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I would like to calculate how the RA/Dec changes when an object has an apparent linear motionin the equatorial sphere. For example, more or less what an artificial satellite would do in a circular orbit.

In particular, what I'm looking for is as follows:

I have the RA/dec of the object at two times, *t1*, and *t2*. Assuming it has a linear motion at constant speed, what RA/dec will it have a *t3*? So it will have an angular separation proportional to the one it had between *t1* and *t2*, and the three points will form a straight line.

This topic contains a selection of units designed to assist you in teaching motion. Units include frames of reference, graphing skills, motion in one dimension, motion in more than one dimension, vectors, and more. Units are not listed in a prescribed order.

### Graphing (12)

#### Lesson Plans:

An excellent lesson plan developed specifically to accompany the PhET simulation "The Moving Man". Students with little prior knowledge of graph interpretation will gain understanding of velocity vs. time graphs and how they differ from position vs. time. Adaptable for both middle school and high school. ** Note:** You must be a registered user to access PhET teacher-contributed resources. Registration is easy and free.

This is a PhET Gold Star winning lesson that helps students build skills in interpreting graphs of motion. It accompanies the PhET simulation "The Moving Man" (see link below) and includes classroom-ready Power Point concept questions, student guide, and assessments. ** Note:** This lesson is appropriate for more advanced students than the lesson listed directly above. You must be a registered user to access it, but registration is free.

#### Activities:

This free digital grapher is especially user-friendly for beginning learners. Students can choose from five graph types: bar, line, area, pie, and X/Y. Various patterns, colors, grids, and label choices are available to allow customization. Minimum and maximum values can be set to limit the scale of the graph.

Blend a motion sensor lab with student-generated graph modeling. Students use the online graph sketching tool to predict the motion of a person walking back & forth over a 40-meter line. Next, they do a motion sensor lab to collect actual data. Last, they analyze differences between their predictions and the real-world data. *Pair this lab with the one directly below, "Motion on a Ramp" for a great 3-day experience.*

This activity blends a motion sensor lab with digital graph modeling. Students use the online graph-sketching tool to predict graphs of distance vs. time and velocity vs. time. Next, they use a motion sensor to collect data on a real toy being pushed up a ramp. Last, they analyze differences between their predictions and the actual data. *Highly recommended by the editors. For beginners, try first introducing the activity directly above.*

Maneuver a simulated man and watch simultaneous graphs of his position, velocity, and acceleration. For beginning learners, the acceleration graph may be closed. Try teaming this simulation with the great companion lessons by PhET teacher-fellows, found under "Lesson Plans" above. *Highly versatile resource adaptable to a broad spectrum of abilities/levels.*

This interactive graphing activity explores the effects of gravity on light and heavy objects. It gives learners a means to make predictions, quickly compare their predictions with real data, and analyze why the predictions were right or wrong. It's one of the few resources we've seen in which universal gravitation, constant acceleration, slope of the line, and graphing of free-fall motion can all be meaningfully explored in one class period. Includes lesson plan & assessment w/answer key.

#### References and Collections:

This set of lessons investigates the language of kinematics (the physics of motion). It is designed to help students understand that the scientific meaning of words like "velocity" and "acceleration" is different from their use in everyday language.

This robust browser-based tool lets users perform calculations, share designs, and create interactive Whiteboard resources. Choose from Cartesian or polar grids, select angle measurement in degrees or radians, and view parabolic graphs in standard, vertex, or intercept form. You can zoom in/out, drag a graph onto the page, and change axes. The platform also supports conic sections, Fourier expansions, and polar graphing. When you finish, you can publish your resource or embed the file into a virtual learning space.

#### Content Support For Teachers:

A very well-organized tutorial on how to construct and interpret three basic kinematic graphs: P/T, V/T and A/T. It includes animated examples, links to five worksheets, and related problems for student exploration.

#### Student Tutorials:

Excellent self-guided tutorial promotes understanding of "position" as a physics concept. Contains multiple graphs, animations, and interactive opportunities for students to test their comprehension.

#### Assessment:

Here is a set of free assessment tools for grades 6-12 on topics related to physics and biology. Within the sub-topic of Force and Motion, you'll find assessments to gauge understanding of kinematics, the relationship of motion and force, and momentum. Diagnoser assessments are aligned with the NGSS, and include elicitation (warm-ups) and developmental lessons. After completing the lesson, students answer digital question sets on specific topics, with immediate feedback provided to both teachers and learners. The final step in the process is "Prescriptive Activities", designed to target specific problematic areas located by Diagnoser.

