Ch5_GrunfeldM

**Chapter 5** Circular Motion and Satellite Motiontoc


 * Lesson 1: Motion Characteristics for Circular Motion **

=1a: Speed and Velocity =

Uniform circular motion is the motion of an object in a circle with a constant speed. The distance of one complete cycle around the circle is the circumference, and it equals (2)(pi)(radius). Because we know average speed to equal distance/time, in circular motion, it is the same as saying circumference/time. The period (T) is the time it takes for an object to make one cycle around the circle. Combining some of the above equations, we derive a new equation: Average Speed = [(2)(pi)(R)]/T. Although objects moving in uniform circular motion travel at constant speed, that does not mean that the velocity will remain constant. The direction of velocity is pointed at the same direction where the object is moving, and because it is traveling in a circle, the direction of velocity changes at each instant while keeping the same magnitude.
 * Still Dealing With Speed and Velocity?! **

=1b: Acceleration =

The equation for acceleration is Ave. acceleration = change in velocity/change in time. In order to calculate the change in velocity, you subtract your initial velocity from your final. Because we know that circular motion requires a changing velocity, and acceleration essentially means a change in velocity, we know that circular motion requires acceleration as well. Even when an object travels at constant speed, because the velocity is changing, it accelerates towards the center of the circle. The acceleration of the object is in the same direction as the velocity change vector.
 * Accelerate Your Knowledge **

=1c: The Centripetal Force Requirement =

**No More Centripetal Confusion** For an object traveling in circular motion, there must be an inward force acting upon it to cause its inward acceleration. This is referred to as the centripetal force. It is simply a physical force that pushes or pulls an object in circular motion towards the circle's center. The centripetal force for circular motion alters the direction of the object without altering its speed. As the force acts upon an object moving at constant speed, it always acts inward as the velocity of the object is directed tangent to the circle.

=1d: The Forbidden F-Word =

Centrifugal, opposite of centripetal, means away from the center or outward. The word "centrifugal" creates a common misconception, that objects in circular motion are experiencing an outward force. People tend to believe that there have been instances where they have been thrown outward away from the center of the circle, such as on a roller coaster. This is directly related to Newton's law of inertia and the misconception that someone is being pulled forward when a car breaks suddenly. Centripetal is the correct force to believe in when considering circular motion, NOT centrifugal!
 * Don't Drop the F-Bomb! **

=1e: Mathematics of Circular Motion =

<span style="font-family: 'Comic Sans MS',cursive;">**Math, Math, and More Math** <span style="font-family: 'Comic Sans MS',cursive;">In one of the previous lessons, we learned the modified equation for speed in circular motion. There is a new equation for acceleration, and we can substitute the expression for speed into this equation: <span style="font-family: 'Comic Sans MS',cursive;">Furthermore, we use our original equation to calculate net force, but we can substitute the expression for acceleration into this equation:


 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 140%;">Lesson 2: Applications of Circular Motion **

=<span style="font-family: 'Comic Sans MS',cursive;">2a: Newton's Second Law - Revisited =


 * **<span style="font-family: 'Comic Sans MS',cursive;">How can we revisit Newton's second law and apply it to circular motion? **
 * <span style="font-family: 'Comic Sans MS',cursive;">Fnet = ma or a = Fnet/m
 * <span style="font-family: 'Comic Sans MS',cursive;">Consider a car moving in a horizontal circle on a level surface:
 * <span style="font-family: 'Comic Sans MS',cursive;">Net force acting upon the car is directed inwards (centripetal force requirement)
 * <span style="font-family: 'Comic Sans MS',cursive;">If the car is positioned on the left side, the net force is directed rightward
 * <span style="font-family: 'Comic Sans MS',cursive;">3 forces acting on it: [[image:u6l2a3.gif width="142" height="120"]]
 * <span style="font-family: 'Comic Sans MS',cursive;">Friction force supplies the centripetal force requirement for the car to move in a horizontal circle
 * <span style="font-family: 'Comic Sans MS',cursive;">Solve the rest of the problem using the given information with the known equations

=<span style="font-family: 'Comic Sans MS',cursive;">2b: Amusement Park Physics =


