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Circular Motion

=Motion Characteristics for Circular Motion; Lesson 1: A-E = December 13, 2011

LEARN ALL ABOUT THE CALCULATION OF AVERAGE SPEED! Before you can learn how to calculate the average speed of an object moving in a circular motion, there is one big thing you will need to know first. An object that is moving at constant speed in a circular motion is said to be covering that much distance on the perimeter. Let's say you have a train moving at a constant 10 m/s along the train tracks, which are in a circular shape, that train is said to be cover 10 meters along the perimeter of the circle ever second in time. Now, with this information, you can calculate average speed using this equation: To find circumference, you would need to use this formula: Circumference = 2*pi*Radius SO in esscence, the formula to find average speed for an object moving in a circular motion is:

VECTORS, VECTORS, AND MORE VECTORS! Just because an object that is moving in a circular motion can have a constant speed, it doesn't mean that it has a constant velocity. While speed is a scalar quantity, velocity is a vector quantity, meaning it has both magnitude and direction. So for an object moving in a circular path, the velocity is always changing! This vector is known as tangential. It represents the velocity vector at any instant in the direction of a tangent line drawn to the circle at the object's location.



TOP STORY ON THE DIRECTION OF ACCELERATION! For an object moving in a circular motion, the acceleration of that object is dependent upon thee velocity change and is in the same direction as the velocity change vector. This acceleration of object measured using an accelerometer. This tool consists of an object immersed in water. To test the vecot of acceleration for an object moving in a circle, the jar can be inverted and attached to the end of a short section of a wooden 2x4. 

 When the flame inside the jar deflects from its upright position, this signifies that there is an acceleration when the flame moves in a circular path t constant speed.

INERTIA, FORCE AND ACCELERATION FOR AN AUTOMOBILE PASSENGER! Newton's second law, which deals with inertia is something people experience in the passenger seat of a car. When sitting at an intersection and the light turns green, the car is accelerating from rest. Although the car accelerates forward, a persons body is leaning backward. The direction which the passengers lean is opposite the direction of the acceleration, and this is the result of the passenger's inertia. This is different when the car is making a turn at constant speed. It may feel like you are being accelerated outwards away from the center of the circle, but you are actually continuing in your straight line inertial path tangent to the circle while the car is accelerating out from under youl. The equation that would be used to find acceleration is:

CENTRIPETAL FORCE AND DIRECTION CHANGE! Any object moving in a circle experiences a centripetal force where there is a physical force pushing or pulling the object towards the center of the circle. The centripetal force for uniform circular motion actually changes the vector for the object without altering its speed. To find the amount of work done upon an object can be found with the equation: Work = Force * displacement * cosine (Theta)



CENTRIFUGAL! WHAT?! The centrifugal of an object means away from the center or outward and it is sometimes a common misconception with people. However, the truth is that there is an inward-directed acceleration that demands an inward force and without it, the object would maintian a straight line motion tangent to the perimeter of the circle. Here is an example:

EQUATIONS AS A GUIDE TO THINKING! Newton's second law is of help when figuring out how acceleration is related to the net force and the mass of an object. The equation is a = Fnet/m. This shows that the acceleration of an object is directly proportional to the net force acting upon it. Also, the acceleration of an object is inversely proportional to mass of the object. For an object moving in uniform circular motion the equation used is:

EQUATIONS AS A RECIPE FOR PROBLEM SOLVING! To select the proper equation for a problem there are several things to put into consideration. These are reading the problem, the identification of the known and required information in variable for, the selection of the relevant equations, substitution of known values into the equation, and using math to determine the answer.

= Circular Motion and Satellite Motion; Lesson 3: A-E = January 4, 2011

Directed reading method

1. What (specifically) did you read that you already understood well from our class discussion? Describe at least 2 items fully. The force of gravity and the acceleration of gravity (g) are two different things. When the only force acting upon an object is gravity, it the factor that causes the object to speed up if it is dropped, and slow down if it is going up.

