Ch5_FavaA

= = =Lesson 1 - Motion Characteristics for Circular Motion toc =
 * 1) Not Quite Done with Speed and Velocity Yet
 * Linear speed and velocity concepts can be applied to circular motion. Uniform circular motion is when an object is moving at constant speed, where average speed is calculated by distance/time though since circle it is circumference/time. Although constant speed, not constant velocity. Velocity is a vector and is tangent to the circle; since tangent is facing different direction as object moves, the velocity is constantly changing.
 * 1) Acceleration: Circular vs. Linear
 * Circular acceleration depends on the change of velocity and is in the same direction as the velocity. It is calculated by the change of velocity over time (vfinal-vinitial/time). Constant circular speed accelerates toward the center of the circle. Accelerometer is used to measure acceleration and is made up of an object suspended in a fluid; it determines inertia of the object.
 * 1) Here Comes Centripetal Force
 * Centripetal force requirement is that an object moving in a circle must have an inward force acting upon it to cause the inward acceleration. Therefore, centripetal force is the force pushing or pulling object towards center of circle. Newton's first law states must be an unbalanced force to move in circular motion. Centripetal force changes direction of circle but not necessarily the speed. Force is perpendicular to tangential velocity, so doesn't change magnitude only direction.
 * 1) Centri-what?
 * Centripetal force is towards the circle and is necessary for circular motion since no outward force. Centrifugal force is away from the circle. Do not confuse the two!
 * 1) Let's Do Some Math!
 * The circular motion of objects can be described by speed, acceleration, and force. Average speed is calculated by circumference/time (2(pi)r/t). Acceleration is calculated by 4(pi sq)r/time. Net force is mass x acceleration.

=Lesson 2 - Applications of Circular Motion= Newton's second law states that the acceleration of an object is directly proportional to the net force acting upon the object and inversely proportional to the mass of the object. The law is often expressed in the form of the following two equations. In this Lesson, we will use Unit 2 principles (free-body diagrams, Newton's second law equation, etc.) and circular motion concepts in order to analyze a variety of physical situations involving the motion of objects in circles or along curved paths.
 * Newton's Second Law - Revisited **

Applying the concept of a centripetal force requirement, we know that the net force acting upon the object is directed inwards. Since the car is positioned on the left side of the circle, the net force is directed rightward. An analysis of the situation would reveal that 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). It is the friction force that supplies the centripetal force requirement for the car to move in a horizontal circle. Without friction, the car would turn its wheels but would not move in a circle (as is the case on an icy surface).

In this part of Lesson 2, we will focus on the centripetal acceleration experienced by riders within the circular-shaped sections of a roller coaster track. These sections include the clothoid loops (that we will approximate as a circle), the sharp 180-degree banked turns, and the small dips and hills found along otherwise straight sections of the track.
 * Roller Coasters and Amusement Park Physics **

The most obvious section on a roller coaster where centripetal acceleration occurs is within the so-called **clothoid loops**. A clothoid is a section of a spiral in which the radius is constantly changing. The radius at the bottom of a clothoid loop is much larger than the radius at the top of the clothoid loop. The amount of curvature at the bottom of the loop is less than the amount of curvature at the top of the loop. To simplify our analysis of the physics of clothoid loops, we will approximate a clothoid loop as being a series of overlapping or adjoining circular sections. Our analysis will focus on the two circles that can be matched to the curvature of these two sections of the clothoid. As a roller coaster rider travels through a clothoid loop, she experiences a centripetal acceleration due to both a change in speed and a change in direction. As energy principles would suggest, an increase in height (and in turn an increase in potential energy) results in a decrease in kinetic energy and speed, and vice versa. In the case of a rider moving through a noncircular loop at non-constant speed, the acceleration of the rider has two components. There is a component that is directed towards the center of the circle ( **ac** ) and attributes itself to the direction change; and there is a component that is directed tangent ( **at** ) to the track (either in the opposite or in the same direction as the car's direction of motion) and attributes itself to the car's change in speed.

As depicted in the free body diagram, the magnitude of Fnorm (or whatever centripetal force) is always greater at the bottom of the loop than it is at the top to overcome Fgrav. The normal force must always be of the appropriate size to combine with the Fgrav in such a way to produce the required inward or centripetal net force.

When at the top of the loop, a rider will __feel__ partially weightless if the normal forces become less than the person's weight. And at the bottom of the loop, a rider will feel very "weighty" due to the increased normal forces.

The diagram below shows the various directions of accelerations that riders would experience along these hills and dips.

