Ch5_HallR

=Lesson 1: Motion Characteristics for Circular Motion= toc

12/13/11: A-E Summary
__Speed and Velocity of objects experiencing uniform circular motion__ To find the average speed of an object in uniform circular motion, the equation is used. Since the circumference of a circle is 2xPixRadius, the average speed an object travels a period (T) in is. Objects moving in uniform circular motion have constant speed, but not constant velocity. The direction of the velocity vector is in the direction that the object moves, and since an object is moving in a circle, its direction is always changing. While the magnitude of the velocity vector is constant, the direction of the velocity vector is tangential, meaning that the direction of the velocity vector at any instant is in the direction of a tangent line drawn to the circle at the object's location.
 * A: Speed and Velocity**

__Acceleration of objects experiencing uniform circular motion__ The idea uniformly moving objects have no acceleration is NOT true. Since an accelerating object is one that changes velocity, and since an object experiencing uniform circular motion is constantly changing its direction of velocity, it is in fact accelerating. Since acceleration is the rate tht the velocity of an object changes, average acceleration can be found using the formula: . The numerator of the equation is one vector ( **vi** ) from a second vector ( **vf** ). The acceleration of an object, often measured with an accelerometer, is in the same direction as the velocity change vector, and also directed towards the center of the circle. Objects moving in circles at a constant speed accelerate towards the center of the circle.
 * B: Acceleration**

__Objects experiencing uniform circular motion also experience centripetal force__ Since objects moving in uniform circular motion are accelerating inward, they are acting upon by an inward force, known as centripetal force, centripetal meaning center-seeking. According to Newton's 1st law of motion, moving objects naturally continue in motion in the same direction that they are moving, unless an unbalanced force acts upon the object. An unbalanced force is required to cause a turn in an object's path, and therefore, an unbalanced force is required in a circle. The feeling of an outward force while turning in a car is only due to inertia, and there is no actual force pushing outwards. A centripetal force alters the direction of the object without altering its speed, and always acts inward as the velocity of the object is directed tangent to the circle. A force causes displacement, and the amount of work done upon an object is found using the equation **Work = Force * displacement * cosine (Theta)**, with **Theta** being the angle between the force and the displacement. Since the force is perpendicular to the direction that the object is being displaced, Theta is 90 degrees, and the cosine of 90 degrees is 0, meaning the work done by the centripetal force in the case of uniform circular motion is 0 Joules.
 * C: The Centripetal Force Requirement**

__Centrifugal Force vs. Centripetal Force__ **Centrifugal** and **centripetal** are two different words. Centrifugal means away from the center or outward, the opposite of centripetal. The common misconception is that objects in circular motion are experiencing an outward force. However, this is incorrect, and the inward-directed acceleration requires an inward force. Without this inward force, an object would stay in a straight-line motion tangent to the perimeter of the circle, meaning circular motion would be impossible. The feeling of experiencing an outward force while turning in a car is only due to inertia, and in fact there is no force pushing outwards, meaning a centrifugal or outward net force does not exist. An object moving in circular motion is constantly moving tangent to the circle, so the velocity vector for the object is directed tangentially. To cause circular motion, it requires net unbalanced centripetal force towards the center of the circle in to change the object from it's tangential path.
 * D: The Forbidden F-Word**

__Finding Speed, Acceleration, and Net Force of uniform circular motion__ The speed of an object moving in a circle is given by the equation: , the acceleration of an object moving in a circle can be determined by either of the following equations (by substituting the expression for speed): , and the net force can be found by any of the following (by substituting the expressions for acceleration): . Equations, such as Newton's 2nd law, let predictions to be made about how altering one quantity with affect a 2nd quantity. For example, the following equation relates the net force ( **Fnet** ) to the speed ( **v** ) of an object experiencing uniform circular motion. This shows that the net force required for an object to move in a circle is directly proportional to the square of the speed of the object. For a constant mass and radius, the **Fnet** is proportional to the **speed** squared. The factor by which the net force is altered is the square of the factor by which the speed is altered, so if the speed of the object is doubled, the net force needed for that object's circular motion is quadrupled.
 * E: Mathematics of Circular Motion**

