Lesson A:
Speed and Force are related when talking about circular motion:
The formula for finding average speed for circular motion is: . When calculating the average speed of an object in circular motion we use circumference. We can define uniform circular motion as any object that moves in a circular pattern, this objects motion will occur at a constant or uniform speed. The velocity of the vector is tangential, meaning that at any place on the circle the direction of a drawn tangent line is the direction of the velocity vector.
Lesson B:
Acceleration:
An object that experiences uniform circular motion is constantly changing its direction of velocity, and thus it is accelerating. An object that is moving in a circular pattern at a constant rate is accelerating toward the center of the circle. If we were to do a lab with this concept we might want to us an accelerometer which can measure the acceleration of an object.
Lesson C:
The Centripetal Force Requirement:
We can define the centripetal force requirement by determining if an object has inward acceleration and if it does it also has an inward acting force. An example would be when a car accelerates or decelerates. The person is forced the opposite way of the car (accelerate=backward decelerate=forward). This occurs because of inertia. Work is a force acting upon an object that causes some form of displacement. A formula for work would be Work= Force*displacement*cosine(theta). The centripetal force is perpendicular to the tangential velocity of the object that is traveling in the circular path and thus the force can affect the direction of its velocity vector without changing its magnitude.
Lesson D:
The Centrifugal:
Centrifugal means away from the center or outward. The most common misconception is that objects in circular motion are experiencing an outward force. However this is not the case, the inward-directed acceleration requires an inward force. Without this inward force the object would continue its motion straight ahead tangent to the perimeter of the circle and thus would not form a circular pattern.
Lesson E:
The Math of Circular motion.
When completing circular motion problems it is important to understand the formulas that go along with it. For acceleration the formula that we will be using is acceleration=velocity^2/radius or simply a=v2/R. The equations below are also important when we are looking to get net force.
Physics Classroom Lesson 2 12/22/11:
Method 1:
Lesson A: Newton's Second Law - Revisited
Topic Sentence: When solving circular motion problems the use of free body diagrams is essential.
Newtons 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 law is often expressed in the form of the following two equations.
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.
Circular motion principles can be combined with Newton's second law to analyze a physical situation, consider a car moving in a horizontal circle on a level surface.
Suggested Method of Solving Circular Motion Problems
construct a free-body diagram
Use circular motion equations to determine any unknown information
Lesson B: Roller Coasters and Amusement Park Physics
Topic Sentence: Roller coasters have centripetal acceleration on dips, hills, and clothoid loops.
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.
A change in direction is one characteristic of an accelerating object. In addition to changing directions, the rider also changes speed.
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.
Dips and Hills:
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.
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 a = v2 / R
Lesson C: Athletics:
Topic Sentence: Any turn that an athlete makes is considered a part of a circle.
Athletes also experience centripetal force, which is characterized by an inward acceleration and caused by an inward net force.
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.
With a skater there is a reaction force of the ice pushing upward and inward upon the skate.
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.
Physics Classroom Lesson 3 1/4/12:
Method 1:
Lesson A: Gravity is More Than a Name
Topic Sentence: Gravity causes an acceleration of our bodies during a brief trip off the earths surface and back.
Gravity must be understood in terms of its cause, its source, and its far-reaching implications on the structure and the motion of the objects in the universe.
When we jump up the force of gravity slows us down. And as we fall back to Earth after reaching the peak of our motion, the force of gravity speeds us up. In this sense, the force gravity causes an acceleration of our bodies during this brief trip away from the earth's surface and back.
In fact, many students of physics have become accustomed to referring to the actual acceleration of such an object as the acceleration of gravity.
Lesson B: The Apple, the Moon, and the Inverse Square Law
Topic Sentence: The inverse square law explains why the force of gravity between Earth and any other object is inversely proportional to the square of the distance that separates the object from Earth's core center.
German mathematician and astronomer Johannes Kepler mathematically analyzed known astronomical data in order to develop three laws to describe the motion of planets about the sun. Kepler's three laws emerged from the analysis of data.
- 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)
It is often claimed that the notion of gravity as the cause of all heavenly motion was instigated when he was struck in the head by an apple while lying under a tree in an orchard in England. This led him to his notion of universal gravitation.
Newton knew that the force of gravity must somehow be "diluted" by distance.
The force of gravity between the earth and any object is inversely proportional to the square of the distance that separates that object from the earth's center. The moon, being 60 times further away than the apple, experiences a force of gravity that is 1/(60)2 times that of the apple. The force of gravity follows an inverse square law.