The magnitude of the normal forces along these various regions is dependent upon how sharply the track is curved along that region (the radius of the circle) and the speed of the car. These two variables affect the acceleration according to the equation **a = v2 / R** and in turn affect the net force. As suggested by the equation, a large speed results in a large acceleration and thus increases the demand for a large net force. And a large radius (gradually curved) results in a small acceleration and thus lessens the demand for a large net force.

The circular motion of athletics - however brief or prolonged they may be - is characterized by an inward acceleration and caused by an inward net force. The most common example of the physics of circular motion in sports involves the turn. The motion around a turn can be approximated as part of a circle or a collection of circles. When a person makes a turn on a horizontal surface, the person often //leans into the turn//. By leaning, the surface pushes upward at an angle //to the vertical//. As such, there is both a horizontal and a vertical component resulting from contact with the surface below. 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. The upward component of the contact force is sufficient to balance the downward force of gravity and the horizontal component of the contact force pushes the person towards the center of the circle. =Lesson 3 - Universal Gravitation= **Fgrav** is a force that exists between the Earth and the objects that are near it. ** The acceleration of gravity ** ( **g** ) is the acceleration experienced by an object when the only force acting upon it is the force of gravity. On and near Earth's surface, the value for the acceleration of gravity for all objects is approximately 9.8 m/s/s.
 * Athletics **
 * Gravity is More Than a Name **

German mathematician and astronomer Johannes 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. Three laws of planetary motion can be briefly described as follows: 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 were 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.Newton's ability to relate the cause for heavenly motion (the orbit of the moon about the earth) to the cause for Earthly motion (the falling of an apple to the Earth) that led him to his notion of **universal gravitation**. A survey of Newton's writings reveals an illustration similar to the one shown at the right. The illustration was accompanied by an extensive discussion of the motion of the moon as a projectile. Cannonball fired at speed such that the trajectory of the falling cannonball matched the curvature of the earth, then would fall around the earth instead of into it as an orbiting satellite ( **path C** ). And then at even greater launch speeds, a cannonball would once more orbit the earth, but in an elliptical path ( **path D** ). The motion of the cannonball orbiting to the earth under the influence of gravity is analogous to the motion of the moon orbiting the Earth and to falling apple. The same force that causes objects on Earth to fall to the earth also causes objects in the heavens to move along their circular and elliptical paths. The same law of mechanics can also be used. Of course, Newton's dilemma was to provide reasonable evidence for the extension of the force of gravity from earth to the heavens. Newton knew that the force of gravity must somehow be "diluted" by distance. The force of gravity between the earth and any object is inversely proportional to the square of the distance that separates that object from the earth's center. The force of gravity follows an **inverse square law**. **Using Equations as a Guide to Thinking** An increase in the separation distance causes a decrease in the force of gravity and a decrease in the separation distance causes an increase in the force of gravity. Furthermore, the factor by which the force of gravity is changed is the square of the factor by which the separation distance is changed. So if the separation distance is doubled (increased by a factor of 2), then the force of gravity is decreased by a factor of four (2 raised to the second power).
 * The Apple, the Moon, and the Inverse Square Law **
 * 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)

Distance is not the only variable affecting the magnitude of a gravitational force. Consider Newton's famous equation **Fnet = m • a** So for Newton, the force of gravity acting between the earth and any other object is directly proportional to the mass of the earth, directly proportional to the mass of the object, and inversely proportional to the square of the distance that separates the centers of the earth and the object. But Newton's law of universal gravitation extends gravity beyond earth. **ALL** objects attract each other with a force of gravitational attraction. 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 The proportionalities expressed by Newton's universal law of gravitation are represented graphically by the following illustration.
 * Newton's Law of Universal Gravitation **

Another means of representing the proportionalities is to express the relationships in the form of an equation using a constant of proportionality. This equation is shown below. The constant of proportionality (G) in the above equation is known as the **universal gravitation constant**. 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 **m1• m2** units and divided by **d2** units, the result will be Newtons - the unit of force.

The first conceptual comment is the inverse relationship between separation distance and the force of gravity (or in this case, the weight of the student). The student weighs less at the higher altitude. However, a mere change of 40 000 feet further from the center of the Earth is virtually negligible. The distance of separation becomes much more influential when a significant variation is made. The second conceptual comment to be made is that the use of Newton's universal gravitation equation to calculate the force of gravity (or weight) yields the same result as when calculating it using the equation presented in Unit 2: **Fgrav = m•g = (70 kg)•(9.8 m/s2) = 686 N**

**Cavendish and the Value of G** The constant of proportionality in this equation is **G** - the universal gravitation constant. The value of G was not experimentally determined until nearly a century later (1798) by Lord Henry Cavendish using a torsion balance. Cavendish's apparatus for experimentally determining the value of G involved a light, rigid rod about 2-feet long. 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, 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 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.

