Ch5_KostmanL

= = Chapter 5: Circular Motion toc

Lesson 1: Motion Characteristics for Circular Motion
Lauren Kostman 12/13/11 Homework- summarizing method 5 __**Lesson a: Speed and Velocity**__ //Learn the difference between speed and velocity, it's simple!// Uniform circular motion is the motion of an object in a circle with a constant speed. Although the //speed// remains constant, there still is an acceleration. However, it's not due to a change in speed (since this is constant), but rather due to the change in direction (since velocity is a vector quantity). It's average speed can be calculated using the formula 2*pi*radius/time. The //direction// of the velocity at each instant in a direction is __tangent__ to the circle. __**Lesson b: Acceleration**__ //Discover the truth as to why there's an acceleration!// Objects moving in a uniform circular path have an acceleration due to the change in direction of the velocity vector. By subtracting the initial velocity from the final velocity and dividing this by the amount of time, one can calculate the average acceleration. The object accelerates towards the center of the circle. Having less inertia, the smaller mass experiences the greater acceleration, and so it leans in the direction of the acceleration. __**Lesson c: The Centripetal Force Requirement**__ //Read all about the need for an inward force!// Objects moving in a uniform circular path have a "centripetal force requirement"; this means that there must be a force that causes it to accelerate inward. This force can be friction, tension, weight, etc. Due to Newton's Law of Inertia, there must be an unbalanced force for the object to move in a circle (since there's acceleration).This force is directed perpendicular to the tangential velocity, allowing the direction of the object's velocity vector to change its direction without changing its magnitude. __**Lesson d: The Forbidden F-word**__ //Get the idea of an **out**ward for **out** of your head!// Centrifugal means away from the center, or outward. Such a force does NOT exist for objects moving in a circular path; it's the "forbidden f-word". But, there is an **//inward// -directed acceleration that demands an inward force (centripetal force). Without this inward force, an object would maintain a straight-line motion tangent to the perimeter of the circle (circular motion would be impossible). It may //seem// as though an object is being pushed outward, but this is only relative and isn't in fact true; in actuality and due to inertia, there is't any force pushing it outwards while it moves in a circle. ** __**Lesson e: Mathematics of Circular Motion**__ //Use these formulas to calculate all you need to know about circular motion!// The speed of an object moving in a circle can be calculated by multiplying the diameter (or the radius times 2) by pi, and then dividing this by the amount of time. The acceleration can be calculated by multiplying 4 *pi^2*radius, divided by the amount of time squared. The net force acting upon an object moving in circular motion (directed inwards) can be calculated by multiplying the mass by (4*pi^2*radius) divided by the amount of time squared. As acceleration increases, so does net force, but as acceleration decreases, the mass increases. For a constant mass and radius,the net force is proportional to the speed^2.

Lauren Kostman 12/22/11 Summarizing Method 3

Lesson 2: Applications of Circular Motion
__Lesson a: Newton's Second Law- Revisited__ How does Newton's Second Law apply to circular motion? - This law states that the net force equals the mass times the acceleration; this same equation can be applied to problems relating to circular motion. - Applying the concept of a centripetal force requirement, one knows that the net force acting on the object is directed inwards; forces on this axis (ex: friction or tension) are part of the overall (centripetal) net force. So, one can use the same equation (net force=m*a) to solve for variables. - Additionally, the same equation used to solve for friction still applies: f= µ*N. __Lesson b: Ammusement Park Physics__ How does physics relate to amusement park rides, such as roller coasters? - Roller coasters usually have either a loop, dips/hills, and/ or banked turns, all of which are part of circular motion, - The velocity, acceleration, and other aspects to these rides ride can be calculated using the equations stated above. - The centripetal force always points towards the center of the loop or hill. __Lesson c: Athletics__ Explain the connection between physics and athletics. - There are different types of turns (sharp, gradual, changing radii, widely varying radii, etc.), all of which can be considered part of a circle; many of these are done in sports. - 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. - Example: ice skater: ice supplies both an upward and inward component of force on the skater. The vertical component balances the force of gravity and the horizontal component proves the inward force. Circular motion is very applicable to everyday life, including roller coasters and sports. By applying Newton's Second Law, one can calculate many values, including velocity, acceleration, and different forces. While riding on a roller coaster or ice skating, for example, there are many forces acting upon the person, including a centripetal force that draws them inward towards the center of their circular path.
 * Summary of Lesson 2:**

