Ch5+LambertM

toc = __Chapter 5__ =

Homework Assignment 1-Motion Characteristics for Circular Motion (a-e):Method 5
The concepts of speed and velocity also play a big role in circular circular motion. An object can have uniform circular motion as it is going at a constant speed and the average speed can still be calculated by distance over time, but now that it is a circle, it is written as circumference/time (2 X pi X radius/time). However, while the speed of the object is constant, its velocity is changing. Velocity, being a vector, has a constant magnitude but a changing direction, which is always directed tangent to the circle. As the object turns the circle, the tangent line is always pointing in a new direction.
 * a) Speed and Velocity, Back in the Game**

Average acceleration is calculated by the change of velocity over time. The acceleration of the object is dependent upon the velocity change and is in the same direction as this velocity change. Objects moving in circles at a constant speed accelerate towards the center of the circle.The acceleration of an object is often measured using a device known as an accelerometer, consisting of an object immersed in a fluid such as water.
 * b) Who Could FORGET About Acceleration?!**

According to the Centripetal Force Requirement, for an object to move in a circle, there must be an inward force acting upon it causing an inward acceleration. And in accordance with Newtons first law, the presence of an unbalanced force is required for objects to move in circles. The physical force pushing or pulling an object towards the center of the circle alters the direction of the object without altering its speed.
 * c) Old Ideas Coming Alive**

The word centrifugal means away from the center of the circle and is unfortunately very commonly confused with the word centripetal! Objects in circular motion do not experience an outward force. They must undergo an inward force or circular motion would not be possible.
 * d) Foul Language In The Classroom**


 * e) Expressions To Intense For The Face**

Homework Assignment 2-Application of Circular Motion (a-c)

 * a) Newton's Second Law - Revisited**

Survey Question Read Recite Review
 * 1) How do you solve circular motion problems?
 * 2) What is an example that combines Newton's second law and circular motion?
 * 1) **Solving circular motion problems:** 1)Construct a FBD. 2) Identify given and unknown information. 3) If any forces are at angles, use vector principles to resolve them into horizontal and vertical components. 4) Use circular motion equations to find unknown information.
 * 2) **Real life situation:** A car moving along a curve.


 * b) Roller Coasters and Amusement Park Physics**

Survey Question Read Recite Review
 * 1) What circular-shaped sections of a roller coaster track make riders experience centripetal acceleration?
 * 2) The centripetal acceleration of an object moving around a clothoid loop has two components. What are the characteristics of these components?
 * 3) What are clothoid loops?
 * 1) **Types of track****:** Loops, small dips and hills, and banked turns.
 * 2) **Component Characteristics:**
 * **ac** - directed towards center of circle, causes object's change in direction.
 * **at** - directed tangent to the track, causes object's change in speed.
 * Speed decreases leads to at directed opposite object's motion
 * Speed increases leads to at directed same way as object's motion.
 * 1) **Clothoid Loops**: These types of loops are tear-drop shaped, and therefore the radius is constantly changing. The radius at the bottom is significantly larger than the radius at the top


 * c) Athletics**

Survey Question Read Recite Review
 * 1) What is contact force?
 * 2) What does contact force do?
 * 1) **Contact force**: Occurs when a surface pushes upward on an object at an angle //to the vertical//. Both a horizontal and a vertical component result from contact with the surface below.
 * 2) **Role of contact force**: It balances the downward force of gravity and meets the centripetal force requirement for an object in uniform circular motion.

Homework Assignment 3-Circular Motion and Satellite Motion (a-e):Method 1
The acceleration of gravity (**g**) is the acceleration experienced by an object when the only force acting upon it is the force of gravity. Approximately 9.8 m/s/s is the same acceleration value for all objects, regardless of their mass.
 * Gravity is More Than a Name **

In the early 1600's, German mathematician and astronomer Johannes Kepler's three laws emerged from the analysis of data carefully collected by his Danish predecessor and teacher, Tycho Brahe. Kepler's three laws of planetary motion can be briefly described as follows: To Kepler, the planets were somehow "magnetically" driven by the sun to orbit in their elliptical trajectories.
 * The Apple, the Moon, and the Inverse Square Law **
 * 1) 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)
 * 2) 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)
 * 3) 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)

To Newton, there must be some cause for such elliptical motion. 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. These celestial motions required a cause in the form of an unbalanced force. Whether it is a myth or a reality, the fact is certain that it was Newton's ability to relate the cause for 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**.

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 is inversely related to the distance. The force of gravity follows an **inverse square law**.



