http://sites.science.oregonstate.edu/portfolioswiki/ 2020-01-27T00:47:16-08:00 http://sites.science.oregonstate.edu/portfolioswiki/ http://sites.science.oregonstate.edu/portfolioswiki/lib/images/favicon.ico text/html 2015-12-08T11:03:25-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:rflec:rflec2d?rev=1449601405 ROTATIONS IN THE PLANE INTRODUCTION In Cartesian coordinates, the natural orthonormal basis is $\{\ii,\jj\}$, where $\ii\equiv\xhat$ and $\jj\equiv\yhat$ denote the unit vectors in the $x$ and $y$ directions, respectively. The position vector from the origin to the point ($x$,$y$) takes the form $$\rr = x \,\ii + y \,\jj$$ Note that $\ii$ and $\jj$ are constant. text/html 2012-07-08T13:57:12-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:rflec:rflec2drot?rev=1341781032 ROTATIONS IN THE PLANE II Recall the expression for $d\rr$ in polar coordinates, namely $$d\rr = dr\,\rhat + r\,d\phi\,\phat$$ ``Dividing'' this expression by $dt$ yields an expression for the velocity in polar coordinates which you may already have seen, namely $$\dot{\rr} = \dot{r}\,\rhat + r\,\dot\phi\,\phat$$ But recall that the position vector in polar coordinates takes the form $$\rr = r\,\rhat$$ and we can differentiate this directly to obtain $$\dot{\rr} = \dot{r}\,\rhat + r\,\rhatdot$… text/html 2016-05-02T15:09:42-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:rflec:rflec2dtime?rev=1462226982 2-d Relative Time Derivatives (Lecture: 30 minutes) Lecture notes (alternate derivation) Reflections This lecture consists largely of a mathematical derivation of the modified second law, and the identification of the Coriolis and centrifugal accelerations. How much detail to present will depend on the students' (and instructor's) comfort with this style of derivation. text/html 2011-07-14T21:13:02-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:rflec:rflec3d?rev=1310703182 ROTATIONS IN SPACE INTRODUCTION In Cartesian coordinates, the natural orthonormal basis is $\{\ii,\jj,\kk\}$, where $\ii\equiv\xhat$, $\jj\equiv\yhat$, $\kk\equiv\zhat$ denote the unit vectors in the $x$, $y$, $z$ directions, respectively. The position vector from the origin to the point ($x$,$y$,$z$) takes the form $$\rr = x \,\ii + y \,\jj + z \,\kk$$ Note that $\ii$, $\jj$, $\kk$ are constant. text/html 2012-07-08T13:58:32-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:rflec:rflec3drot?rev=1341781112 ROTATIONS IN SPACE II Recall the expression for $d\rr$ in spherical coordinates, namely $$d\rr = dr\,\rhat + r\,d\theta\,\that + r\,\sin\theta\,d\phi\,\phat$$ ``Dividing'' this expression by $dt$ yields an expression for the velocity in polar coordinates which you may already have seen, namely $$\dot{\rr} = \dot{r}\,\rhat + r\,\dot\theta\,\that + r\,\sin\theta\,\dot\phi\,\phat$$ Recall that the position vector in spherical coordinates takes the form $\rr = r\,\rhat$, which we can differentia… text/html 2012-04-11T15:49:14-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:rflec:rflec3dtime?rev=1334184554 3-d Relative Time Derivatives (Lecture: 30 minutes) Lecture notes (alternate derivation) Reflections This lecture generalizes the results in the 2-dimensional case to 3 dimensions. The comments made there apply even more so in this case -- apart from the value in reviewing spherical basis vectors, students will have little interest in, effectively, repeating the same derivation in a slightly different context. text/html 2011-07-27T10:17:23-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:rflec:rflec4velocity?rev=1311787043 4-velocity and 4-momentum (Lecture: 10 minutes) See (ss)9.2 of the text. Reflections The argument that ordinary velocity transforms in a complicated way is a repeat of the algebraic derivation of the addition formula in a previous lecture. text/html 2011-07-27T10:17:00-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:rflec:rflecaddition2?rev=1311787020 Addition Formula from Lorentz Transformation (Lecture: 10 minutes) Reflections This is a nice exercise in manipulating differentials, providing an algebraic derivation of the Einstein addition formula, providing a nice supplement to the geometric derivation in a previous lecture. text/html 2011-07-04T15:45:58-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:rflec:rflecangles?rev=1309819558 Angles and Speeds (Lecture: 20 minutes) See (ss)5.2 and (ss)6.1 of the text. Reflections This lecture finally relates hyperbolic trig to velocity, showing that Lorentz transformations are nothing more than hyperbolic rotations --- and the Einstein addition formula is nothing more than the addition formula for hyperbolic trig functions. text/html 2011-07-04T15:45:56-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:rflec:rflecapps?rev=1309819556 Applications (Lecture: 10 minutes) text/html 2015-08-15T13:24:02-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:rflec:rfleccircle?rev=1439670242 Circle Trig (Lecture: 10 minutes) Based on (ss)3 of the text, especially (ss)3.2. Reflections Having just given students a chance to reflect on the essential ideas of circle trigonometry (in this SWBQ), this is an opportunity to