Lauren Kostman 1/3/12 Summarizing Method 3

Lesson 3: Universal Gravitation
__Lesson a: Gravity is More than a Name__ What is the difference between the force of gravity and the acceleration of gravity? - The force of gravity is a force that exists between the Earth and the objects that are near it. It's the force that acts upon objects when they're thrown upward (from the earth); as the object goes up, the force of gravity slows it down, and speeds it up when the objects falls back down. It causes an acceleration during this "trip" away from the earth's surface and back. - Such an acceleration can be referred to as the acceleration of gravity. This is the acceleration experienced by an object when the only force acting upon it is the force of gravity. On earth, this value is 9.8 m/s/s (g), regardless of mass. __Lesson b: The Apple, the Moon, and the Inverse Square Law__ What is the inverse square law? - The laws of mechanics that govern the motions of objects on Earth also govern the movement of objects in the heavens. - The force of gravity is lessened by more distance; it follows an inverse square law. - According to this equation, the force of gravity is //inversely// related to the distance. __Lesson c: Newton's Law of Universal Gravitation__ What is Newton's Law of Universal Gravitation? - The force of gravity between the earth and an 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. - This law extends gravity beyond earth (**universality** of gravity). - Force of gravity is directly proportional to the mass of both interacting objects, so more massive objects will attract each other with a greater gravitational force (as mass increases, so does gravitational force). __Lesson d: Cavendish and the Value of G__ Briefly describe Cavendish's apparatus that determined the value of "G". - This device involved a light and a rod with 2 small lead spheres attached to each end suspenmded by a thin wire. When the rod is twisted, it exerts a torsional force that's 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 bought the larger spheres near the smaller ones, until they exerted a gravitational force on the smaller ones. When the object came to rest, Cavendish was able to determine the gravitational force of attraction between the masses. This was the value of "G". __Lesson e: The Value of g__ What is the meaning of the value of "g"? - The acceleration of gravity (g) is dependent upon the mass of the earth and the distance that an object is from earth's center. - The value of g is inversely proportional with the distance form the center of the earth. - 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 that that planet). The force of gravity can be applied to planets other than the earth.The value of "G" is the universal gravitational constant, whereas "g" represents the acceleration of gravity. Newton, Kepler, and Cavendish all contributed to the knowledge and discoveries of universal gravitation.
 * Summary(a-e):**

Lesson 2: The Clockwork Universe
Lauren Kostman 1/4/12 Summarizing Method 3 __**Mechanics and Determinism**__ __Part 1:__ What were the 3 views of planetary motion? 1.) Geocentric- the earth is the center of the universe. 2.) Heliocentric- the sun is the center of the universe (Copernican system). 3.) Keplerian system- planets follow an elliptical orbit, with the sun at one focus of the ellipse. __Part 2:__ What were the new mathematical discoveries that underpinned Kepler's ideas? - People realized that problems in geometry can be done in algebra. The problem can be displayed on a grid (points would have x and y coordinates), but a circle can also be characterized by an equation,, by taking the square root of the x and y coordinates each squared. - This is the beginning of //coordinate geometry// (represents geometrical shapes by equations). __Part 3:__ What are the key components of Newton's laws about the world? - He believed that all the motion we see around us can be explained in terms of a single set of laws. 3 key points of these include: 1.) Newton focused on //deviation from steady motion// (example: deviation that occurs when an object speeds up, slows down, or goes in a new direction). 2.) Wherever deviation from steady motion occurred, Newton looked for a cause (example: slowing down could be caused by braking). He described such as cause as a force. 3.) He produced a quantitative link between force and deviation from steady motion and (in the case of gravity) quantified the force by proposing his famous law of universal gravitation. __Part 4:__ How did Newton's discoveries go beyond those of Kepler and lead to the study of mechanics and determinism? - Newton was able to mathematically show that a planet moved around the sun in an elliptical orbit (as Kepler said). Mechanics is the study of force and motion; Newton's discoveries became the basis of this. It led to a mechanical world-view regarding the universe as something that went according to mathematical laws with precision and inevitability of a well-made clock. The detailed part of Newton's laws was that once this clockwork was set in motion, its future development was entirely predictable; this is called determinism. Due to the discoveries of many scientists, such as Kepler and Newton, many new understanding in terms of universal gravitation were made. The world began applying mathematics and scientific principles to the universe/ solar system, and discovered that it didn't function as they initially believed (geocentric model, etc.).
 * Kepler had no real reason to expect that plants moved in ellipses. but he speculated that they were driven by some kind of magnetic influence sent from the sun.
 * Summary:**