**Cavendish and the Value of G** Isaac Newton's law of universal gravitation proposed that the gravitational attraction between any two objects is expressed as follows: The constant **G** - the universal gravitation constant. 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. When the rod becomes twisted, the torsion of the wire begins to exert a torsional force that is proportional to the angle of rotation of the rod. The more twist of the wire, the more the system pushes //backwards// to restore itself towards the original position. Cavendish had calibrated his instrument to determine the relationship between the angle of rotation and the amount of torsional force. Cavendish then brought two large lead spheres near the smaller spheres attached to the rod. Since all masses attract, the large spheres exerted a gravitational force upon the smaller spheres and twisted the rod a measurable amount. 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 because the force of gravitational attraction is only appreciable for objects with large mass.

**The Value of g** For calculating the force of gravity with which an object is attracted to the earth: d represents the distance from the center of the object to the center of the earth g is referred to as the acceleration of gravity (9.8 m/s2) There are slight variations in the value of g about earth's surface, resulting 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 resulting 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. If the value 6.38x106 m (a typical earth radius value) is used for the distance from Earth's center, then g will be calculated to be 9.8 m/s2. The value of g on any other planet can be calculated from the mass of the planet and the radius of the planet. The equation takes the following form: The value of g is independent of the mass of the object and only dependent upon //location.//

The Clockwork Universe (1-4): Method 1

 * a) Part 1**
 * Topic Sentence:** By removing the Earth, and with it humankind, from the centre of creation, Copernicus had set the scene for a number of confrontations between the Catholic church and some of its more independently minded followers.

In 1543 Nicolaus Copernicus launched a scientific revolution by rejecting the prevailing Earth-centred view of the Universe in favour of a heliocentric view in which the Earth moved round the Sun. One of the most famous confrontations between the Catholic church and its independent minded was Galileo, who was summoned to appear before the Inquisition in 1633, on a charge of heresy, for supporting Copernicus' ideas. As a result Galileo he was invited to renounce his declared opinion that the Earth moves around the Sun. In the Protestant countries of Northern Europe the German-born astronomer Johannes Kepler devised a modified form of Copernicanism. According to Kepler, the planets//did// move around the Sun, but their orbital paths were ellipses rather than collections of circles.

This shows the two-dimensional case, with a grid extending over part of the page. The grid is calibrated (in centimetres) so the position of any point can be specified by giving its //x-// and //y-// coordinates on the grid.
 * b) Part 2**
 * Topic Sentence: Kepler's ideas were underpinned by new discoveries in mathematics.**
 * [[image:http://physicalworld.org/restless_universe/figs/fig_1_4.gif width="175" height="188" align="left" caption="figure 1.4, locate the position of any point in terms of its x and y coordinates"]] ||  ||

This idea becomes more powerful when we consider lines and geometrical shapes. The straight line shown is characterized by the fact that, at each point along the line, the //y-//coordinate is half the x-coordinate. Thus, the //x-// and y- coordinates of each point on the line obey the equation //y// = 0.5//x//, and this is said to be the equation of the line.

Similarly, the circle is characterized by the equation
 * [[image:http://physicalworld.org/restless_universe/figs/fig_1_5.gif width="175" height="137" align="left" caption="figure 1.5, a 2-D coordinate system can represent lines and other geometrical shapes by equationsrep"]] ||  ||

**c) Part 3 ** **3.** Newton produced a quantitative link between force and deviation from steady motion and, at least in the case of gravity, quantified the force by proposing his famous law of universal gravitation
 * Topic Sentence: At the core of Newton's world-view is the belief that all the motion we see around us can be explained in terms of a single set of laws.**
 * 1.** Newton concentrated on deviation from steady motion
 * 2.** Wherever deviation from steady motion occurred, Newton looked for a cause. He described such a cause as a force

**d) Part 4 ** **Topic Sentence: The detailed character of Newton's laws was such that once this majestic clockwork had been set in motion, its future development was entirely predictable. **  This property of Newtonian mechanics is called determinism.Given an accurate description of the character, position and velocity of every particle in the Universe at some particular moment, and an understanding of the forces that operated between those particles, the subsequent development of the Universe could be predicted with as much accuracy as desired. French scientist Pierre Simon Laplace used Newton's discoveries which became the basis for a detailed and comprehensive study of mechanics (the study of force and motion).