 * **<span style="font-family: 'Comic Sans MS',cursive;">How can we use roller coasters as examples of circular motion? **
 * <span style="font-family: 'Comic Sans MS',cursive;">The most obvious section of a roller coaster where centripetal acceleration occurs is within the clothoid loops
 * <span style="font-family: 'Comic Sans MS',cursive;"><span style="color: #ff0000; font-family: 'Comic Sans MS',cursive;">**Clothoid** = a section of a spiral in which the radius is constantly changing [[image:u6l2b2.gif width="208" height="176"]]
 * <span style="font-family: 'Comic Sans MS',cursive;">Continuous change in direction and speed
 * <span style="font-family: 'Comic Sans MS',cursive;">Greatest speeds at the bottom, lowest speeds at the top
 * <span style="font-family: 'Comic Sans MS',cursive;">The force of gravity acting upon the passenger (or car) can then be easily found using Fgrav = mg

=<span style="font-family: 'Comic Sans MS',cursive;">2c: Athletics =


 * **<span style="font-family: 'Comic Sans MS',cursive;">How can we relate athletics to circular motion? **
 * <span style="font-family: 'Comic Sans MS',cursive;">Circular motion of athletics is characterized by an inward acceleration and caused by an inward net force
 * <span style="font-family: 'Comic Sans MS',cursive;">The most common example is a turn (not all involve a complete circle or have a perfectly circular shape)
 * <span style="font-family: 'Comic Sans MS',cursive;">There is both a horizontal and vertical component resulting from contact with the surface below
 * <span style="font-family: 'Comic Sans MS',cursive;">The <span style="color: #ff0000; font-family: 'Comic Sans MS',cursive;">**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: 'Comic Sans MS',cursive; font-size: 140%;">**Lesson 3: Universal Gravitation**

=<span style="font-family: 'Comic Sans MS',cursive;">3a: Gravity is More than a Name =


 * <span style="font-family: 'Comic Sans MS',cursive;">**What does gravity represent?**
 * <span style="font-family: 'Comic Sans MS',cursive;">What goes up must come down
 * <span style="font-family: 'Comic Sans MS',cursive;">Gravity can be a force or acceleration
 * <span style="font-family: 'Comic Sans MS',cursive;"> **Acceleration of gravity** **(g)** = the acceleration experienced by an object when the only force acting upon it is the force of gravity

=<span style="font-family: 'Comic Sans MS',cursive;">3b: The Apple, the Moon, and the Inverse Square Law =


 * <span style="font-family: 'Comic Sans MS',cursive;">**What is important about the apple and the moon?**
 * <span style="font-family: 'Comic Sans MS',cursive;">The moon is falling
 * <span style="font-family: 'Comic Sans MS',cursive;">The type of force that acts on the apple is the same as the force that acts on the Moon
 * <span style="font-family: 'Comic Sans MS',cursive;">**What is the inverse square law?**
 * <span style="font-family: 'Comic Sans MS',cursive;">Kepler had 3 laws of planetary motion
 * <span style="font-family: 'Comic Sans MS',cursive;">Law of Ellipses: the paths of the planets about the sun are elliptical in shape, with the center of the sun being located at one focus
 * <span style="font-family: 'Comic Sans MS',cursive;">Law of Equal Areas: 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
 * <span style="font-family: 'Comic Sans MS',cursive;">The Law of Harmonies: 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
 * <span style="font-family: 'Comic Sans MS',cursive;">Newton was not satisfied and discovered the notion of ** universal gravitation ,** which reflected his ability to relate heavenly motion to the motion of the Earth
 * <span style="font-family: 'Comic Sans MS',cursive;">The **inverse square law** = the relationship between the force of gravity between the Earth and any other object and the distance that separates the two centers can be described as Fgrav:1/d^2

=<span style="font-family: 'Comic Sans MS',cursive;">3c: Newton's Law of Universal Gravitation =


 * <span style="font-family: 'Comic Sans MS',cursive;">**How can we go more in depth with Newton's law of universal gravitation?**
 * <span style="font-family: 'Comic Sans MS',cursive;">Newton's law of universal gravitation extends gravity beyond Earth
 * <span style="font-family: 'Comic Sans MS',cursive;">All objects have gravitational attraction towards one another[[image:u6l3c1.gif width="324" height="156"]]
 * <span style="font-family: 'Comic Sans MS',cursive;">Another means of representing the proportionalities is to express the relationships in the form of an equation using a constant of proportionality:[[image:u6l3c3.gif width="309" height="95"]]
 * <span style="font-family: 'Comic Sans MS',cursive;">Knowing the value of G allows us to calculate the force of gravitational attraction between any two objects of known mass and known separation distance

=<span style="font-family: 'Comic Sans MS',cursive;">3d: Cavendish and the Value of G =