Lesson C: Newton's Law of Universal Gravitation
Topic Sentence: Universal Gravitational constant is equal to (G) G = 6.673 x 10-11 N m2/kg2.
This comparison led him to conclude that the force of gravitational attraction between the Earth and other objects is inversely proportional to the distance separating the earth's center from the object's center. But distance is not the only variable affecting the magnitude of a gravitational force. Consider Newton's famous equation
Fnet = m • a
Newton knew that the force that caused the apple's acceleration (gravity) must be dependent upon the mass of the apple. And since the force acting to cause the apple'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. 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.
Newton's law of universal gravitation is about the universality of gravity. ALL objects attract each other with a force of gravitational attraction. Gravity is universal.
Since gravitational force is inversely proportional to the separation distance between the two interacting objects, more separation distance will result in weaker gravitational forces.
Another means of representing the proportionalities is to express the relationships in the form of an equation using a constant of proportionality. This equation is shown below.
The constant of proportionality (G) in the above equation is known as the universal gravitation constant. G = 6.673 x 10-11 N m2/kg2
Lesson D:
Cavendish and the Value of G
Topic Sentence: The constant of proportionality is “G” or the universal gravitational constant.
Isaac Newton's law of universal gravitation proposed that the gravitational attraction between any two objects is directly proportional to the product of their masses and inversely proportional to the distance between their centers.
The constant of proportionality in this equation is G - the universal gravitation constant.
Lesson E:
The Value of g
Topic Sentence: We can combine two equation to calculate the force of gravity with which an object is attracted to the earth.
Now in this unit, a second equation has been introduced for calculating the force of gravity with which an object is attracted to the earth.
where d represents the distance from the center of the object to the center of the earth.
The Clock Work Universe parts 1-4 1/5/12:
Method 5: Part 1: Mechanics and determinism part 1:
The men who challenged the Geocentric model:
Nicolaus Copernicus launched a scientific revolution by rejecting the prevailing Earth-centred view of the Universe in favor of a heliocentric view in which the Earth moved round the Sun.
The most famous of these must surely have been Galileo, who was summoned to appear before the Inquisition in 1633, on a charge of heresy, for supporting Copernicus' ideas.
Johannes Kepler (1571-1630) devised a modified form of Copernicanism that was in good agreement with the best observational data available at the time. According to Kepler, the planets did move around the Sun, but their orbital paths were ellipses rather than collections of circles. He did speculate that they might be impelled by some kind of magnetic influence emanating from the Sun.
Part 2: Mechanics and determinism part 2:
The Birth of Coordinate Geometry:
Kepler's ideas were underpinned by new discoveries in mathematics. Chief among these was the realization, by René Descartes, that problems in geometry can be recast as problems in algebra.
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 3: Mechanics and determinism part 3:
Newton changed science as we know it:
For years before Newton, people had been trying to understand the world from a scientific perspective, discovering laws that would help explain why things happen in the way that they do. He discovered a convincing quantitative framework that seemed to underlie everything else.
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 and determinism part 4:
The Law that applies to everything:
In keeping with his grand vision, Newton proposed just one law for gravity - a law that worked for every scrap of matter in the Universe. By combining this law with his general laws of motion, Newton was able to demonstrate mathematically that a single planet would move around the Sun in an elliptical orbit, just as Kepler claimed each of the planets did.
Newton’s discoveries became the basis for a detailed and comprehensive study of mechanics (the study of force and motion). Determinism: the utilization of Newtonian mechanics to predict numerical values for the questionable components of the universe
1. The orbit of the planets about the sun is an ellipse, with the sun being located at one focus (Law of Ellipses)
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)
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)
Circular motion of Satellites
Gravity is the only force that acts on a satellite
In order to enter earths orbit a satellite needs to be launched at a specific speed
Because the satellite is not as large as the earth just like the moon it orbits it
The math of Satellite motion
When doing Satellite motion problems use the following equations:
Orbital Weightlessness
Weightlessness is a sensation experienced when all contact forces are removed
The result is a state of perpetual free fall
Satellites and Energy
Work-Energy Theorem- the initial amount of total mechanical energy of a system plus the work done by external forces on that system is equal to the final amount of total mechanical energy of the system
Work Energy Theorem:
When the external work=0 The equation is simplified to:
The total mechanical energy of the system is conserved
If speed and height are constant, than potential, kinetic, and mechanical energy will be as well
In elliptical orbits, speed and height changes, and therefore potential and kinetic energy changes, but total mechanical energy remains the same
Joey Miller Chapter 5 Wikispace:
Table of Contents
Physics Classroom Lesson 1 Sections A-E Method 5 12/13/11:
Lesson A:
Speed and Force are related when talking about circular motion:
The formula for finding average speed for circular motion is:
Lesson B:
Acceleration:
An object that experiences uniform circular motion is constantly changing its direction of velocity, and thus it is accelerating. An object that is moving in a circular pattern at a constant rate is accelerating toward the center of the circle. If we were to do a lab with this concept we might want to us an accelerometer which can measure the acceleration of an object.