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Homework Assignment 4-Planetary and Satellite Motion (a-e)
 **Planetary and Satellite Motion (a-c)**

1. What are Kepler's Laws?


 * **Law of Ellipses****:** path of planets about sun is elliptical in shape, with center of sun being located at one focus
 * **Law of Equal Areas****:** even though planets move fastest when closest to the sun and their speeds are constantly changing, if an imaginary line is drawn from center of sun to center of planet will sweet out equal areas in equal intervals of time
 * **Law of Harmonies****:** ratio of squares of periods of any two planets is equal to ratio of cubes of their average distances from sun

2. What are the mathematics of satellite motion? <span style="font-family: Tahoma,Geneva,sans-serif;">3. What are circular motion principles for satellites?
 * <span style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0.5em; padding-bottom: 0px; padding-left: 3em; padding-right: 0px; padding-top: 0px;"><span style="font-family: Tahoma,Geneva,sans-serif;">G=6.673 X 10^-11
 * <span style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0.5em; padding-bottom: 0px; padding-left: 3em; padding-right: 0px; padding-top: 0px;"><span style="font-family: Tahoma,Geneva,sans-serif;">Mcentral=mass of central body about which satellite orbits
 * <span style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0.5em; padding-bottom: 0px; padding-left: 3em; padding-right: 0px; padding-top: 0px;"><span style="font-family: Tahoma,Geneva,sans-serif;">R=radius of orbit for satellite
 * <span style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0.5em; padding-bottom: 0px; padding-left: 3em; padding-right: 0px; padding-top: 0px;"><span style="font-family: Tahoma,Geneva,sans-serif;">The force of gravity = (G*m1*m2)/d <span style="font-family: Tahoma,Geneva,sans-serif; vertical-align: super;">2
 * <span style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0.5em; padding-bottom: 0px; padding-left: 3em; padding-right: 0px; padding-top: 0px;"><span style="font-family: Tahoma,Geneva,sans-serif;">Velocity = sqrt((G*Mcentral)/R)
 * <span style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0.5em; padding-bottom: 0px; padding-left: 3em; padding-right: 0px; padding-top: 0px;"><span style="font-family: Tahoma,Geneva,sans-serif;">Acceleration= (G*Mcentral)/R <span style="font-family: Tahoma,Geneva,sans-serif; vertical-align: super;">2
 * <span style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0.5em; padding-bottom: 0px; padding-left: 3em; padding-right: 0px; padding-top: 0px;"><span style="font-family: Tahoma,Geneva,sans-serif; vertical-align: super;">the period, speed, and acceleration of a satellite are only dependent upon radius of orbit and mass of central body that satellite is orbiting


 * <span style="font-family: Tahoma,Geneva,sans-serif;">Satellites are any objects that are orbiting the Earth, sun, or other massive body and can be natural or man made; move in an orbit about object (ex: moon)
 * <span style="font-family: Tahoma,Geneva,sans-serif;">Satellites act in similar motion to projectiles because gravity is the only force acting on it
 * <span style="font-family: Tahoma,Geneva,sans-serif;">Motion of satellites can be described by acceleration and velocity
 * <span style="font-family: Tahoma,Geneva,sans-serif;">Velocity is directed tangent to circular at every point and acceleration is directed towards center of circle
 * <span style="font-family: Tahoma,Geneva,sans-serif;">Satellites moves in elliptical motion with central body being at one focus

<span style="font-family: Tahoma,Geneva,sans-serif;">**Planetary and Satellite Motion (d-e)**

<span style="font-family: Tahoma,Geneva,sans-serif;">1. What does weightlessness in orbit mean?


 * <span style="font-family: Tahoma,Geneva,sans-serif;">Weightlessness is a sensation experience by a person when there are no external objects touching one's body or exerting a push on it
 * <span style="font-family: Tahoma,Geneva,sans-serif;">Exist when all contact forces are removed
 * <span style="font-family: Tahoma,Geneva,sans-serif;">Momentarily in free fall, where gravity is the only force
 * <span style="font-family: Tahoma,Geneva,sans-serif;">Force of gravity supplies centripetal force to allow the inward acceleration of circular motion (orbit)
 * <span style="font-family: Tahoma,Geneva,sans-serif;">Earth orbiting astronauts are weightless in orbit

<span style="font-family: Tahoma,Geneva,sans-serif;">2. What are the energy relationships for satellites?


 * <span style="font-family: Tahoma,Geneva,sans-serif;">Motion of satellites is circular or elliptical and they move at constant speed and remain at same height
 * <span style="font-family: Tahoma,Geneva,sans-serif;">Throughout trajectory, the force of gravity acts in a direction perpendicular to direction that satellite is moving
 * <span style="font-family: Tahoma,Geneva,sans-serif;">There is no acceleration in tangential direction so the satellite remains in circular motion at constant speed
 * <span style="font-family: Tahoma,Geneva,sans-serif;">Work energy theorem says that the initial amount of total mechanical energy of a system plus the work done by external forces on a system is equal to the final amount of total mechanical energy on the system