The planets rotations are caused by some unbalanced force acting upon them that cause their motions. 2. What (specifically) did you read that you were a little confused/unclear/shaky about from class, but the reading helped to clarify? Describe the misconception you were having as well as your new understanding The effect of mass on the force of gravity and the effect of distance on the force of gravity. Properly used with this equation:

3. What (specifically) did you read that you still don’t understand? Please word these in the form of a question. Is the distance (d) in the equation below also the radius? 4. What (specifically) did you read that was not gone over during class toda y? Kepler's laws. They are: - The Law of Ellipses - The law of Equal Areas - The law of Harmonies

= The Clockwork Universe: Lesson 2; 1-4 = January 5, 2012

Rules-Based Summaries

Topic Sentence: Kepler had discovered that planets moved in an elliptical fashion around the sun.

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. By removing the Earth, and with it humankind, from the centre of creation, Copernicus had set the scene for a number of confrontations between the Catholic church and some of its more independently minded followers.

Galileo was 'shown the instruments of torture', and invited to renounce his declared opinion that the Earth moves around the Sun. This he did, though tradition has it that at the end of his renunciation he muttered 'Eppur si muove' ('And yet it moves').

<span style="color: #800000; font-family: Verdana,Arial,Helvetica,sans-serif;">**Figure 1.3 Three views of planetary motion**


 * [[image:http://physicalworld.org/restless_universe/figs/fig_1_3a.jpg width="145" height="125" caption="The Earth-centred view of the ancient Greeks and of the Catholic church in the sixteenth century"]] || <span style="color: #800000; font-family: Verdana,Arial,Helvetica,sans-serif;">(a) The Earth-centred view of the ancient Greeks and of the Catholic church in the sixteenth century. ||


 * <span style="color: #800000; font-family: Verdana,Arial,Helvetica,sans-serif;">(b) The Copernican system, in which the planets move in collections of circles around the Sun. || [[image:http://physicalworld.org/restless_universe/figs/fig_1_3b.jpg width="156" height="125" caption="The Copernican system in which the planets move in collections of circles around the Sun"]] ||


 * [[image:http://physicalworld.org/restless_universe/figs/fig_1_3c.gif width="142" height="124" caption="The Keplerian system in which a planet follows an elliptical orbit with the Sun at one focus of the ellipse"]] || <span style="color: #800000; font-family: Verdana,Arial,Helvetica,sans-serif;">(c) The Keplerian system in which a planet follows an elliptical orbit, with the Sun at one focus of the ellipse. ||

<span style="color: #800000; font-family: Verdana,Arial,Helvetica,sans-serif;">According to Kepler, the planets //did// move around the Sun, but their orbital paths were ellipses rather than collections of circles.

<span style="color: #800000; font-family: Verdana,Arial,Helvetica,sans-serif;">Kepler had no real reason to //expect// that the planets would move in ellipses, though he did speculate that they might be impelled by some kind of magnetic influence emanating from the Sun.

<span style="color: #800000; font-family: Verdana,Arial,Helvetica,sans-serif;">Topic Sentence: Coordinate geometry is helpful in deciphering the placements of planets.

<span style="color: #800000; font-family: Verdana,Arial,Helvetica,sans-serif;">Imagine a giant grid extending over the whole of space. <span style="color: #800000; font-family: Verdana,Arial,Helvetica,sans-serif;">The grid is calibrated (in centimetres) so the position of any point can be specified by giving its //x-// and //y-// coordinates on the grid. For example, the coordinates of point A are //x// = 3cm and //y// = 4cm.
 * [[image:http://physicalworld.org/restless_universe/figs/fig_1_4.gif width="175" height="188" align="left" caption="figure 1.4, locate the position of any point in terms of its x and y coordinates"]] || <span style="color: #800000; font-family: Verdana,Arial,Helvetica,sans-serif;">Fig 1.4 A two-dimensional coordinate system can be used to locate the position of any point in terms of its x- and y-coordinates. ||

The //x-// and y- coordinates of each point on the line obey the equation //y// = 0.5//x//, and this is said to be the equation of the line.

