Ch5_solomonk

** Summary of Chapter 5, Lesson 1: Motion Characteristics for Circular Motion **
a. Speed and Velocity: Familiar Terms, New Ideas The same concepts that were used in previous units can be applied to describe the motion of objects that mtoc ove in circular motion. Uniform circular motion is the motion of an object in a circle with a constant speed. The formula for calculating the average speed of circular motion is distance/time, which is equal to circumference/time, and because circumference equals 2(pi)r, the formula can be converted to ((2)pi)(r))/T. Additionally, the average speed and the radius of the circle are directly proportional. Also interesting is that, while the magnitude of the velocity vector may be constant, the direction of the velocity is constantly changing, and is tangential to the circle.

b. Watch Out, It's Accelerating! Because velocity is changing direction, there is acceleration in circular motion. Accelerometers are devices that are commonly used for measuring acceleration.

c. It's Centripetal! Not Centrifugal! The centripetal force requirement means center seeking. For objects moving in circular motion, there is a net force acting towards the center which causes the object to seek the center. Objects moving in a circle experience centripetal force. The centripetal force for uniform circular motion alters the direction of the object without altering its speed. Work = (force)(displacement)(cosine theta).

d. Don't Fall into the Trap! Centrifugal means away from the center or outward. Without an inward force, an object would maintain a straight-line motion tangent to the perimeter of the circle. There are many misconceptions. Being thrown outward from the center of a circle does not mean that there was an outward force.

e. There is so Much to Learn from Equations The equation for Average Speed = 2piR/T. The equation for acceleration is V^2/R. The equation for net force is 4pi^2R/T^2. The acceleration of an object is directly proportional to the net force acting upon it. The acceleration of an object is inversely proportional to the mass.

**Summary of Lesson 2 a-c (method 1)**
a. Newton's Second Law - Revisited 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 net force must be equal to the mass times the acceleration.Subsequently, the acceleration of an object can be found if the mass of the object and the magnitudes and directions of each individual force are known. And the magnitude of any individual force can be determined if the mass of the object, the acceleration of the object, and the magnitude of the other individual forces are known. The process of analyzing such physical situations in order to determine unknown information is dependent upon the ability to represent the physical situation by means of a free-body diagram. 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. Applying the concept of a centripetal force requirement, we know that the net force acting upon the object is directed inwards.

b. Amusement Park Physics The thrill of roller coasters is not due to their speed, but rather due to their accelerations and to the feelings of weightlessness and weightiness that they produce. The most obvious section on a roller coaster where centripetal acceleration occurs is within the so-called **clothoid loops**. Roller coaster loops assume a tear-dropped shape that is geometrically referred to as a clothoid. A clothoid is a section of a spiral in which the radius is constantly changing. Unlike a circular loop in which the radius is a constant value, the radius at the bottom of a clothoid loop is much larger than the radius at the top of the clothoid loop. A mere inspection of a clothoid reveals that the amount of curvature at the bottom of the lop is less than the amount of curvature at the top of the loop. This change in speed as the rider moves through the loop is the second aspect of the acceleration that a rider experiences. Neglecting friction and air resistance, a roller coaster car experiences the force of gravity and the normal force.

c. Athletics Circular motion is common to almost all sporting events. The most common example of the physics of circular motion in sports involves the turn. Because turning a corner involves the motion of an object that is momentarily moving along the path of a circle, both the concepts and the mathematics of circular motion can be applied to such a motion. Conceptually, such an object is moving with an inward acceleration - the inward direction being towards the center of whatever //circle//the object is moving along. 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.

Summary of Chapter 5, Lesson 3 (method 1): Universal Gravitation
Gravity is a force that exists between the Earth and the objects that are near it. In this sense, the force gravity causes an acceleration of our bodies during this brief trip away from the earth's surface and back. the acceleration of gravity 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 (and assuming that the only significant force is gravity)
 * Lesson 3a Gravity is More than a Name**

Kepler’s 3 laws of motion: that it was 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 gravity. 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. Quite amazingly, the laws of mechanics that govern the motions of objects on Earth also govern the movement of objects in the heavens.
 * 3b 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)

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. Newton's law of universal gravitation is about the universality of gravity. 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. 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. Since gravitational force is inversely proportional to the separation distance between the two interacting objects, more separation distance will result in weaker gravitational forces. 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. 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. 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.
 * 3c Newton’s Laws of Universal Gravitation**
 * 3d Cavendish and the Value of G**

There are slight variations in the value of g about earth's surface. These variations 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. As one proceeds further from earth's surface - say into a location of orbit about the earth - the value of g changes still. the value of g will change as an object is moved further from Earth's center. As is evident from both the equation and the table above, the value of g varies inversely with the distance from the center of the earth. 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 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.
 * 3e The Value of G**