### Vectors (5)

#### Activities:

It can be difficult for beginning students to understand what a vector represents. This fun simulation allows them to watch vectors change as they drive a virtual car. Speed vs. Time is also displayed in a real-time companion graph. ** Note:** This resource requires Flash.

This very simple simulation can help beginners understand what vector arrows represent. It was designed by the PhET team to target specific areas of difficulty in student understanding of vectors. Learners can move a ball with the mouse or let the simulation control the ball in four modes of motion (two types of linear, simple harmonic, and circular). Two vectors are displayed -- one green and one blue. Which color represents velocity and which acceleration? *This resource requires Java.*

#### Content Support For Teachers:

This page is an interactive environment where subjects are organized in flow charts allowing easy movement from one topic to a related item. Vector resolution, addition, and product are covered in-depth. Great background information for teachers.

This award-winning web tutorial is a great choice for the crossover teacher who wants a refresher on vectors and their properties. Included is an introduction to free-body diagrams, example problems, a series of self-paced questions, and related interactive simulations.

#### Student Tutorials:

As instructors, we may forget that certain representations (like vector arrows) seem like a foreign language to beginning students. This thoughtfully-crafted tutorial introduces vector diagrams in kid-friendly language and extends the learning to interactive practice problems with answers provided.

### Motion in One Dimension (8)

#### Lesson Plans:

In this inquiry-based lesson plan for grades K-2, students record data as they roll different objects down a ramp whose height is variable. It is the first of a two-part lesson on ramps and their mechanical advantages.

This project-based lesson for grades K-2 is designed as a follow-up to Ramps 1 (see item above.) In this activity, students experiment with a variety of materials as they design, build, and test their own ramps. Included is a printable student data sheet.

#### Activities:

A simulation to explore the motion of a model car with constant acceleration. The student sets values for initial position, velocity and acceleration -- the simulation creates the real-time graphs. A pair of timers can be placed anywhere along the path of the car to measure the motion at intervals. Available in HTML5 or Java. *Can be adapted for grades 6-7 by using only the velocity and position fields.*

This interactive graphing activity explores the effects of gravity on light and heavy objects. It gives learners a means to make predictions, quickly compare their predictions with real data, and analyze why the predictions were right or wrong. It's one of the few resources we've seen in which universal gravitation, constant acceleration, slope of the line, and graphing of free-fall motion can all be meaningfully explored in one class period. Includes lesson plan & assessment w/answer key.

#### References and Collections:

A must-read for teachers of K-8 science and 9th grade physical science. A physics education researcher studied groups of students in grades 4, 6, and 8. The research took a deep look at how students at these grade levels distinguish between speed and changing speed. The findings will help teachers in constructing effective lessons. *Editor's Note: Why is this Important? Research reveals that children in elementary school form and maintain conceptions about the physical world that remain deeply entrenched into adulthood. Inaccurate conceptions can be very difficult to reverse.*

#### Content Support For Teachers:

This resource offers support in understanding the concept of acceleration as a rate of change. It includes example problems with solutions, homework problems, and a fun section that provides sample accelerations of selected events. Great content support for middle school teachers or solid tutorial for high school physics.

This page offers a clear explanation of the equations that can be used to describe the motion of an object in a straight line. A comprehensive set of algebraic, statistical, and conceptual problems are included. Provides content support for middle school teachers. also appropriate for high school physics students.

#### Student Tutorials:

This is a web-based homework problem that helps students understand velocity vs. time graphs (v vs. t). A sequence of user-activated questions takes beginners through a full conceptual analysis before introducing the math. It was developed using principles of physics education research. Appropriate for gifted/talented middle school students.

### Motion in More Than One Dimension (6)

#### Lesson Plans:

An excellent two-day lesson to accompany the PhET simulation *Projectile Motion* (see Activities below). It was crafted by educators to provide robust support to both teachers and learners on projectile motion with and without air resistance. You will find scripted teacher discussion, explanations of fluid properties of air, and modifiable worksheets. *Created for middle school, but can be easily adapted to Physics First or Physics Prep courses.*

#### Activities:

Students can have fun exploring projectile motion as they interactively fire objects of varying mass from a cannon. Users may set initial velocity, angle, and air resistance. This resource would be teamed well with the Physics Classroom student tutorial on projectile motion (below).

