as the object moves in a circular motion. Inertia is involved with this, because the motion around the circle makes the perception that acceleration is outside the circle, but it really is inside the circle. Also, work can be calculated because this allows the change of direction in velocity; it is calculated by work=force*displacement*cos(theta).

__The Forbidden F Word__ Centrifugal means outside of the circle, as opposed to centripetal. Never use this word in physics as these don't exist. A force feeling like it is going outward is only due to relativity; the object moves outward relative to another, but the forces are still moving inward.

__Mathematics of Circular Motion__ Acceleration can be calculated either by velocity squared over radius, or 4 pi squared times radius all over time squared. Thus, using F=ma, F equals m times one of the two acceleration equations.

Lesson 2: Applications of Circular Motion
__Newton Didn't Disappear Just Because It's Circular__ The process is easy. We start with doing a free body diagram, finding the direction of the centripetal force, and using the equation fc=mac. We know centripetal acceleration already, so the equation just becomes fc=mv^2/R. If other information needs to found (such as friction), use the same equation for other axes. Components may have to be made.

__RollerCoaster Loops and Dips, Oh My!__ There are 2 turns on a rollercoaster: loops and dips/hills. In a loop, speed and direction changes when going up and down (slower at the top), so the centripetal force changes. One feels more weight at the bottom due to increased Normal Force. The same applies for hills and dips. Normal force is also dependent on the radius of the circle (the bigger the radius, the smaller the acceleration, the smaller the normal force).

__We're not the Only Ones Who Need to Know This!__ Circle problems also apply to athletics, like how sharp to make a turn in skating, racecar driving, and skiing. For the turn to occur there has to be a centripetal force, usually friction. If there is leaning onto the ground at an angle, there will be components used, and there will be tangent involved.

Lesson 3: Universal Gravitation
__What Goes Up, Must Go Down__ Gravity is the force that validates the statement above. Gravity, Fgrav, is what pulls us to the center of the Earth. That is why, when we jump up, we get pulled back down. On Earth, the value for acceleration of gravity is 9.8 m/s/s.

__And Then The Apple Hit My Head__ Kepler explained how planets moved in an elliptical path, as well as the inverse square law, but it was unknown how such a path existed. Newton reasoned that it was gravity, the same force that caused an apple to hit his head, and this led to his universal laws of gravitation. Newton also realized that the ratio of acceleration between the moon and the acceleration of the apple is 1:3600. That is because the distance between the moon and the center of Earth is 60 times greater than the apple and center of Earth. This verifies the inverse square law, where the force of gravity equals 1 over the square of the distance.

__Laws Regulate Planets, Too!__ According to Newton, the force of gravity is equal to a mass times a second mass divided by their distance squared. Increasing mass increases gravity, and increasing the distances decreases gravity. We must, however, add the universal gravitation constant to make it apply to all objects. This is G, or 6.673 x 10-11 N m2/kg2, which gets put on the numerator.

__Why So Small?__ In 1798, Lord Cavendish discovered the universal gravitation constant. To do this he used a rod 2-feet long. Two small lead spheres were attached to the ends of the rod. The rod was suspended by a thin wire. Then he attached two large lead spheres by the smaller spheres. The large spheres exerted a gravitational force upon the smaller spheres and twisted the rod. 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. G was determined.

__The Well-Known 9.8 Doesn't Always Work__ The acceleration of gravity is 9.8 at the surface of Earth, the sea level. The farther away one is from the surface, the lower this value becomes. This is derived from g=(G*Mearth**)/**d^2. Since G and the mass of the Earth are constant, distance is the only factor determining the acceleration, and it is an inverse relationship (the farther away the lower the acceleration). The acceleration value for other planets can be derived by g=(G*Mplanet)/(Rplanet^2). The larger the planet the larger g is.

The Clockwork Universe
__What was the revolutionary statement that Copernicus made, and what were its reactions?__ __Why was Kepler able to press his statements without regret, and what were his ideas?__ __Which scientist was needed before Kepler to allow him to do as he did?__ __What did Descartes do?__ __How did Newton verify and go beyond Kepler's ideas?__ __What is determinism?__
 * He rejected the Earth-centered view of the universe and suggested a heliocentric model of the universe where the Earth moved around the sun. Mostly religious people disagreed with this model because of their belief that people and Earth was the center of creation by God. They could not accept that there was another way.
 * Kepler was in England, where it was Protestant and not controlled by the Pope. He suggested that planets move in ellipses around the sun, not in circles like Copernicus said. He speculated the reason for this was magnetic (we know this is false).
 * Renee Descartes
 * Descartes realized that geometry could be solved using algebra. He used coordinate geometry and rearranged equations to see how they fit on the graph.
 * Newton mathematically demonstrated (with calculus) that the planets moved around the sun in an ellipse. Newton also proposed a law of gravitation. It was gravity that pulled the planets toward the sun.
 * Once a clockwork has been set for the universe, its future motion and development is entirely predictable. In other words, our actions and the actions of the universe have already been predetermined in the past. This is an idea by Newton.