=Lesson 2: Applications of Circular Motion=

12/22/11: A-C Summary (Method 1)
__Topic Sentence: Newton's Second Law can also be applied to objects experiencing uniform circular force.__ >> *Eg: Fgrav >> *Eg: f the speed and the radius are known, then the acceleration can be determined. If the period and radius are known, then the acceleration can be found.
 * A: Newton's Second Law - Revisted**
 * 2nd law says 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.
 * [[image:http://www.physicsclassroom.com/Class/circles/u6l2a1.gif width="240" height="72" align="bottom"]] A free-body diagram is a vector diagram that depicts the relative magnitude and direction of all the individual forces that are acting upon the object
 * The 2nd law can be applied to circular motion as well. [[image:Car.png]]
 * There are three forces acting on the car - 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.
 * Steps for solving centripetal force problems:
 * 1) Construct a free-body diagram
 * 2) Identify the given and the unknown information, use vector principles to put forces in horizontal/vertical components.
 * 3) Determine the magnitude of any known forces
 * 1) Use circular motion equations to determine any unknown information.
 * 1) Use the remaining information to solve for the requested information.
 * If the problem requests the value of an individual force, then use the kinematic information (R, T and v) to determine the acceleration and the Fnet ; then use the free-body diagram to solve for the individual force value.
 * If the problem requests the value of the speed or radius, then use the values of the individual forces to determine the net force and acceleration; then use the acceleration to determine the value of the speed or radius.

__Topic Sentence: Roller Coasters experience centripetal acceleration on clothoid loops, dips and hills, and banked turns.__ __Clothoid Loops__: __Dips and Hills:__ > **a = v2 / R**
 * B: Amusement Park Physics**
 * Roller-coaster rides experience centripetal acceleration within the circular-shaped sections of a roller coaster track, such as clothoid loops, sharp 180-degree banked turns, and the dips and hills on the track.
 * [[image:http://www.physicsclassroom.com/Class/circles/u6l2b1.gif width="362" height="131" align="center"]]
 * The most obvious section on a roller coaster where centripetal acceleration occurs is in the clothoid loops, tear-dropped shapes. The radius at the bottom of a clothoid loop is much larger than the radius at the top of the clothoid loop, and the curvature at the bottom is less than the amount at the top.
 * [[image:Clothoid.jpg]]
 * Approximate a clothoid loop as being a series of overlapping or adjoining circular sections. The radius of these circular sections is decreasing as one approaches the top of the loop. Furthermore, we will limit our analysis to two points on the clothoid loop - the top of the loop and the bottom of the loop. For this reason, our analysis will focus on the two circles that can be matched to the curvature of these two sections of the clothoid.
 * A change in direction is one characteristic of an accelerating object. In addition to changing directions, the rider also changes speed. As the rider begins to ascend (climb upward) the loop, she begins to slow down. An increase in height (and in turn an increase in potential energy) results in a decrease in kinetic energy and speed, and vice versa. A rider experiences the greatest speeds at the bottom of the loop - both upon entering and leaving the loop - and the lowest speeds at the top of the loop.
 * There is an acceleration 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. This tangential component would be directed opposite the direction of the car's motion as its speed decreases (on the ascent towards the top) and in the same direction as the car's motion as its speed increases (on the descent from the top). At the very top and the very bottom of the loop, the acceleration is primarily directed towards the center of the circle. At the top, this would be in the downward direction and at the bottom of the loop it would be in the upward direction.
 * The inwards acceleration of an object is caused by an inwards net force. Circular motion (or merely motion along a curved path) requires an inwards component of net force. If all the forces that act upon the object were added together as vectors, then the net force would be directed inwards. Neglecting friction and air resistance, a roller coaster car will experience two forces: the force of gravity (Fgrav) and the normal force (Fnorm). The normal force is directed in a direction perpendicular to the track and the gravitational force is always directed downwards
 * The normal force must be sufficiently large to overcome this Fgravand supply some excess force to result in a net inward 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. It is important to realize that the force of gravity and the weight of your body are not changing. Only the magnitude of the supporting normal force is changing.
 * Riders often feel heavy as they ascend the hill. Near the crest of the hill, their upward motion makes them feel as though they will fly out of the car; often times, it is only the safety belt that prevents such a mishap. As the car begins to descend the sharp drop, riders are momentarily in a state of free fall, and finally, as they reach the bottom of the sharp dip, there is a large upwards force that slows their downward motion.
 * [[image:u6l2b8.png]]
 * At various locations along these hills and dips, riders are momentarily traveling along a circular shaped arc, which are parts of circles. In each of these regions there is an inward component of acceleration, and there also be a force directed towards the center of the circle.
 * 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
 * Large speed results in a large acceleration and thus increases the demand for a large net force, and a large radius results in a small acceleration and thus lessens the demand for a large net force.