This simulation would be a good follow-up to the PhET projectile motion applet (above). This item takes the learner to the next level by calculating maximum height, horizontal distance, magnitude of velocity, and total energy of a projected object. Students will set initial height, speed, angle, and mass before firing their projectile. *Appropriate for high school or gifted/talented middle school students.*

This popular sim has been converted to HTML5 and dressed up with cool new features. As with the older version, students will set initial speed & angle, and choose from a variety of projectiles (pumpkin, piano, cannonball, person). Now they can also set drag coefficient and easily see how air resistance affects the trajectory. You can also adjust the gravitational constant and manually adjust the diameter of the projectile. *Very robust, very easy to adapt to different learning levels!*

#### Content Support For Teachers:

Kinematics is the science of describing the motion of objects using words, diagrams, numbers, graphs, and equations. The goal of any study of kinematics is to develop sophisticated mental models which serve to describe (and ultimately, explain) the motion of real-world objects.

#### Student Tutorials:

This seven-part resource is an excellent introduction to the characteristics of projectile motion. Through in-depth explanations and animations, it explores vertical acceleration and explains why there are no horizontal forces acting upon projectiles, a common student misconception. The last two sections are devoted to problem solving. Try teaming it with the PhET Projectile Motion activity above.

### Circular Motion (4)

#### Activities:

For the teacher planning a unit on amusement park physics, this tutorial can double as a student classroom activity. It offers an overview of the forces acting upon a roller coaster as it travels on a straight, curved, or looped track. Free body diagrams and animations depicting kinetic/potential energy also enhance student understanding of a complex set of interactions. (Includes a self-test.)

Can an amusement park Merry-Go-Round be designed to be dangerous? This simple model lets kids discover for themselves how rotational speed and radial distance interact to create a more thrilling ride. Don't miss the page link to "Physiological impact of G-forces". Setting the speed & radial distance at the highest points will result in g-forces that exceed space shuttle re-entry and high speed fighter jets!

#### Content Support For Teachers:

One of the most deeply entrenched misconceptions among beginning physics students is that centrifugal motion (away from the center) is a "force" in itself. In this tutorial, part of *Physics Classroom*, the author explains why the direction of force is viewed from an inertial frame of reference in a classical mechanics course and thus why centrifugal motion is not a force in a Newtonian framework.

#### Student Tutorials:

This student tutorial illustrates how circular motion principles can be combined with Newton's Second Law to analyze physical situations. Two algebraic problems and detailed solutions are provided, plus a five-step model for solving circular motion problems.

### Planetary Motion (3)

#### Lesson Plans:

This 40-minute lesson, created by a veteran high school teacher, gives kids explicit guidance in using the PhET simulation *My Solar System* to explore orbital motion and gravitational attraction. Great concept building activity. ** Note:** Registration is required to access PhET teacher-contributed materials (it's free).

#### Activities:

With this orbit simulator, you can set initial positions, velocities, and masses of 2, 3, or 4 bodies, and then watch them orbit each other. The simulation is especially effective at helping students understand how distance and mass are related to orbit. Scroll down on the page for related lesson plans developed by middle school and high school teachers. *Requires Flash*

How did scientists first directly demonstrate that the Earth rotates? This short video, seen through the eyes of a child, explores the work of French scientist Leon Foucault -- a pendulum seems to rotate as it swings, but there is no external force that would cause the rotation (clockwise in the Northern Hemisphere, counterclockwise in the Southern). Through experiments, Foucault showed that it's not the pendulum doing the rotating. It's the steady, predictable movement of the Earth's rotation on its axis.

### Special Collections (4)

#### Lesson Plans:

This page contains procedures for setting up 20 demonstrations relating to motion. All demos have been fully tested in the classroom and were selected for inclusion because they are engaging, require minimal set-up, and are highly illustrative of key concepts taught in introductory classical mechanics. Historical anecdotes and commentary add to the depth of this unique resource.

This 8-day instructional unit for middle school integrates engineering practice into a study of the energy of motion. Through investigations of waterwheels, roller coasters, bouncing balls, and a pendulum, students get a solid introduction to energy transformation in a mechanical system. The unit also introduces static and kinetic friction, drag, elastic/inelastic collision, and students learn to calculate frictional force. *Don't have time to do the full unit? Lessons can be pulled out individually.*

#### Activities:

This free collection will impress teachers in the way it promotes depth of understanding about graphs. Learners use interactive digital tools to predict how a motion graph will look, then they watch as the computer simulates process in real time. Next, they place inputs on the graphs and use language to explain what is happening. Finally, they compare their own predictions with the simulated process to analyze why the graphs appear as they do. As with all Concord Consortium materials, the resources are subjected to rigorous classroom testing to ensure their effectiveness.