Lesson C:
The Centripetal Force Requirement:
We can define the centripetal force requirement by determining if an object has inward acceleration and if it does it also has an inward acting force. An example would be when a car accelerates or decelerates. The person is forced the opposite way of the car (accelerate=backward decelerate=forward). This occurs because of inertia. Work is a force acting upon an object that causes some form of displacement. A formula for work would be Work= Force*displacement*cosine(theta). The centripetal force is perpendicular to the tangential velocity of the object that is traveling in the circular path and thus the force can affect the direction of its velocity vector without changing its magnitude.
Lesson D:
The Centrifugal:
Centrifugal means away from the center or outward. The most common misconception is that objects in circular motion are experiencing an outward force. However this is not the case, the inward-directed acceleration requires an inward force. Without this inward force the object would continue its motion straight ahead tangent to the perimeter of the circle and thus would not form a circular pattern.
Lesson E:
The Math of Circular motion.
When completing circular motion problems it is important to understand the formulas that go along with it. For acceleration the formula that we will be using is acceleration=velocity^2/radius or simply a=v2/R. The equations below are also important when we are looking to get net force.
Physics Classroom Lesson 2 12/22/11:
Method 1:Lesson A:
Newton's Second Law - Revisited
Topic Sentence: When solving circular motion problems the use of free body diagrams is essential.
Newtons 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 law is often expressed in the form of the following two equations.
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.
Circular motion principles can be combined with Newton's second law to analyze a physical situation, consider a car moving in a horizontal circle on a level surface.
Suggested Method of Solving Circular Motion Problems
construct a free-body diagram
Use circular motion equations to determine any unknown information
Lesson B:
Roller Coasters and Amusement Park Physics
Topic Sentence: Roller coasters have centripetal acceleration on dips, hills, and clothoid loops.
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.
A change in direction is one characteristic of an accelerating object. In addition to changing directions, the rider also changes speed.
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.
Dips and Hills:
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.
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 a = v2 / R
Lesson C:
Athletics:
Topic Sentence: Any turn that an athlete makes is considered a part of a circle.
Athletes also experience centripetal force, which is characterized by an inward acceleration and caused by an inward net force.
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.
With a skater there is a reaction force of the ice pushing upward and inward upon the skate.
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.
Physics Classroom Lesson 3 1/4/12:
Method 1:Lesson A:
Gravity is More Than a Name
Topic Sentence: Gravity causes an acceleration of our bodies during a brief trip off the earths surface and back.
Gravity must be understood in terms of its cause, its source, and its far-reaching implications on the structure and the motion of the objects in the universe.
When we jump up the force of gravity slows us down. And as we fall back to Earth after reaching the peak of our motion, the force of gravity speeds us up. In this sense, the force gravity causes an acceleration of our bodies during this brief trip away from the earth's surface and back.
In fact, many students of physics have become accustomed to referring to the actual acceleration of such an object as the acceleration of gravity.
Lesson B:
The Apple, the Moon, and the Inverse Square Law
Topic Sentence: The inverse square law explains why the force of gravity between Earth and any other object is inversely proportional to the square of the distance that separates the object from Earth's core center.
German mathematician and astronomer Johannes Kepler mathematically analyzed known astronomical data in order to develop three laws to describe the motion of planets about the sun. Kepler's three laws emerged from the analysis of data.
- 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)
It is often claimed that the notion of gravity as the cause of all heavenly motion was instigated when he was struck in the head by an apple while lying under a tree in an orchard in England. This led him to his notion of universal gravitation.
Newton knew that the force of gravity must somehow be "diluted" by distance.
The force of gravity between the earth and any object is inversely proportional to the square of the distance that separates that object from the earth's center. The moon, being 60 times further away than the apple, experiences a force of gravity that is 1/(60)2 times that of the apple. The force of gravity follows an inverse square law.
Lesson C:
Newton's Law of Universal Gravitation
Topic Sentence: Universal Gravitational constant is equal to (G) G = 6.673 x 10-11 N m2/kg2.