<span style="color: #800000; font-family: Verdana,Arial,Helvetica,sans-serif;">Similarly, the circle in Figure 1.5 is characterized by the equation
 * [[image:http://physicalworld.org/restless_universe/figs/fig_1_5.gif width="175" height="137" align="left" caption="figure 1.5, a 2-D coordinate system can represent lines and other geometrical shapes by equationsrep"]] || <span style="color: #800000; font-family: Verdana,Arial;">Fig 1.5 A two-dimensional coordinate system can be used to represent lines and other geometrical shapes by equations. ||

<span style="color: #800000; font-family: Verdana,Arial,Helvetica,sans-serif;">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.

<span style="color: #800000; font-family: Verdana,Arial,Helvetica,sans-serif;">Topic Sentence: Motion is described through three laws, written by Joseph Lagrange, but derived from Newton.

Newton's great achievement was to provide a synthesis of scientific knowledge. For the first time, scientists felt they understood the fundamentals, and it seemed that future advances would merely fill in the details of Newtonís grand vision.

The great Italian-French scientist Joseph Lagrange remarked: **'There is only one Universe... It can happen to only one man in the world's history to be the interpreter of its laws.'**

The motion we see around us can be explained in terms of a single set of laws.

**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.

<span style="color: #800000; font-family: Verdana,Arial,Helvetica,sans-serif;">**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.

Topic Sentence: Newton was able to create a basis of thought of which people had free will.

<span style="color: #800000; font-family: Verdana,Arial,Helvetica,sans-serif;">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, for moons and planets as well as for apples and the Earth.

<span style="color: #800000; font-family: Verdana,Arial,Helvetica,sans-serif;">Newton was able to demonstrate mathematically that a single planet would move around the Sun in an elliptical orbit.

<span style="color: #800000; font-family: Verdana,Arial,Helvetica,sans-serif;">Newtonian physics was able to predict that gravitational attractions between the planets would cause small departures from the purely elliptical motion that Kepler had described.

<span style="color: #800000; font-family: Verdana,Arial,Helvetica,sans-serif;">Newtonís discoveries became the basis for a detailed and comprehensive study of **mechanics** (the study offorce and motion).

<span style="color: #800000; font-family: Verdana,Arial,Helvetica,sans-serif;">This property of Newtonian mechanics is called **determinism**. It had an enormously important implication.

<span style="color: #800000; font-family: Verdana,Arial,Helvetica,sans-serif;">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.

<span style="font-family: Verdana,Arial,Helvetica,sans-serif;">If you accepted the proposition that humans were entirely physical systems, composed of particles of matter obeying physical laws of motion, then in principle, every future human action would be already determined by the past.
 * [[image:http://physicalworld.org/restless_universe/figs/fig_1_10.jpg width="175" height="119" align="left" caption=" An orrery (a mechanical model of the Solar System) "]] || <span style="color: #800000; font-family: Verdana,Arial,Helvetica,sans-serif;">Figure 1.10 An orrery (a mechanical model of the Solar System) can be taken as a metaphor for the clockwork Universe of Newtonian mechanics. ||

<span style="color: #800000; font-family: Verdana,Arial,Helvetica,sans-serif;">For some this was the ultimate indication of God: where there was a design there must be a Designer, where there was a clock there must have been a Clockmaker.

<span style="color: #800000; font-family: Verdana,Arial,Helvetica,sans-serif;">For others it was just the opposite, a denial of the doctrine of **free will** which asserts that human beings are free to determine their own actions.

= Circular and Satellite Motion: Lesson 4; A-C = January 8, 2012

Rules-Based method

Topic Sentence:

Kepler's Three Laws Kepler's first law - An ellipse is a special curve in which the sum of the distances from every point on the curve to two other points is a constant. The two other points (represented here by the tack locations) are known as the **foci** of the ellipse.

The closer together that these points are, the more closely that the ellipse resembles the shape of a circle. Kepler's first law is rather simple - all planets orbit the sun in a path that resembles an ellipse, with the sun being located at one of the foci of that ellipse.