#### References and Collections:

This collection of short videos explores the basic physics of football in a way that's sure to spark interest among kids. Each video features an NFL player, file footage of games, slow-motion video captured with a super high-speed Phantom Cam. Physicists appear in each video to explain the concepts and clarify the connection to physics. Topics: Newton's Laws, momentum, inertia, vectors, center of mass, projectile motion, and more.

### Velocity and Acceleration (6)

#### Lesson Plans:

This inquiry-based lesson for grades K-2 is similar to Galileo's classic experiment with inclined planes. Children roll spherical objects of different masses down ramps of varying heights. As they record data, they are building a conceptual base for understanding the constant nature of acceleration due to gravity. See the item below for Part 2 of the lesson.

This project-based lesson for grades K-2 is designed as a follow-up to Ramps 1 (see item above.) In this activity, students experiment with a variety of materials as they design, build, and test their own ramps. Included is a printable student data sheet.

#### Activities:

This robust activity from Concord Consortium lets kids deeply explore the meaning behind the slopes of velocity/time and position/time graphs. It blends interactive graph sketching, data analysis, and digital Q&A as learners explore the motion of an animated car. It will help students understand why motion graphs appear as they do, rather than mimic the pathway of an object's motion.

This interactive graphing activity explores the effects of gravity on light and heavy objects. It gives learners a means to make predictions, quickly compare their predictions with real data, and analyze why the predictions were right or wrong. It's one of the few resources we've seen in which universal gravitation, constant acceleration, slope of the line, and graphing of free-fall motion can all be meaningfully explored in one class period. Includes lesson plan & assessment w/answer key.

#### References and Collections:

A must-read for teachers of K-8 science and 9th grade physical science. A physics education researcher studied groups of students in grades 4, 6, and 8. The research took a deep look at how students at these grade levels distinguish between speed and changing speed. The findings will help teachers in constructing effective lessons. **Editor's Note:** Why is this important? Research reveals that children in elementary school form and maintain conceptions about the physical world that remain deeply entrenched into adulthood. Inaccurate conceptions can be very difficult to reverse.

In this study, students in an inquiry-based classroom were videotaped to detect how they made sense of concepts relating to force and motion. The analysis revealed that focused "sense-making activities", free-body diagrams, energy diagrams, and related real-world activities produced deeper student understanding. *Free download*

### Assessments (1)

#### Assessment:

Here is a set of free assessment tools for grades 6-12 on topics related to physics and biology. Within the sub-topic of Force and Motion, you'll find assessments to gauge understanding of kinematics, the relationship of motion and force, and momentum. Diagnoser assessments are aligned with the NGSS, and include elicitation (warm-ups) and developmental lessons. After completing the lesson, students answer digital question sets on specific topics, with immediate feedback provided to both teachers and learners. The final step in the process is "Prescriptive Activities", designed to target specific problematic areas located by Diagnoser.

### Modeling Motion (1)

#### Activities:

We like the simplicity of this model for introducing free fall and gravitational acceleration. Students can control the initial height, set initial velocity from -20 to 20 m/s and change the gravitational constant. The free fall is displayed as a motion diagram, while graphs are simultaneously displayed showing position, velocity, and acceleration vs. time.

### The Case of Roller Coasters (10)

#### Lesson Plans:

Roller coasters offer an inherently interesting way to study energy transformation in a system. This simulation lets students choose from 5 track configurations or create their own design, then watch the resulting motion. Energy bar graphs are simultaneously displayed as the coaster runs its course. Students can adjust the initial speed and friction, or switch to stepped motion to see exact points where kinetic and potential energy reach maximum and minimum levels. *Includes lesson plan and student guide.*

Students build understanding of kinetic and potential energy as they design a physical model of a roller coaster with foam pipe insulation and marbles. The lesson is almost completely turn key: scripted teacher introduction, detailed illustrated instructions, student worksheet, scoring rubric, and post-activity assessment. Which track configuration works best? What can be done to reduce friction?