__Topic Sentence: Athletes also experience centripetal force, and Newton's laws, vector principles, and circular motion equations can be used to describe the force-mass-acceleration relationship; the relationship between individual forces and any angular forces; and the speed-radius-acceleration relationship.__
 * C: Athletics**


 * Athletes also experience centripetal force, which is characterized by an inward accelerationand caused by an inward net force.
 * The most common example of circular motion in sports involves any turn; any turn can be approximated as being a part of a larger circle or a part of several circles of varying size.
 * [[image:u6l2c1.png width="361" height="215"]]
 * 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.
 * These two vector components are a vertical normal force and a horizontal friction force. The normal force is the result of the stable surface providing support for any object pushing downward against it. The friction force is the result of the static friction force resulting from the ice-skate interaction. As the skater leans into the turn, she pushes downward and //outward// upon the ice.
 * The blade pushes outward upon the vertical wall of the groove in the ice, and downward upon the floor of the groove. Due to the 3rd law, there is a //reaction force// of the ice pushing upward and inward upon the skate. If this blade-ice action does not occur, the skater could still lean and still try to push outward upon the ice. However, the blade would not get a //grip// upon the ice and the skater would be at risk of not making the turn. As a result, the ice skater's skates would move out from under her, she would fall to the ice, and she would travel in a straight-line inertial path. Without an inward force, the skater cannot travel through the turn.
 * **A turn is only possible when there is a component of force directed towards the center of the circle about which the person is moving.**
 * Regardless of the athletic event, the analysis of the circular motion remains the same. Newton's laws describe the force-mass-acceleration relationship; vector principles describe the relationship between individual forces and any angular forces; and circular motion equations describe the speed-radius-acceleration relationship.

=Lesson 3: Universal Gravitation=

1/3/12: A-E Summary (Method 1)
__Topic Sentence: Gravity from the Earth acts on all objects near it, with acceleration due to gravity being 9.8 m/s/s/__
 * A: Gravity is More than a Name**
 * Certain gravity/force of gravity (Fgrav) is the force of gravity exerted by the Earth on objects near it.
 * When we jump, gravity brings us back down to the ground.
 * Acceleration due to 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 is approximately 9.8 m/s/s.
 * It is the same acceleration value for all objects, regardless of their mass.

__Topic Sentence: The laws of mechanics that govern the motions of objects on Earth also govern the movement of objects in out space.__
 * B**: **The Apple, the Moon, and the Inverse Square Law**
 * Johannes Kepler analyzed astronomical data to develop three laws to describe the motion of planets about the sun, which emerged from the analysis of data by his Danish predecessor and teacher, Tycho Brahe.
 * These three laws are:
 * 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)
 * However. there was no explanation for why such paths existed.
 * 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, possibly "magnetically" driven by the sun to orbit in their elliptical trajectories
 * There was however no interaction between the planets themselves.
 * Newton though there must be some cause for such elliptical motion, and even more troubling was the circular motion of the moon about the earth.
 * Newton knew that 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 (centripetal force).
 * According to legend, a breakthrough came in an apple orchard, when an apple dropped on Newton's head, causing him to think of the notion of gravity as the cause of all heavenly motion
 * This enabled him 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.
 * [[image:u6l3b2.gif]]
 * Path A = no gravity, Path B = no gravity (greater speed), Path C = velocity matches curvature of the Earth, so it orbits the Earth in circular motion, Path D = greater velocity, so it orbits the Earth in an elliptical orbit.
 * 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.
 * 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 force of gravity causes earthbound objects to accelerate towards the earth at a rate of 9.8 m/s2, and it was also known that the moon accelerated towards the earth at a rate of 0.00272 m/s2.
 * If the same force that causes the acceleration of the apple to the earth also causes the acceleration of the moon towards the earth, then there must be a plausible explanation for why the acceleration of the moon is so much smaller than the acceleration of the apple; the more distant moon to accelerate at a rate of acceleration that is approximately 1/3600-th the acceleration of the apple?
 * This was solved by a comparison of the distance from the apple to the center of the earth with the distance from the moon to the center of the earth.
 * The moon in its orbit about the earth is approximately 60 times further from the earth's center than the apple is; so 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 moon, being 60 times further away than the apple, experiences a force of gravity that is 1/(60)2 times that of the apple, showing the inverse square law.
 * The relationship between the force of gravity ( **Fgrav** ) between the earth and any other object and the distance that separates their centers ( **d** ) can be expressed by the following relationship
 * Since the distance **d** is in the denominator of this relationship, it can be said that the force of gravity is inversely related to the distance. And since the distance is raised to the second power, it can be said that the force of gravity is inversely related to the square of the distance.