**Summary of Lesson 2 parts 1-4: The Clockwork Universe (method 1) 1/5/12**
In 1543, a century before Newton's birth, 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. According to Kepler, the planets //did// move around the Sun, but their orbital paths were ellipses rather than collections of circles.
 * Part 1:**

Kepler's ideas were underpinned by new discoveries in mathematics. This is the beginning of a branch of mathematics, called //coordinate geometry//, which represents geometrical shapes by equations, and which establishes geometrical truths by combining and rearranging those equations.
 * Part 2:**

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.
 * Part 3:**
 * 1) Newton concentrated not so much on motion, as on //deviation from steady motion// - deviation that occurs, for example, when an object speeds up, or slows down, or veers off in a new direction.
 * 2) Wherever deviation from steady motion occurred, Newton looked for a cause. Slowing down, for example, might be caused by braking. He described such a cause as a force. We are all familiar with the idea of applying a force, whenever we use our muscles to push or pull anything.
 * 3) Finally 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.


 * Part 4:**
 * mechanics ** (the study of force and motion). The upshot of all this was a mechanical world-view that regarded the Universe as something that unfolded according to mathematical laws with all the precision and inevitability of a well-made clock. its future development was, in principle, entirely predictable. This property of Newtonian mechanics is called **determinism**. It had an enormously important implication. Given an accurate description of the character, position and velocity of every particle in the Universe at some particular moment (i.e. the //initial condition// of the Universe), 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.

**Summary of Chapter 5 Lesson 4a-c** (method 1)

 * Lesson 4a: Kepler’s Three Laws **

Kepler's three laws of planetary motion can be described as follows:


 * The path of the planets about the sun is 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)
 * describes the speed at which any given planet will move while orbiting the sun. The speed at which any planet moves through space is constantly changing. A planet moves fastest when it is closest to the sun and slowest when it is furthest from the sun. Yet, if an imaginary line were drawn from the center of the planet to the center of the sun, that line would sweep out the same area in equal periods of time
 * 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)
 * The comparison being made 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


 * Lesson 4b: Circular Motion Principles for Satellites **

A satellite is any object that is orbiting the earth, sun or other massive body. Satellites can be categorized as natural satellites or man-made satellites. A satellite is a projectile.

So what launch speed does a satellite need in order to orbit the earth? The answer emerges from a basic fact about the curvature of 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. For this reason, 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.

The motion of an orbiting satellite can be described by the same motion characteristics as any object in circular motion. The velocity of the satellite would be directed tangent to the circle at every point along its path. The acceleration of the satellite would be directed towards the center of the circle - towards the central body that it is orbiting. And this acceleration is caused by a net force that is directed inwards in the same direction as the acceleration.

This centripetal force is supplied gravity. Were it not for this force, the satellite in motion would continue in motion at the same speed and in the same direction. It would follow its inertial, straight-line path. Observe that the inward net force pushes (or pulls) the satellite (denoted by blue circle) inwards relative to its straight-line path tangent to the circle.

Occasionally satellites will orbit in paths that can be described as ellipses. In such cases, the central body is located at one of the foci of the ellipse. The velocity of the satellite is directed tangent to the ellipse. The acceleration of the satellite is directed towards the focus of the ellipse. Once more, this net force is supplied by the force of gravitational attraction between the central body and the orbiting satellite. In the case of elliptical paths, there is a component of force in the same direction as (or opposite direction as) the motion of the object. Unlike uniform circular motion, the elliptical motion of satellites is not characterized by a constant speed.


 * Lesson 4c: Mathematics of Satellite Motion **

The motion of objects is governed by Newton's laws. The same simple laws that govern the motion of objects on earth also extend to the //heavens// to govern the motion of planets, moons, and other satellites. The mathematics that describes a satellite's motion is the same mathematics presented for circular

This net centripetal force is the result of the gravitational force that attracts the satellite towards the central body. Expressions for centripetal force and gravitational force can be set equal to each other. Observe that the mass of the satellite is present on both sides of the equation; thus it can be canceled by dividing through by m. Then both sides of the equation can be multiplied by R, leaving the following equation. Taking the square root of each side, leaves the equation for the velocity of a satellite moving about a central body in circular motion

Similar reasoning can be used to determine an equation for the acceleration of our satellite that is expressed in terms of masses and radius of orbit. The acceleration value of a satellite is equal to the acceleration of gravity of the satellite at whatever location that it is orbiting. The final equation that is useful in describing the motion of satellites is Newton's form of Kepler's third law.