Kepler's second law - 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.

Kepler's third law - sometimes referred to as 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.

**(s)** ||< **Average** **Dist. (m)** ||< T2/R3 **(s2/m3)** ||
 * < Planet ||< **Period**
 * < Earth ||< 3.156 x 107s ||< 1.4957 x 1011 ||< 2.977 x 10-19 ||
 * < Mars ||< 5.93 x 107 s ||< 2.278 x 1011 ||< 2.975 x 10-19 ||

**(yr)** ||< **Ave.** **Dist. (au)** ||< **T2/R3** **(yr2/au3)** ||
 * < **Planet** ||< **Period**
 * < Mercury ||< 0.241 ||< 0.39 ||< 0.98 ||
 * < Venus ||< .615 ||< 0.72 ||< 1.01 ||
 * < Earth ||< 1.00 ||< 1.00 ||< 1.00 ||
 * < Mars ||< 1.88 ||< 1.52 ||< 1.01 ||
 * < Jupiter ||< 11.8 ||< 5.20 ||< 0.99 ||
 * < Saturn ||< 29.5 ||< 9.54 ||< 1.00 ||
 * < Uranus ||< 84.0 ||< 19.18 ||< 1.00 ||
 * < Neptune ||< 165 ||< 30.06 ||< 1.00 ||
 * < Pluto ||< 248 ||< 39.44 ||< 1.00 ||

Kepler's third law provides an accurate description of the period and distance for a planet's orbits about the sun.

Additionally, the same law that describes 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.

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**.

The moon, the planets and comets are examples of natural satellites.

The fundamental principle to be understood concerning satellites is that a satellite is a __ [|projectile] __. That is to say, a satellite is an object upon which the only force is gravity

A projectile launched horizontally with a speed of about 8000 m/s will be capable of orbiting the earth in a circular path.

**Velocity, Acceleration and Force Vectors** 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 by __ [|gravity - the force that universally] __ acts at a distance between any two objects that have mass.

Like any projectile, gravity alone influences the satellite's trajectory such that it always falls below its __ [|straight-line, inertial path] __. This is depicted in the diagram below.

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.

**Elliptical Orbits of Satellites** The velocity of the satellite is directed tangent to the ellipse.

The acceleration of the satellite is directed towards the focus of the ellipse.

And in accord with __ [|Newton's second law of motion] __, the net force acting upon the satellite is directed in the same direction as the acceleration - towards the focus of the ellipse. 

Mathematics of Satellite Motion

Consider a satellite with mass Msat orbiting a central body with a mass of mass MCentral. The central body could be a planet, the sun or some other large mass capable of causing sufficient acceleration on a less massive nearby object. **Fnet = ( Msat • v2 ) / R** This net centripetal force is the result of the __ [|gravitational force] __ that attracts the satellite towards the central body and can be represented as **Fgrav = ( G • Msat • MCentral ) / R2** Since Fgrav = Fnet, the above expressions for centripetal force and gravitational force can be set equal to each other. Thus, **(Msat • v2) / R = (G • Msat • MCentral ) / R2** Observe that the mass of the satellite is present on both sides of the equation; thus it can be canceled by dividing through by **Msat**. Then both sides of the equation can be multiplied by **R**, leaving the following equation. **v2 = (G • MCentral ) / R** Taking the square root of each side, leaves the following equation for the velocity of a satellite moving about a central body in circular motion where **G** is 6.673 x 10-11 N•m2/kg2, **Mcentral** is the mass of the central body about which the satellite orbits, and **R** is the radius of orbit for the satellite.  The equation for the acceleration of gravity was given as **g = (G • Mcentral)/R2** The acceleration of a satellite in circular motion about some central body is given by the following equation where **G** is 6.673 x 10-11 N•m2/kg2, **Mcentral** is the mass of the central body about which the satellite orbits, and **R** is the average radius of orbit for the satellite.