__Topic Sentence: All objects in the universe attract each other with a force of gravitational attraction, which is directly dependent upon the masses of both objects and inversely proportional to the square of the distance that separates their centers.__ > **G = 6.673 x 10-11 N m2/kg2**
 * C: Newton's Law of Universal Gravitation**
 * Distance is not the only variable affecting the magnitude of a gravitational force.
 * It is also dependent upon the mass of the object, and since the force acting to cause the objects's downward acceleration also causes the earth's upward acceleration (Newton's third law), that force must also depend upon the mass of the earth.
 * Therefore, 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.
 * Newton's law of universal gravitation is about the universality of gravity, so **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
 * Since the gravitational force is directly proportional to the mass of both interacting objects, more massive objects will attract each other with a greater gravitational force.
 * So as the mass of either object increases, the force of gravitational attraction between them also increases, and if the mass of one of the objects is doubled, then the force of gravity between them is doubled, etc.
 * Since gravitational force is inversely proportional to the separation distance between the two interacting objects, more separation distance will result in weaker gravitational forces, so as two objects are separated from each other, the force of gravitational attraction between them also decreases.
 * If the separation distance between two objects is doubled (increased by a factor of 2), then the force of gravitational attraction is decreased by a factor of 4 (2 raised to the second power), and if the separation distance between any two objects is tripled (increased by a factor of 3), then the force of gravitational attraction is decreased by a factor of 9 (3 raised to the second power).
 * [[image:http://www.physicsclassroom.com/Class/circles/u6l3c3.gif width="312" height="98" align="center"]]
 * The constant of proportionality (G) in the above equation is known as the universal gravitation constant. The value of G is found to be
 * 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.
 * Knowing that all objects exert gravitational influences on each other, the small perturbations in a planet's elliptical motion can be easily explained. As the planet Jupiter approaches the planet Saturn in its orbit, it tends to deviate from its otherwise smooth path; this deviation, or perturbation, is easily explained when considering the affect of the gravitational pull between Saturn and Jupiter.

__The value of the universal gravitation constant, G, is 6.67259 x 10-11 N m2/kg2.__
 * D: Cavendish and the Value of G**
 * Newton's law of universal gravitation proposed:
 * [[image:http://www.physicsclassroom.com/Class/circles/u6l3d1.gif width="144" height="52" align="center"]]
 * The constant of proportionality in this equation is G - the universal gravitation constant, which was not experimentally determined until nearly a century later by Lord Henry Cavendish using a torsion balance.
 * This torsion balance 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, which resulted in an experimentally determined value of 6.75 x 10-11 N m2/kg2 (the currently accepted value is 6.67259 x 10-11 N m2/kg2).
 * The smallness of gravity accounts for the fact that the force of gravitational attraction is only appreciable for objects with large mass.

__Topic Sentence: 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.__
 * E: The Value of g**

> Mass can be canceled from the equation. This leaves us with an equation for the acceleration of gravity. >
 * To understand why the value of g is so location dependent, two equations are set equal to each other to find an equation for the value of g.
 * [[image:http://www.physicsclassroom.com/Class/circles/u6l3e2.gif width="151" height="49" align="center"]]
 * The above equation shows 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 will change as an object is moved further from Earth's center.
 * For instance, if an object were moved to a location that is two earth-radii from the center of the earth - that is, two times 6.38x106 m - then a significantly different value of g will be found. As shown below, at twice the distance from the center of the earth, the value of g becomes 2.45 m/s2.
 * G varies inversely with the distance from the center of the earth.
 * The variation in g with distance follows an the inverse square law, where g is inversely proportional to the distance from earth's center.
 * This inverse square relationship means that as the distance is doubled, the value of g decreases by a factor of 4.
 * 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, using the mass of the planet and the radius of the planet. The equation takes the following form:

1/5/12: The Clockwork Universe Lesson 2
__Method 1:__ __Topic Sentence: Using the previous workings of Copernicus, Galileo, and Kepler, Newton discovered that gravity is universal, and caused planets to orbit the sun in imperfect, elliptical orbits.__
 * Nicolaus Copernicus thought of the heliocentric idea of the universe, which stated that the Earth revolved around the Sun.
 * Galileo continued his ideas, which caused debate in the Catholic church.
 * Johannes Kepler modified Copernicus's ideas based on observational data.
 * Kepler said that the planets orbited the sun in ellipses, based on observations
 * He did not know why, but thought that it may be due to magnetic influence from the sun.
 * Coordinate Geometry: represents geometrical shapes by equations, and combines/rearranges such equations to establish other geometrical truths.
 * Newton then followed up on the work, and focused on deviation from steady motion, and why.
 * He created a quantitative link between force and deviation from steady motion.
 * Newton thought that gravity was universal and worked for the whole universe,was able to show mathematically that planets orbit the sun in ellipses due to gravity, and predicted that gravity between the planets would cause small changes from the actual elliptical paths.
 * Pierre Simon Laplace studied mechanics (the study of force and motion).
 * Determinism: the fact that the universe worked like a clock, and was entirely predictable.

=Lesson 4: Planetary and Satellite Motion=

1/6/12: A-C Summary (Method 1)
__Topic Sentence: Kepler though of three laws, described the universal properties of the orbits of planets__
 * A: Kepler's 3 Laws**
 * Law #1 :The orbit of the planets about the sun is an ellipse, with the sun being located at one focus (Law of Ellipses)
 * Law #2: A hypothetical line drawn from the center of the sun to the center of a planet will cover equal areas in equal intervals of time (Law of Equal Areas)
 * [[image:http://www.physicsclassroom.com/Class/circles/u6l4a2.gif width="306" height="313" align="bottom"]]
 * <span style="font-family: 'Times New Roman',Times,serif;">Law #3: The ratio of the squares of the periods of any 2 planets is equal to the ratio of the cubes of their average distances from the sun (Law of Harmonies)
 * <span style="font-family: 'Times New Roman',Times,serif;">In an ellipse, the sum of the distances from any point on the outline of the ellipse to the foci, with one being the sun in this case, is equal to that of any other point on the ellipse.
 * The closer together the two points, the more circular the ellipse is.
 * <span style="font-family: 'Times New Roman',Times,serif;">The 2nd Law describes the speed of a planet in a position while orbiting the Sun.
 * <span style="font-family: 'Times New Roman',Times,serif;">A planet moves faster when closer to the Sun, and slower when further away.
 * The 3rd law compares the orbital periods and radius of an orbit of a planet to those of other planets.
 * This comparison is that the ratio of the squares of the periods to the cubes of their average distances from the sun is the same for every one of the planets.
 * It also states that the T2/R3 ratio for the planets' orbits about the sun also accurately describes the T2/R3 ratio for any satellite (whether a moon or a man-made satellite) about any planet.

__Topic Sentence: Satellites are projectiles that orbit around a central massive body instead of falling into it, and are acted upon solely by gravity.__
 * B: Circular Motion Principles for Satellites**
 * Natural satellites: Moon, planets, comets
 * Man-made satellites: Communications/weather/research/intelligence built satellites
 * All follow same physics, and are projectiles, since they are only acted upon by gravity.
 * Satellites must attain a minimum speed in order to not fall back to Earth and instead orbit.
 * At every point along its trajectory, a satellite is falling toward the earth, but due to the curvature of the Earth, it never reaches the earth.
 * For every 8000 meters measured along the horizon of the earth, the earth's surface curves downward by approximately 5 meters.
 * For a projectile to orbit the earth, it must travel horizontally a distance of 8000 meters for every 5 meters of vertical fall. A horizontally launched projectile would fall approximately 5 meters (0.5*g*t2) in its first second, so a projectile launched horizontally with a speed of about 8000 m/s will be capable of orbiting the earth in a circular path.
 * If shot with a speed greater than 8000 m/s, it would orbit the earth in an elliptical path.
 * Like in a normal circle, a satellite's velocity would be tangental to the circle, and its acceleration will be towards the center of the circle, as is the net force acting upon the object.
 * [[image:http://www.physicsclassroom.com/Class/circles/u6l4b3.gif width="187" height="236" align="center"]]
 * In an elliptical orbit, the body that the satellite is orbiting won't be in the middle of the orbit, but instead one of the two foci, with the other being empty space.
 * In an elliptical orbit, there is a component of force in the same direction as (or opposite direction as) the motion of the object, and there is not constant speed.