A four-day lesson that explores the same physics concepts as roller coaster design, but breaks the learning into two distinct segments to ensure that beginners understand the basics. In Part I, kids build a very simple curved track to explore kinetic and potential energy for a gumball moving downhill. Part II becomes more complex: build and test a gumball machine with loops and specific design constraints.

#### Activities:

This short video does a great job of demonstrating centripetal force and how it acts to keep objects moving along a curved path. What makes a rider on a roller coaster feel a sensation of being thrown outward from the center during a loop, although there is no outward net force? The video serves to help beginners understand the dynamics of circular motion.

Exactly what IS centripetal force and what does it do? An astronaut on board the International Space Station demonstrates this force in ways students cannot observe in daily life. The environment is "almost" weightless, making it easy to observe the center-seeking motion without the complicating effects of gravity.

Want to do a quick lesson on energy transfer in a roller coaster, but can't devote more than one class period? This simulation lets kids design a very simple roller coaster, then it evaluates the design based on physical principles, safety, and "fun factor". Good springboard for further inquiry into energy transformation.

This self-paced multimedia tutorial explores how cars move along a roller coaster track as a result of energy transformation. Part of the Middle School Literacy Project, it is designed to develop literacy skills in the context of a focused science or math lesson. Students read informational text, build vocabulary, view videos and interactive simulations, and create written responses in both short and extended forms. Registered teachers may set up student accounts for tracking progress.

#### References and Collections:

The heart of this website for engineering education is its wonderful collection of videos that feature kids doing activities that illuminate key concepts in science and engineering. Each video is well-produced to allow the actor-students to apply concepts of science, then experience the excitement that goes with real discovery. Here's a taste: one video takes kids on a field trip to the Etnies shoe headquarters to learn the biomechanics of skateboarding from engineers who are also skateboarders. Another video takes the kids to Epcot Center to design and plan their own roller coaster configurations. The site also offers interactive games and simulations.

#### Content Support For Teachers:

If you need a refresher in the basics, this is a nicely organized tutorial that addresses four topics: uniform circular motion, centripetal force, applications and mathematics of circular motion, and amusement park physics. Includes free-body diagrams, animations, and problem sets with answers.

Short tutorial that uses an animation to illustrate the work/energy relationship in a roller coaster. The author breaks down the associated equation to show how total mechanical energy is conserved in the system.

## Linear polarization

A stereoscopic 3D display technology that separates the stereo frames by polarization. Using polarizer overlays, the movie projector or TV screen emits the left frame in a different polarization than the right frame, and the viewer wears polarized glasses to filter the frames to the appropriate eyes.

Polarized 3D is also called "passive 3D," because the glasses have no built-in electronics and do not perform any processing (contrast with active 3D). See 3D visualization, anaglyph 3D, lenticular 3D, parallax 3D and 3D glasses.

**Linear and Circular Polarization**

Dating back to the 1950s, the linear polarized method was the first 3D technology used for major motion pictures such as the 1953 horror film "House of Wax." The theater projector emitted stereo frames polarized 90 degrees apart. Later on, the technology changed to a clockwise/counterclockwise orientation like the RealD method widely used today. Polarized 3D glasses with two different lens polarizations filter the images to each eye respectively.

The circular polarized method migrated to TVs that alternate the lines of resolution: one line polarized to the left eye, the next line to the right. However, because the lines are interlaced, the viewer sees half the resolution (only 540 lines reach each eye on a 1080p TV). With 2160p 3D TVs, the full 1080p resolution can be delivered (see 4K TV).

## Flying a Drone Inside a Moving Vehicle

If we fly a drone inside a vehicle that is moving in a constant speed and direction, the drone will hover normally because it preserves the motion it gets from the vehicle. It will not abruptly move toward the rear of the vehicle.

Flat-Earthers claim that it should not be possible for drones to hover if Earth is rotating. In reality, drones follow Newton’s first law of motion and will preserve their inertia. It is easy to demonstrate the phenomenon by flying a drone inside a moving vehicle. The fact that drones can hover still in mid-air is not evidence of stationary Earth.

## VibraScreener™ for All Your Needs

Whether you decide on linear or circular movement while choosing a vibrating screen, we’ve got you covered with quality products at VibraScreener™. Our products have been trusted and re-ordered by leading companies around the globe time and time again. Find out why — request a free quote now. If you have questions about how the vibratory screener or sieve process works with our equipment, fill out our talk to an expert form and a representative will assist you.