The final equation that is useful in describing the motion of satellites is Newton's form of Kepler's third law. where **T** is the period of the satellite, **R** is the average radius of orbit for the satellite (distance from center of central planet), and **G** is 6.673 x 10-11 N•m2/kg2.

 Equation (1) was derived __ [|above] __. Equation (2) is a __ [|general equation for circular motion] __. Either equation can be used to calculate the acceleration. The use of equation (1) will be demonstrated here. a = (G •Mcentral)/R2a = ( 6.673 x 10-11 N m2/kg2 ) • ( 5.98 x 1024 kg ) / ( 6.47 x 106 m )2 **a = 9.53 m/s2** Observe that this acceleration is slightly less than the 9.8 m/s2 value expected on earth's surface. __ [|As discussed in Lesson 3] __, the increased distance from the center of the earth lowers the value of g. Finally, the period can be calculated using the following equation: The equation can be rearranged to the following form T = SQRT [(4 • pi2 • R3) / ( G*Mcentral) ]The substitution and solution are as follows: T = SQRT [(4 • (3.1415)2 • (6.47 x 106 m)3) / ( 6.673 x 10-11 N m2/kg2 ) • ( 5.98x1024 kg ) ] **T = 5176 s = 1.44 hrs**

=Circular Motion and Satellite Motion: Lesson 4; D-E= January 10, 2012

Rules Based Method:

Weightlessness in Orbit It is only a false perception that you are weightless in orbit.

**Contact versus Non-Contact Forces** - **contact forces** and **action-at-a-distance forces**.

As you sit in a chair, you experience two forces - the force of the Earth's gravitational field pulling you downward toward the Earth and the force of the chair pushing you upward.

__ [|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.

The force of gravity can never be felt. Yet those forces that result from contact can be felt.

And in the case of sitting in your chair, you can feel the chair force; and it is this force that provides you with a sensation of weight. 

**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.

When in free fall, the only force acting upon your body is the force of gravity - a non-contact force. Since the force of gravity cannot be felt without any other opposing forces, you would have no sensation of it.

You would feel weightless when in a state of free fall.

 **Scale Readings and Weight** Technically speaking, a scale does not measure one's 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.

However, if you stand on the scale and bounce up and down, the scale reading undergoes a rapid change. As you undergo this bouncing motion, your body is accelerating.

For instance, the value of the normal force (Fnorm) on Otis's 80-kg body could be predicted if the acceleration is known. This prediction can be made by simply __ [|applying Newton's second law as discussed in Unit 2] __.

 The normal force is greater than the force of gravity when there is an upward acceleration (B), less than the force of gravity when there is a downward acceleration (C and D), and equal to the force of gravity when there is no acceleration (A). Since it is the normal force that provides a sensation of one's weight, the elevator rider would feel his normal weight in case A, more than his normal weight in case B, and less than his normal weight in case C. In case D, the elevator rider would feel absolutely weightless; without an external contact force, he would have no sensation of his weight. In all four cases, the elevator rider weighs the same amount - 784 N. Yet the rider's sensation of his weight is fluctuating throughout the elevator ride.

**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.

Energy Relationships for Satellites The orbits of satellites about a central massive body can be described as either circular or elliptical.

Since __ [|perpendicular components of motion are independent] __ of each other, the inward force cannot affect the magnitude of the tangential velocity.

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. This force is capable of doing __ [|work] __ upon the satellite.

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 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. These principles are depicted in the diagram below. 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 work-energy theorem is expressed in equation form as **KEi + PEi + Wext = KEf + PEf** 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. **KEi + PEi = KEf + PEf**

**Energy Analysis of Circular Orbits** When in circular motion, a satellite remains the same distance above thesurface of the earth; that is, its radius of orbit is fixed.

**Energy Analysis of Elliptical Orbits** the total amount of mechanical energy of a satellite in elliptical motion also remains constant. The speed of this satellite is greatest at location A (when the satellite is closest to the earth) and least at location C (when the satellite is furthest from the earth).

An energy analysis of satellite motion yields the same conclusions as any analysis guided by Newton's laws of motion.