Lauren Kostman 1/6/12 Summarizing Method 3

Lesson 4: Planetary and Satellite Motion
__Lesson a: Kepler's Three Laws__ What are Kepler's 3 laws of planetary motion? 1. The Law of Ellipses: paths of planets around the sun are elliptical in shape, with the center of the sun being located at one focus. 2. The 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. 3. The Law of Harmonies: ratio of squares of the periods of any two planets is equal to the ratio of the cubes of their average distances from the sun. - Kepler believed the planets were "magnetically" driven by the sun to orbit in their elliptical trajectories. There wasn't any interaction between the planets themselves. __Lesson b: Circular Motion Principles for Satellites__ What is the circular motion principle for satellites? - A satellite is any object that orbits the earth, sun, or other massive body. They can be categorized as natural (ex: moon, planets) or man-made. - A satellite is a projectile (the only force acting upon it is gravity). If launched with sufficient speed, it will fall towards the earth at the same rate that the earth curves, causing it to stay at the same height above the earth, and orbit around it (in a circular path), and thus never actually hit the earth. - For every 8,000 meters along the horizon, the earth curves downward by 5 meters.For a projectile to orbit the earth, it must travel 8,000 meters horizontally for every 5 meters of vertical fall. - Velocity of a satellite is directly tangent to the circle at every point along its path. - The acceleration is directed towards the center of the circle (the center of whatever it's orbiting). - Satellites encounter inward forces and accelerations and tangential velocities.An orbiting satellite requires a centripetal force. - Even moving in elliptical motion, there is a tangential velocity and an inward acceleration and force. __Lesson c: Mathematics of Satellite Motion__ Describe the mathematics of the motion of satellites. - Equation for the velocity of a satellite moving about a central body in circular motion: - Equation for the acceleration of a satellite in circular motion: - The period of a satellite (T) and the mean distance from the central body (R) are related as following: Kepler created 3 laws to describe planetary motion- the law of ellipses, equal areas, and harmonies. Satellites are projectiles; the only force acting upon them is that of gravity. Three equations can be used to solve for the velocity, acceleration, and period of satellites.
 * The period, speed, and acceleration of an orbiting satellite are NOT dependent upon the mass of the satellite!
 * Summary (a-c):**

Lauren Kostman 1/9/12

Lesson 4 (d-e):
__Lesson d: Weightlessness in Orbit__ What is the meaning and cause of weightlessness? - Weightlessness is a sensation experienced by someone when there are no external objects touching one's body and exerting a push or pull on it. It exists when all contact forces are removed. They are common to situations when the object is in a state of free fall (only force acting upon object is gravity- not a contact force). - Weightlessness has very little to do with weight and mostly to do with the presense or absence of contact forces (example: riding on roller coaster and momentarily lifted out of seat). Describe scale readings and weight. - Scale's don't measure "weight", they measure the upward force applied by the scale to balance the downward force of gravity acting upon the object. It measures the external contact force that is being applied to your body. Describe weightlessness in orbits. - There must be a force of gravity in order for their to be an orbit. - Earth-orbiting astronauts are weightless because there is no external contact force puashing or pulling upon their bodyies (only gravity acts upon them). - Since gravity is NOT a contact force, it cannot be felt and so it doesn't provide a sensation of their weight (so they feel weightless). __Lesson e: Energy Relationships for Satellites__ What are the energy relationships for satellites? - Orbits of satellites about a central massive body can be described as either circular or elliptical. - Work-energy theorem: initial amount of total mechanical energy plus the work done by external forces on that system is equal to the final amount of total mechanical energy. Describe the energy of circular orbits and of elliptical orbits. - Circular: constant radius and constant speed. Since kinetic energy is dependent on the speed of an object, the amount of kinetic energy will be constant throughout the satellite's motion. Potential energy depends on the height of an object, and since this remains constant in circular orbits, this remains constant too. Since kinetic and potential energy are both constant, the total mechanical energy remains constant. - Elliptical: total amount of mechanical energy remains constant. Speed changes, so kinetic energy changes.It speeds up as its height (or distance from the earth) is decreasing and slow down as its height (or distance from the earth) is increasing. Weightlessness doesn't actually mean that the object doesn't weigh anything; it is the sensation of feeling :weightless" because the only force acting upon the object is gravity, which isn't a contact force and so the object feels like there are not forces acting on it. Satellites can move in circular and elliptical orbits; for circular, the TME remains constant; for elliptical, the speed changes, so the kinetic and potential energy are NOT constant.
 * Summary (d-e):**

Lauren Kostman