Lesson 4: Planetary and Satellite Motion
__Part A:__ __Part B:__ __Part C:__
 * What is Kepler's Law of Ellipses?
 * This law explains how planets orbit the sun in an ellipse. That is, there are two foci, and the sum of the distances from the two foci to a point on the ellipse will always be equal. One of the foci is the sun, and there is nothing on the other.
 * What is Kepler's Law of Equal Areas?
 * This law states how the speed of planets orbiting the sun always changes. It is slowest when furthest from the sun and fastest when closest to the sun.Triangles can be formed from the sun to the Earth at two point during its revolution; do this for equal intervals all around the ellipse. The areas of the triangles will be equal, showing that Earth travels faster when closer to the sun.
 * What is Kepler's Law of Harmonies?
 * This law states that the ratio of the square of the period and the cube of the average distance from the sun is the same for all of the planets.
 * What is a Satellite?
 * A satellite is any object that orbits around a massive object (does not fall in). They can be manmade or artificial. They follow parabolic trajectories. Since the satellite is launched with enough speed, it falls towards Earth at the same rate Earth curves (cancelling each other out). So it seems as though the satellite is orbiting Earth.
 * What are the Velocity, Acceleration, and Force vectors of satellites?
 * The velocity vector is tangent to the circle of Earth along its path. The acceleration is pointed towards the center of Earth. The centripetal force is supplied by gravity. Without gravity, the satellite would continue its path in a straight line.
 * How do elliptical satellite patterns work?
 * Earth is at one foci. The acceleration and velocity are in the same way. Gravity also supplies the centripetal force. There is, however, an additional force in the same or opposite direction as the motion of the object.It can make the satellite speed up or slow down in different places.
 * How do we calculate the net force of satellites?
 * Taking Newton's second law equation F=ma, we can say that mass is the mass of the satellite, and R is the distance between the center of the planet and the satellite.
 * Fnet = ( Msat • v2 ) / R
 * How do we calculate the gravity of satellites?
 * Taking Newton's law of gravitation equation and the constant G, we can use the masses of the planet, Mcentral, and the mass of the satellite, Msatellite. So we use the equation Fg=(G*M1*M2)/R^2.
 * Fgrav = ( G • Msat • MCentral ) / R2
 * How do we calculate the velocity of satellites?
 * Since Fg=Fnet,( Msat • v2 ) / R = ( G • Msat • MCentral ) / R2. This leaves us with the the equation for velocity:
 * [[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 do we calculate the acceleration due to gravity of satellites?
 * We can simply derive this from the equation for the value of g, which is the same equation as the below.
 * [[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 do we calculate the period of satellites?
 * Using the velocity 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"]], we can square this to make it easier. Then we can say that velocity is 2*pi*R over T, and plug that in for v, then square it. That equals the velocity equation squared, and make this an equation. That is how we get this equation to find the period.
 * [[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"]]
 * This also verifies Kepler's Law of Harmonies.

__Part D:__
 * What is the difference between a contact and non-contact force?
 * Contact forces only come from direct contact between two surfaces. Gravity is a non-contact force, or an at-a-distance force. I can be all the way in outer space and still be pulled by Earth with gravity. It does not matter because it is not a contact force.
 * What causes weightlessness?
 * Weightlessness is only caused by the sensation of no object pulling or pushing on it, and there is no contact. This would be in the case of freefalling. Weightlessness is only a sensation; it does not mean you have lost weight.
 * Why do scales read our "weights" differently in outer space.
 * It reads our weight differently because it reads the upwards force on our body - the normal force, because that is where there is contact with the scale. The Normal force changes if there is acceleration, but weight remains the same. If there is upwards acceleration, you would have a higher "weight" number, and lower if you are accelerating down.
 * Why is there "weightlessness" in outer space?
 * Just like in free fall, astronauts feel no external forces on them. There still is gravity from planets in the universe. However, there is no contact with any other object, so one feels weightless.

__Part E:__
 * What is the work-energy theorem?
 * There is work that can be done on a satellite going in an elliptical orbit. It makes the satellite move slower as it moves away from the planet.
 * KEi + PEi + Wext = KEf + PEf
 * The initial 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.
 * In satellites, the only work done is on gravity, and since it is conservative we can get rid of this from the equation.
 * What is a work-energy bar chart?
 * This shows the amount of energy in an object in a vertical bar. A bar is made for each form of energy (mechanical, etc).
 * [[image:jitskyhonorsphysics/Screen_shot_2012-01-09_at_1.20.29_PM.png caption="Screen_shot_2012-01-09_at_1.20.29_PM.png"]]
 * What is different about the energy analysis between circles and ellipses?
 * In circular motion the speed remains constant, as well as the height above Earth. Kinetic energy is dependent on speed, so kinetic energy is constant. Since potential energy depends on height, potential is constant throughout the satellite's motion. If KE and PE is constant, TME remains constant.
 * In elliptical motion, TME also remains constant. Since the only force doing work upon the satellite is conservative, Wext is zero and mechanical energy is conserved. However, the quantities will change in the forms.The force of gravity slow down the satellite while moving away from Earth, and vice versa. The kinetic energy will also be changing. When going faster, the satellite gains speed and loses height, thus a gain of kinetic energy and a loss of potential energy. Yet throughout the entire elliptical trajectory, the total mechanical energy of the satellite remains constant.