 * **<span style="font-family: 'Comic Sans MS',cursive;">How was the value of G discovered? **
 * <span style="font-family: 'Comic Sans MS',cursive;">Lord Henry Cavendish used a torsion balance to determine the value of G
 * <span style="font-family: 'Comic Sans MS',cursive;">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
 * <span style="font-family: 'Comic Sans MS',cursive;">By measuring m1, m2, d, and Fgrav, the value of G could be determined
 * <span style="font-family: 'Comic Sans MS',cursive;">Today, the currently accepted value is 6.67259 x 10-11 N m2/kg2

=<span style="font-family: 'Comic Sans MS',cursive;">3e: The Value of g =

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 * **<span style="font-family: 'Comic Sans MS',cursive;">What is significant about the value of g? **
 * <span style="font-family: 'Comic Sans MS',cursive;">The symbol g is referred to as the acceleration of gravity (9.8 m/s/s)
 * <span style="font-family: 'Comic Sans MS',cursive;">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)

<span style="font-family: 'Comic Sans MS',cursive; font-size: 140%;">**Lesson 4: Planetary and Satellite Motion**

=<span style="font-family: 'Comic Sans MS',cursive;">**4a: Kepler's Three Laws** =


 * <span style="font-family: 'Comic Sans MS',cursive;">**What are Kepler's three laws of planetary motion?**
 * <span style="font-family: 'Comic Sans MS',cursive;">Law of Ellipses: the paths of the planets about the sun are elliptical in shape, with the center of the sun being located at one focus
 * <span style="font-family: 'Comic Sans MS',cursive;">Law of Equal Areas: 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
 * <span style="font-family: 'Comic Sans MS',cursive;">The Law of Harmonies: 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
 * <span style="font-family: 'Comic Sans MS',cursive;">Kepler believed the planets were "magnetically" driven by the sun to orbit in their elliptical trajectories

=<span style="font-family: 'Comic Sans MS',cursive;">**4b: Circular Motion Principles for Satellites** =


 * <span style="font-family: 'Comic Sans MS',cursive;">**What are the circular motion principles for satellites?**
 * <span style="font-family: 'Comic Sans MS',cursive;">**Natural satellites** = __ex:__ the moon, planets, comets
 * <span style="font-family: 'Comic Sans MS',cursive;">**Man-made satellites** = __ex:__ host of satellites launched from the Earth for purposes of communication, scientific research, weather forecasting, intelligence, etc.
 * <span style="font-family: 'Comic Sans MS',cursive;">Satellites are projectiles because the only force acting upon them is gravity
 * <span style="font-family: 'Comic Sans MS',cursive;">Satellites have elliptical orbits or other orbital shapes, orbiting around a central massive body

=<span style="font-family: 'Comic Sans MS',cursive;">4c: Mathematics of Satellite Motion =


 * <span style="font-family: 'Comic Sans MS',cursive;">**How can you calculate the:**
 * <span style="font-family: 'Comic Sans MS',cursive;">net force?
 * <span style="color: #ff0000; font-family: 'Comic Sans MS',cursive;">**Fnet = ( Msat • v2 ) / R**
 * <span style="font-family: 'Comic Sans MS',cursive;">force of gravity?
 * <span style="color: #ff0000; font-family: 'Comic Sans MS',cursive;">**Fgrav = ( G • Msat • MCentral ) / R2**
 * <span style="font-family: 'Comic Sans MS',cursive;">After setting these two equations equal to each other, you can eventually derive the equation for velocity:
 * [[image:Screen_shot_2012-01-09_at_9.48.15_PM.png width="143" height="61"]]
 * <span style="font-family: 'Comic Sans MS',cursive;">acceleration?
 * <span style="font-family: 'Comic Sans MS',cursive;">[[image:u6l4b6.gif width="104" height="40"]]
 * <span style="font-family: 'Comic Sans MS',cursive;">period?
 * <span style="font-family: 'Comic Sans MS',cursive;">[[image:u6l4c1.gif width="121" height="48"]]

=<span style="font-family: 'Comic Sans MS',cursive;">4d: Weightlessness in Orbit =


 * <span style="font-family: 'Comic Sans MS',cursive;">**What is the meaning and cause of weightlessness?**
 * <span style="font-family: 'Comic Sans MS',cursive;"> **Weightlessness** = a sensation experienced by an individual when there are no external objects touching one's body and exerting a push or pull upon it
 * <span style="font-family: 'Comic Sans MS',cursive;">Exist when all contact forces are removed
 * <span style="font-family: 'Comic Sans MS',cursive;">These sensations are common to any situation in which you are momentarily in a state of free fall
 * <span style="font-family: 'Comic Sans MS',cursive;">Gravity is the only force acting upon your body; since gravity cannot be felt without any other opposing forces, you would have no sensation of it (weightless)