## Answers and Replies

You have observed that ball A has a non-zero final velocity. But you have not measured the difference between A's initial velocity and B's final velocity. How can you assert that total linear momentum has increased?

For a ball that is rolling without slipping, there is a fixed ratio between the ball's angular momentum about its center and the ball's linear momentum. Since the two balls are identical, this ratio will be the same for each. For a ball that whose rolling rate and linear speed do not match, it will require a rotational impulse (a torque applied over time) to get them to match.

During the collision, linear momentum is be transferred from ball A to ball B.

Claim: The resulting imbalance between linear momentum and rolling speed on each ball will be equal and opposite. It will require equal and opposite rotational impulses to get them to match. Because the balls have equal radius, this means equal and opposite linear impulses. That means that momentum will be conserved while ball A spins down (and speeds up) and ball B spins up (and slows down).

During the collision, steel-on-steel friction will cause both balls to experience a slight backwards torque around their respective centers of mass. This can only act to reduce the final momentum of the system.

## Polarization effects

### 6.3.2 Poincaré-sphere representation

An alternative approach to describing the evolution of the SOP in optical fibers is based on the rotation of the Stokes vector on the Poincaré sphere [27] . In this case, it is useful to write Eqs. (6.3.1) and (6.3.2) in terms of linearly polarized components using Eq. (6.1.14) . The resulting equations are

These equations can also be obtained from Eqs. (6.1.11) and (6.1.12) .

At this point, we introduce the four real variables known as the *Stokes parameters* and defined as

and rewrite Eqs. (6.3.13) and (6.3.14) in terms of them. After considerable algebra, we obtain

It can be easily verified from Eq. (6.3.15) that S 0 2 = S 1 2 + S 2 2 + S 3 2 . As S 0 is independent of *z* from Eq. (6.3.16) , the Stokes vector **S** with components S 1 , S 2 , and S 3 moves on the surface of a sphere of radius S 0 as the CW light propagates inside the fiber. This sphere is known as the Poincaré sphere and provides a visual description of the polarization state. In fact, Eqs. (6.3.16) and (6.3.17) can be written in the form of a single vector equation as [27]

where S = ( S 1 , S 2 , S 3 ) is the Stokes vector with W = W L + W NL defined such that

Eq. (6.3.18) includes linear as well as nonlinear birefringence. It describes evolution of the SOP of a CW optical field within the fiber under quite general conditions.

Fig. 6.6 shows motion of the Stokes vector on the Poincaré sphere in several different cases. In the low-power case, nonlinear effects can be neglected by setting γ = 0 . As W NL = 0 in that case, the Stokes vector rotates around the S 1 axis with an angular velocity Δ*β* (upper left sphere in Fig. 6.6 ). This rotation is equivalent to the periodic solution given in Eq. (6.3.3) obtained earlier. If the Stokes vector is initially oriented along the S 1 axis, it remains fixed. This can also be seen from the steady-state (*z*-invariant) solution of Eqs. (6.3.16) and (6.3.17) because ( S 0 , 0 , 0 ) and ( − S 0 , 0 , 0 ) represent their fixed points. These two locations of the Stokes vector correspond to the linearly polarized incident light oriented along the slow and fast axes, respectively.

Figure 6.6 . Trajectories showing motion of the Stokes vector on the Poincaré sphere. (A) Linear birefringence case (upper left) (B) nonlinear case with Δ*β* = 0 (upper right) (C) mixed case with Δ*β* > 0 and *P*_{0} > *P*_{cr} (lower row). Left and right spheres in the bottom row show the front and back of the Poincaré sphere. (After Ref. [27] ©1986 OSA.)

In the purely nonlinear case of isotropic fibers ( Δ β = 0 ), W L = 0 . The Stokes vector now rotates around the S 3 axis with an angular velocity 2 γ S 3 / 3 (upper right sphere in Fig. 6.6 ). This rotation is referred to as self-induced ellipse rotation, or as nonlinear polarization rotation, because it has its origin in the nonlinear birefringence. Two fixed points in this case correspond to the north and south poles of the Poincaré sphere and represent right and left circular polarizations, respectively.