__Topic Sentence: A series of mathematical equations can be used to find the net force, force of gravity, velocity, acceleration, period, and distance to central body of a satellite.__
 * C: Mathematics of Satellite Motion**
 * For a satellite with mass Msat orbiting a central body in a circular orbit with a mass of mass Mcentral:
 * Net force: **= ( Msat • v2 ) / R**
 * Force of gravity **= ( G • Msat • MCentral ) / R2**
 * Since Fgrav = Fnet, the above expressions for centripetal force and gravitational force can be set equal to each other **(Msat • v2) / R = (G • Msat • MCentral ) / R2** [[image:http://www.physicsclassroom.com/Class/circles/u6l4b5.gif width="155" height="54"]]
 * Velocity
 * Acceleration [[image:http://www.physicsclassroom.com/Class/circles/u6l4b6.gif width="120" height="47" align="bottom"]]
 * Period of Satellite (T) and distance from central body (R) [[image:http://www.physicsclassroom.com/Class/circles/u6l4c1.gif width="137" height="55" align="bottom"]]
 * The period, speed and acceleration of a satellite are only dependent upon the radius of orbit and the mass of the central body that the satellite is orbiting.
 * Just as in the case of the motion of projectiles on earth, the mass of the projectile has no affect upon the acceleration towards the earth and the speed at any instant.

1/9/12: D-E Summary (Method 1)
__Topic Sentence: Astronauts feel weightless because there are no contact forces acting upon them in space, and instead only the force of gravity.__
 * D: Weightlessness in Orbit**
 * Different than sensation in roller coaster, felt because there are no contact forces in space.
 * Contact forces: Normal, friction, tension
 * Action-at-a-distance forces: gravity
 * Weightlessness is a sensation experienced by an individual when there are no external objects touching one's body and exerting a push or pull upon it.
 * Scales measure the external contact force that is being applied to your body.
 * The force of gravity is the only force acting upon their body, so they are in free-fall, keeping them in orbit.

__Topic Sentence: The work-energy theorem dictates that while kinetic and potention energy of satellites may change during orbit, their total mechanical energy always remains constant.__ >
 * E: Energy Relationships for Satellites**
 * Satellites orbit in circular motion with a constant speed at the same height above the surface of the earth.
 * It moves with a tangential velocity that allows it to fall at the same rate at which the earth curves, with the force of gravity acts in a direction perpendicular to the direction that the satellite is moving.
 * Since perpendicular components of motion are independent, the inward force cannot affect the magnitude of the tangential velocity, so there is no acceleration in the tangential direction and the satellite remains in circular motion at a constant speed.
 * A satellite orbiting the earth in elliptical motion will experience a component of force in the same or the opposite direction as its motion, meaning a force is capable of slowing down and speeding up the satellite.
 * When the satellite moves away from the earth, there is a component of force in the opposite direction as its motion, and when the satellite moves towards the earth, there is a component of force in the same direction as its motion.
 * The speed of a satellite in elliptical motion is constantly changing - increasing as it moves closer to the earth and decreasing as it moves further from the earth.
 * Work-energy theorem: the initial amount of total 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. The mechanical energy can be either in the form of potential energy (energy of position - usually vertical height) or kinetic energy (energy of motion).
 * ==** **KEi + PEi + Wext = KEf + PEf** **==
 * The Wext term in this equation is representative of the amount of work done by external forces, which is gravity for satellites. Since gravity is considered an [|internal (conservative) force], the Wext term is zero. The equation can then be simplified to:
 * ==** **KEi + PEi = KEf + PEf** **==
 * Therefore, the total mechanical energy of the system is conserved ( the sum of kinetic and potential energies is unchanging ).
 * <span style="display: block; font-family: 'Times New Roman',Times,serif; text-align: left;">In circular orbits, if speed and height are constant, then potential, kinetic, and mechanical energy are also constant.
 * <span style="display: block; font-family: 'Times New Roman',Times,serif; text-align: left;">In elliptical orbits, speed and height changes, and potential and kinetic energy changes, but total mechanical energy stays the same.