=<span style="font-family: 'Comic Sans MS',cursive;">4e: Energy Relationships for Satellites =


 * <span style="font-family: 'Comic Sans MS',cursive;">**What are some of the energy relationships for satellites?**
 * <span style="font-family: 'Comic Sans MS',cursive;"> **Circular motion** = constant speed, constant height
 * <span style="font-family: 'Comic Sans MS',cursive;">Tangential velocity (falls at same rate)
 * <span style="font-family: 'Comic Sans MS',cursive;">No acceleration
 * <span style="font-family: 'Comic Sans MS',cursive;">Constant radius
 * <span style="font-family: 'Comic Sans MS',cursive;">Kinetic energy (depends on speed) and potential energy (depends on height) are constant
 * <span style="font-family: 'Comic Sans MS',cursive;">Time remains constant
 * <span style="font-family: 'Comic Sans MS',cursive;"> **Elliptical motion** = speed and height are not constant
 * <span style="font-family: 'Comic Sans MS',cursive;">Can accelerate
 * <span style="font-family: 'Comic Sans MS',cursive;">Negative work slows down; positive work speeds up
 * <span style="font-family: 'Comic Sans MS',cursive;">All mechanical energy is conserved
 * <span style="font-family: 'Comic Sans MS',cursive;">Kinetic and potential energy change (speed changes)
 * <span style="font-family: 'Comic Sans MS',cursive;">Time remains constant



=<span style="font-family: 'Comic Sans MS',cursive;">The Clockwork Universe =

<span style="font-family: 'Comic Sans MS',cursive;">**Part 1:**

<span style="font-family: 'Comic Sans MS',cursive;">**Topic Sentence:** The view of the universe has been modified and changed over time.


 * <span style="font-family: 'Comic Sans MS',cursive;">Before Newton was born, Copernicus launched a scientific revolution by rejecting the prevailing Earth-centered view of the universe --> discovered heliocentric theory
 * <span style="font-family: 'Comic Sans MS',cursive;"> **Heliocentric** = the view in which the Earth moves around the Sun
 * <span style="font-family: 'Comic Sans MS',cursive;">Went against the Catholic church
 * <span style="font-family: 'Comic Sans MS',cursive;">Galileo supported him and had to renounce his declared opinion
 * <span style="font-family: 'Comic Sans MS',cursive;">Later, Kepler devised a modern form of Copernicanism that a planet follows an elliptical orbit, with the Sun at one focus of the ellipse

<span style="font-family: 'Comic Sans MS',cursive;">**Part 2:**

<span style="font-family: 'Comic Sans MS',cursive;">**Topic Sentence:** Kepler's ideas were supported by new discoveries in mathematics.

>
 * <span style="font-family: 'Comic Sans MS',cursive;">A two-dimensional coordinate system can be used to locate the position of any point in terms of its x- and y-coordinates.
 * <span style="font-family: 'Comic Sans MS',cursive;">A two-dimensional coordinate system can be used to represent lines and other geometrical shapes by equations.[[image:fig_1_5.gif]]
 * <span style="font-family: 'Comic Sans MS',cursive;"> The circle is characterized by this equation:[[image:equ_pp12.gif]]

<span style="font-family: 'Comic Sans MS',cursive;">**Part 3:**

<span style="font-family: 'Comic Sans MS',cursive;">**Topic Sentence**: New discoveries in astronomy called for a new physics which Newton contributed to greatly.


 * <span style="font-family: 'Comic Sans MS',cursive;">His great achievement was to provide a synthesis of scientific knowledge
 * <span style="font-family: 'Comic Sans MS',cursive;">Concentrated on steady motion
 * <span style="font-family: 'Comic Sans MS',cursive;">Always looking for a cause
 * <span style="font-family: 'Comic Sans MS',cursive;">Found a correlation between force and steady motion

<span style="font-family: 'Comic Sans MS',cursive;">**Part 4:**

<span style="font-family: 'Comic Sans MS',cursive;">**Topic Sentence:** Newton proposed a law of gravity that worked in a universal manner.


 * <span style="font-family: 'Comic Sans MS',cursive;">Combined his laws of motion to discover mathematically that a since planet moves around the Sun in an elliptical orbit, just as Kepler discovered
 * <span style="font-family: 'Comic Sans MS',cursive;">Predicted that gravitational attractions between the planets would cause small departures from the elliptical motion that Kepler described