In the mixed case, the behavior depends on the power of incident light. As long as P 0 < P cr , nonlinear effects play a minor role, and the situation is similar to the linear case. At higher power levels, the motion of the Stokes vector on the Poincaré sphere becomes quite complicated because W L is oriented along the S 1 axis while W NL is oriented along the S 3 axis. Moreover, the nonlinear rotation of the Stokes vector along the S 3 axis depends on the magnitude of S 3 itself. The bottom row in Fig. 6.6 shows motion of the Stokes vector on the front and back of the Poincaré sphere in the case P 0 > P cr . When input light is polarized close to the slow axis (left sphere), the situation is similar to the linear case. However, the behavior is qualitatively different when input light is polarized close to the fast axis (right sphere).

To understand this asymmetry, let us find the fixed points of Eqs. (6.3.16) and (6.3.17) by setting the *z* derivatives to zero. The location and number of fixed points depend on the optical power P 0 launched inside the fiber. More specifically, the number of fixed points changes from two to four at the critical power level P cr defined in Eq. (6.3.10) . For P 0 < P cr , only two fixed points, ( S 0 , 0 , 0 ) and ( − S 0 , 0 , 0 ) , are found these are identical to the low-power case. In contrast, when P 0 exceeds P cr , two new fixed points emerge. The components of the Stokes vector at the location of the new fixed points on the Poincaré sphere are given by [37]

These two fixed points correspond to elliptically polarized light and occur on the back of the Poincaré sphere in Fig. 6.6 (lower right). At the same time, the fixed point ( − S 0 , 0 , 0 ) , corresponding to light polarized linearly along the fast axis, becomes unstable. This is equivalent to the pitchfork bifurcation discussed earlier. If an input beam is polarized elliptically with its Stokes vector oriented as indicated in Eq. (6.3.20) , its SOP will not change inside the fiber. When the SOP is close to the new fixed points, the Stokes vector forms a closed loop around the elliptically polarized fixed point. This behavior corresponds to the analytic solution discussed earlier. However, if the SOP is close to the unstable fixed point ( − S 0 , 0 , 0 ) , small changes in input polarization can induce large changes at the output. This phenomenon is discussed next.

## Perturbed Orbital Dynamics

Enrico Canuto , . Carlos Perez Montenegro , in Spacecraft Dynamics and Control , 2018

### Nonlinear Differential Equations of Relative Motion

To properly describe the relative motion between *P*_{2} and *P*_{1}, we define their relative position r → = r → 2 − r → 1 , with r = | r → | , and we assume that the orbit of the chief satellite is Keplerian, which implies that F → 1 = 0 and F → = F → 2 . Thus, we rewrite Eq. (5.82) as

This equation describes the motion of *P*_{2} relative to *P*_{1} in an inertial frame with origin at *O* = *P*_{0} . In the following, we will choose the perifocal frame P = < P 0 , p → 1 , p → 2 , p → 3 >of the chief satellite as the inertial frame.

A description of the relative motion in the noninertial trajectory frame T = < P 1 , t → 1 , t → 2 , t → 3 >defined by the orbit of *P*_{1} is usually more convenient. To work out such a description, we observe, with the help of Section 6.2.2 , that the inertial acceleration r → ¨ can be written as

where ( r → ˙ ) t and ( r → ¨ ) t are the velocity and acceleration vectors in the trajectory frame T , respectively, and ω → 1 and ω → ˙ 1 are the angular velocity and acceleration of the frame T in the inertial frame J , respectively. If we choose as a trajectory frame the Hill's frame H = < P 1 , h → 1 = r → 1 / | r → 1 | , h → 2 , h → 3 = ω → 1 / | ω → 1 | >of the chief orbit, the subscript *t* is replaced by *h*, and we write the coordinate equations

where **r** = [*x*,*y*,*z*] is the coordinate vector of *P*_{2} in the Hill's frame defined by the vectrix H → = [ h → 1 h → 2 h → 3 ] . The three coordinates are usually known as radial (*x*), longitudinal or along-track (*y*), and cross-track or out-of-plane (*z*). The pair (radial, longitudinal) is known as the in-plane pair. With the help Eq. (5.82) and of Eq. (5.83) , the relative acceleration r → ¨ holds:

In addition, since the chief orbit is Keplerian, by recalling that the orbit pole h → 3 is inertial and by denoting with *h* the magnitude of the chief-orbit angular momentum per unit mass (see Eq. (3.40) in Section 3.3.4 ) , we can write the differential equations:

By replacing Eqs. (5.85–5.87) into Eq. (5.84) and after doing some simplifications, the *nonlinear differential equations of the relative motion* are found to be:

This topic contains a selection of units designed to assist you in teaching motion. Units include frames of reference, graphing skills, motion in one dimension, motion in more than one dimension, vectors, and more. Units are not listed in a prescribed order.