This comparison led him to conclude that the force of gravitational attraction between the Earth and other objects is inversely proportional to the distance separating the earth's center from the object's center. But distance is not the only variable affecting the magnitude of a gravitational force. Consider Newton's famous equation
Fnet = m • a
Newton knew that the force that caused the apple's acceleration (gravity) must be dependent upon the mass of the apple. And since the force acting to cause the apple'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. 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.
Newton's law of universal gravitation is about the universality of gravity. ALL objects attract each other with a force of gravitational attraction. Gravity is universal.
Since gravitational force is inversely proportional to the separation distance between the two interacting objects, more separation distance will result in weaker gravitational forces.
Another means of representing the proportionalities is to express the relationships in the form of an equation using a constant of proportionality. This equation is shown below.
The constant of proportionality (G) in the above equation is known as the universal gravitation constant. G = 6.673 x 10-11 N m2/kg2
Lesson D:
Cavendish and the Value of G
Topic Sentence: The constant of proportionality is “G” or the universal gravitational constant.
Isaac Newton's law of universal gravitation proposed that the gravitational attraction between any two objects is directly proportional to the product of their masses and inversely proportional to the distance between their centers.
The constant of proportionality in this equation is G - the universal gravitation constant.
Lesson E:
The Value of g
Topic Sentence: We can combine two equation to calculate the force of gravity with which an object is attracted to the earth.
Now in this unit, a second equation has been introduced for calculating the force of gravity with which an object is attracted to the earth.
where d represents the distance from the center of the object to the center of the earth.
The Clock Work Universe parts 1-4 1/5/12:
Method 5:
Part 1:
Mechanics and determinism part 1:
The men who challenged the Geocentric model:
Nicolaus Copernicus launched a scientific revolution by rejecting the prevailing Earth-centred view of the Universe in favor of a heliocentric view in which the Earth moved round the Sun.
The most famous of these must surely have been Galileo, who was summoned to appear before the Inquisition in 1633, on a charge of heresy, for supporting Copernicus' ideas.
Johannes Kepler (1571-1630) devised a modified form of Copernicanism that was in good agreement with the best observational data available at the time. According to Kepler, the planets did move around the Sun, but their orbital paths were ellipses rather than collections of circles. He did speculate that they might be impelled by some kind of magnetic influence emanating from the Sun.
Part 2:
Mechanics and determinism part 2:
The Birth of Coordinate Geometry:
Kepler's ideas were underpinned by new discoveries in mathematics. Chief among these was the realization, by René Descartes, that problems in geometry can be recast as problems in algebra.
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 3:
Mechanics and determinism part 3:
Newton changed science as we know it:
For years before Newton, people had been trying to understand the world from a scientific perspective, discovering laws that would help explain why things happen in the way that they do. He discovered a convincing quantitative framework that seemed to underlie everything else.
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 and determinism part 4:
The Law that applies to everything:
In keeping with his grand vision, Newton proposed just one law for gravity - a law that worked for every scrap of matter in the Universe. By combining this law with his general laws of motion, Newton was able to demonstrate mathematically that a single planet would move around the Sun in an elliptical orbit, just as Kepler claimed each of the planets did.
Newton’s discoveries became the basis for a detailed and comprehensive study of mechanics (the study of force and motion). Determinism: the utilization of Newtonian mechanics to predict numerical values for the questionable components of the universe
Physics Classroom Lesson 4 Sections (A-C)&(D-E) 1/8/12:
Kepler’s Three Laws of Planetary Motion:
1. The orbit of the planets about the sun is an ellipse, with the sun being located at one focus (Law of Ellipses)
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)
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)
Circular motion of Satellites
Gravity is the only force that acts on a satellite
In order to enter earths orbit a satellite needs to be launched at a specific speed
Because the satellite is not as large as the earth just like the moon it orbits it
The math of Satellite motion
When doing Satellite motion problems use the following equations:
Orbital Weightlessness
Weightlessness is a sensation experienced when all contact forces are removed
The result is a state of perpetual free fall
Satellites and Energy
Work-Energy Theorem- the initial amount of total mechanical energy of a system plus the work done by external forces on that system is equal to the final amount of total mechanical energy of the system
Work Energy Theorem:
When the external work=0 The equation is simplified to:
The total mechanical energy of the system is conserved
If speed and height are constant, than potential, kinetic, and mechanical energy will be as well
In elliptical orbits, speed and height changes, and therefore potential and kinetic energy changes, but total mechanical energy remains the same