There is an important concept evident in all three of these equations - the period, speed and the acceleration of an orbiting satellite are not dependent upon the mass of the satellite. 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. T he radius of orbit indicates the distance that the satellite is from the center of the earth.

Summary of Chapter 5, Lesson 4d-e 1/9/12 (method 1)
Contact versus Non-Contact Forces The upward chair force is sometimes referred to as a normal force and results from the contact between the chair top and your bottom end. This normal force is categorized as a contact force. [|Contact forces] can only result from the actual touching of the two interacting objects - in this case, the chair and you. The force of gravity acting upon your body is not a contact force; it is often categorized as an [|action-at-a-distance force]. The force of gravity is the result of your center of mass and the Earth's center of mass exerting a mutual pull on each other; this force would even exist if you were not in contact with the Earth. The force of gravity does not require that the two interacting objects (your body and the Earth) make physical contact; it can act over a distance through space. Since the upward normal force would equal the downward force of gravity when at rest, the strength of this normal force gives one a measure of the amount of gravitational pull. If there were no upward normal force acting upon your body, you would not have any sensation of your weight. Without the contact force (the normal force), there is no means of feeling the non-contact force (the force of gravity).
 * Lesson 4d: Weightlessness in Orbit **

Meaning and Cause of Weightlessness
 * Weightlessness ** is simply a sensation experienced by an individual when there are no external objects touching one's body and exerting a push or pull upon it. Weightless sensations exist when all contact forces are removed. Weightlessness is only a sensation.

Scale Readings and Weight While we use a scale to measure one's weight, the scale reading is actually a measure of the upward force applied by the scale to balance the downward force of gravity acting upon an object. When an object is in a state of equilibrium (either at rest or in motion at constant speed), these two forces are balanced. The upward force of the scale upon the person equals the downward pull of gravity (also known as weight). And in this instance, the scale reading (that is a measure of the upward force) equals the weight of the person.

Weightlessness in Orbit Earth-orbiting astronauts are weightless for the same reasons that riders of a free-falling amusement park ride or a free-falling elevator are weightless. They are weightless because there is no external contact force pushing or pulling upon their body. In each case, gravity is the only force acting upon their body. It is the force of gravity that supplies the [|centripetal force requirement] to allow the [|inward acceleration] that is characteristic of circular motion. The force of gravity is the only force acting upon their body. The astronauts are in free-fall. Like the falling amusement park rider and the falling elevator rider, the astronauts and their surroundings are falling towards the Earth under the sole influence of gravity. The astronauts and all their surroundings - the space station with its contents - are [|falling towards the Earth without colliding into it]. Their [|tangential velocity] allows them to remain in orbital motion while the force of gravity pulls them inward.

The orbits of satellites about a central massive body can be described as either circular or elliptical. a satellite orbiting about the earth in circular motion is moving with a constant speed and remains at the same height above the surface of the earth. It accomplishes this feat by moving with a tangential velocity that allows it to fall at the same rate at which the earth curves. At all instances during its trajectory, the force of gravity acts in a direction perpendicular to the direction that the satellite is moving. Since [|perpendicular components of motion are independent] of each other, the inward force cannot affect the magnitude of the tangential velocity. For this reason, there is no acceleration in the tangential direction and the satellite remains in circular motion at a constant speed. Thus, the 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. During this portion of the satellite's trajectory, the force does negative work upon the satellite and slows it down. When the satellite moves towards the earth, there is a component of force in the same direction as its motion. During this portion of the satellite's trajectory, the force does positive work upon the satellite and speeds it up. Subsequently, 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.
 * Lesson 4e: Energy Relationships for Satellites **


 * work-energy theorem **: states that 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).

The Wext term in this equation is representative of the amount of work done by [|external forces]. For satellites, the only force is gravity. Since gravity is considered an [|internal (conservative) force], the Wext term is zero. The equation can then be simplified to the following form.

While energy can be transformed from kinetic energy into potential energy, the total amount remains the same - mechanical energy is // conserved //.

Yet throughout the entire elliptical trajectory, the total mechanical energy of the satellite remains constant. An energy analysis of satellite motion yields the same conclusions as any analysis guided by Newton's laws of motion. A satellite orbiting in circular motion maintains a constant radius of orbit and therefore a constant speed and a constant height above the earth. A satellite orbiting in elliptical motion will speed up as its height (or distance from the earth) is decreasing and slow down as its height (or distance from the earth) is increasing. The same principles of motion that apply to objects on earth - Newton's laws and the work-energy theorem - also govern the motion of satellites in the heavens.
 * Energy Analysis of Elliptical Orbits **