In the first equation above, **g** is referred to as the acceleration of gravity. Its value is **9.8 m/s2** on Earth. When discussing the acceleration of gravity, it was mentioned that the value of g is dependent upon location. There are slight variations in the value of g about earth's surface that 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 To understand why the value of g is so location dependent, we will use the two equations above to derive an equation for the value of g. First, both expressions for the force of gravity are set equal to each other. The above equation demonstrates that the acceleration of gravity is dependent upon the mass of the earth (approx. 5.98x1024 kg) and the distance ( **d** ) that an object is from the center of the earth.  The value of g varies inversely with the distance from the center of the earth. In fact, the variation in g with distance follows an __ [|inverse square law] __ where g is inversely proportional to the distance from earth's center.
 * The Value of g **

The same equation used to determine the value of g on Earth' surface can also be used to determine the acceleration of gravity on the surface of other planets. The equation takes the following form:

The value of g is independent of the mass of the object and only dependent upon //location// - the planet the object is on and the distance from the center of that planet.

=The Clockwork Universe= = = =Lesson 4 - Circular and Satellite Motion=
 * 1) Who were important figures in the discovery of the workings of the universe and what did they do?
 * Copernicus
 * heliocentric universe where Earth revolved around the sun
 * Galileo
 * supported Copernicus
 * objects accelerate at same rate, regardless of mass and size
 * basics of laws of motion
 * Newton
 * expanded on Galileo's laws of motion
 * inertia is resistance to change in velocity
 * acceleration is caused by a force
 * Kepler
 * modified Copernicus by saying planets orbit sun in elliptical motion
 * wrote Astronomia Nova on his observations
 * 1) Who was Renee Descartes and what did he do?
 * important mathematician
 * discovered mathematical equations that linked algebra and geometry in coordinate system
 * 1) Expand upon Newton's discoveries.
 * Newton was able to connect mathematics, astronomy, and physics
 * he was able to combine his laws of motion with gravity to show elliptical pattern of planets around sun mathematically
 * his three key points:
 * there were deviations from standard motion
 * the cause of these deviations
 * law of universal gravitation linked a force and this deviation
 * used physics to show that gravitational attraction would cause planets to veer off from their elliptical track
 * Pierre Simon Lapace expanded on Newton's discoveries with the discovery of mechanics
 * used determinism - once clockwork was set in motion, its future development was entirely predictable
 * 1) What are Kepler's three laws?
 * law of ellipses
 * path of planets around the sun is elliptical, with sun centered at one focii
 * law of equal areas
 * an imaginary line from center of planets to center of sun would create equal areas in equal intervals of time
 * law of harmonies
 * ratio of squares of the periods of any two planets is equal to the ratio of the cubes of average distance to center of sun
 * 1) What are the circular motion principles for satellites?
 * satellite - an object that is orbiting the Earth, sun, or other massive object; can be natural or man made
 * gravity is only force acting on satellite, so similar to projectiles
 * velocity is directed tangent to circle at every point and can describe satellite motion
 * acceleration is directed towards center of circle and can also describe satellite motion
 * satellites also move in elliptical motion, where one foci is object being orbited around
 * 1) What are the mathematics of satellite motion?
 * Fg = G(m1)(m2)/d(sq)
 * v(squared) = G(Mcentral)/R
 * a = G(Mcentral)/R(sq)
 * T(sq)/R(cubed) = 4(pi)(sq)/G(Mcentral)
 * 1) What exactly is weightlessness?
 * sensation felt by a person when all external contact forces are removed
 * though gravity is still acting on it (non-contact), though is the only force, so in free fall momentarily
 * can affect scale readings in moving elevators
 * gravity provides centripetal force to allow inward circular motion acceleration, such as in an orbit
 * for example, astronauts are weightless when orbiting earth
 * 1) What are energy relationships for satellites?
 * satellites move at constant speed and same height, in either elliptical or circular motion
 * direction of force of gravity is perpendicular to direction that satellite is moving
 * since there is no acceleration in tangential direction, satellite goes constant speed
 * work energy theorem - 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.

=Do Now=
 * circular motion requires at least one force - TRUE
 * centrifugal forces are real - FALSE
 * not actually thrown outward, you just straight and environment in circle; no force pushing out just inertia
 * an object moving in a circle with constant speed has no acceleration - FALSE
 * an object moving in a circle will continue in circular motion when released - FALSE
 * an object in circular motion will fly out tangentially when released - TRUE
 * an object moving along a circular path can move with constant tangential velocity - TRUE
 * centrifugal forces push an object in a circular path - FALSE
 * centripetal forces point towards the center of the circular path - TRUE