### Graphing (12)

#### Lesson Plans:

This is a PhET Gold Star winning lesson that helps students build skills in interpreting graphs of motion. It accompanies the PhET simulation "The Moving Man" (see link below) and includes classroom-ready Power Point concept questions, student guide, and assessments. ** Note:** This lesson is appropriate for more advanced students than the lesson listed directly above. You must be a registered user to access it, but registration is free.

High school students can often record data and "plug & chug", but have more difficulty in fitting or interpreting data. This exemplary two-week unit on data analysis introduces students to the statistical method known as least squares regression. Using an online tool to plot data, students then calculate regression lines and fit the data to estimated parameters.

#### Activities:

This website contains a collection of short videos depicting physical processes commonly discussed in beginning courses. Positions of objects in the video frame can be viewed in step motion or real-time, and then mapped onto video analysis software, allowing for more accurate measurement and graphing.

This set of eleven interactive challenges will help students master motion graphing. Each challenge requires the student to match the motion of an animated car to the correct position/time or velocity/time graph. The activity provides enough repetition to help learners construct a meaningful understanding of why the graphs appear as they do.

Maneuver a simulated man and watch simultaneous graphs of his position, velocity, and acceleration. For beginning learners, the acceleration graph may be closed. Try teaming this simulation with the great companion lessons by PhET teacher-fellows, found under "Lesson Plans" above. *Highly versatile resource adaptable to a broad spectrum of abilities/levels.*

#### References and Collections:

This robust browser-based tool lets users perform calculations, share designs, and create interactive Whiteboard resources. Choose from Cartesian or polar grids, select angle measurement in degrees or radians, and view parabolic graphs in standard, vertex, or intercept form. You can zoom in/out, drag a graph onto the page, and change axes. The platform also supports conic sections, Fourier expansions, and polar graphing. When you finish, you can publish your resource or embed the file into a virtual learning space.

#### Content Support For Teachers:

A very well-organized tutorial on how to construct and interpret three basic kinematic graphs: P/T, V/T and A/T. It includes animated examples, links to five worksheets, and related problems for student exploration.

#### Student Tutorials:

This is a web-based homework problem that helps students understand velocity vs. time graphs (v vs. t). A sequence of user-activated questions guides beginners through a full conceptual analysis before introducing the math. Based on principles of physics education research (PER).

Excellent self-guided tutorial promotes understanding of "position" as a physics concept. Contains multiple graphs, animations, and interactive opportunities for students to test their comprehension.

A companion to the resource above, this online tutorial explores the importance of the slope of v-t graphs as a representation of an object's acceleration. Self-guided evaluations help students overcome common misconceptions.

#### Assessment:

A set of homework problems (with answers) written by the PhET team to accompany "The Moving Man" simulation. It assesses student understanding of graphs of position, velocity, and acceleration. ** Note:** You must be a registered user of PhET to access this resource. Registration is easy and free.

Here is a set of free assessment tools for grades 6-12 on topics related to physics and biology. Within the sub-topic of Force and Motion, you'll find assessments to gauge understanding of kinematics, the relationship of motion and force, and momentum. Diagnoser assessments are aligned with the NGSS, and include elicitation (warm-ups) and developmental lessons. After completing the lesson, students answer digital question sets on specific topics, with immediate feedback provided to both teachers and learners. The final step in the process is "Prescriptive Activities", designed to target specific problematic areas located by Diagnoser.

## Linear motion in the equatorial frame - Astronomy

With either round rail or profile rail selection, it's important to choose a type of rail to use before starting the machine component layout. The performance of a linear guide is based upon contact type, rolling element type, inner race geometry and other characteristics such as self-aligning capabilities. It is important to recognize that the options available for each characteristic have performance attributes. Below is a list of considerations to start the conversation. Variables such as dynamic load ratings, preloading and linear guide deflection, resistance and actuation force should be part of your calculations. This analysis is how we go from a functional to an optimal linear motion design.

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Many applications can use either round or square rails depending on the machine design. But often, one works better than the other. Such was the case in a hospital bed where the designer started with a square rail for axial movement. But the assembly would bind because it couldn't move freely unless the mounting bolts were loosened to allow some twisting motion. The bed frame simply wasn't rigid enough. The square rail had to be replaced with the self-aligning round rail.