reinforce the geometry of trig with a clear, unifying lecture. The order is important: this is circle trig, not (yet) triangle trig; the trig functions are defined as coordinates on the unit circle, not as ratios. text/html 2011-07-04T15:45:52-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:rflec:rfleccollision?rev=1309819552 Discussion of Collision Activity (Lecture: 10 minutes) See(ss)10.1 of the text. Reflections Make sure to demonstrate how to solve this problem using a 3--4--5 hyperbolic triangle! text/html 2011-12-08T10:19:48-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:rflec:rflecconsequences?rev=1323368388 Consequences of Special Relativity (Lecture: 30 minutes) See (ss)2 of the text, especially (ss)2.3. If you have access to the Mechanical Universe video, there is a good animation of time dilation @ 8:00 minutes. Reflections This lecture continues the discussion of the consequences of the speed of light being constant by analyzing additional moving train examples. text/html 2012-05-17T20:15:15-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:rflec:rflecemlorentz?rev=1337310915 Lorentz Transformations for Electromagnetism (Lecture: 30 minutes) See(ss)11.2 of the text. Reflections This lecture is a continuation of the previous one. However, the argument is somewhat more complicated; make sure to rehearse. The content of this lecture might make a good activity. A draft version of such an activity can be found here; this activity has not yet been tested in the classroom. This activity is likely to take at least an hour. text/html 2011-07-14T21:11:11-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:rflec:rflecemlorentzact?rev=1310703071 Activity: Lorentz Transformations for Electromagnetism (DRAFT) The electric field of an infinite metal sheet with charge density $\sigma$ points away from the sheet and has the constant magnitude \begin{equation} |E| = \frac{\sigma}{2\epsilon_0} \end{equation} The magnetic field of such a sheet with current density $\vec\kappa$ has constant magnitude \begin{equation} |B| = \frac{\mu}{2} |\vec\kappa| \end{equation} and direction determined by the right-hand-rule. text/html 2011-07-06T10:35:21-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:rflec:rfleceqmo?rev=1309973721 Linear Motion (Lecture: 10 minutes) Reflections This lecture was originally part of the Turntable Hockey activity. The goal is to derive, and then solve, the differential equations which describe linear motion as seen in a rotating frame. The derivation is straightforward: Simply set the true accelaration equal to zero in the modified second law, and write down the components in terms of rotating coordinates. The result is a system of coupled second-order ODEs. text/html 2016-05-02T15:07:44-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:rflec:rflecforces?rev=1462226864 Centrifugal and Coriolis Forces (Lecture: 10 minutes) SWBQ for the 2-d case“”rightleft text/html 2011-12-08T10:19:33-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:rflec:rflecfoucault?rev=1323368373 Foucault Pendulum (Lecture: 20 minutes) Lecture notes Reflections A full treatment of the Foucault pendulum is rather involved, but the simplified treatment in the lecture notes provides an elementary explanation of the basic features of this surprising result, as well as practice solving coupled ODEs. text/html 2011-07-14T21:09:21-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:rflec:rflecfoucaultnotes?rev=1310702961 THE FOUCAULT PENDULUM Marion and Thornton gives the standard treatment of the Foucault pendulum in Example 10.5 on pages 398--401. However, there is an easier way to get the same result. The basic idea is to separate the problem into 2 parts: an ordinary pendulum influenced by gravity, and a Coriolis-like effect acting on the direction of motion of the pendulum. text/html 2015-08-15T13:25:34-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:rflec:rflecgalilean?rev=1439670334 Galilean Transformations (Lecture: 10 minutes) Reflections An explicit derivation of the Galilean transformation between (Newtonian) inertial frames is probably not helpful, although it is worth presenting the simple derivation that, since two such frames differ by a constant velocity, all accelerations are the same in either frame. text/html 2011-07-04T15:45:07-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:rflec:rflecgr?rev=1309819507 Geometry of General Relativity (Lecture: 10 minutes) See (ss)13.3 and (ss)13.4 of the text. Reflections The basic geometry of general relativity can be described simply: Combine hyperbola geometry, with its peculiar distance function, with curvature. text/html 2016-05-02T16:32:52-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:rflec:rflecgravity?rev=1462231972 Direction of Gravity (Lecture: 10 minutes) text/html 2011-12-08T10:18:55-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:rflec:rflechyperbola?rev=1323368335 Hyperbola Trig (Lecture: 15 minutes) “” Based on (ss)4 of the text, especially (ss)4.1 and (ss)4.2. Reflections Having just reviewed circle trigonometry (in this lecture), this lecture repeats this familiar derivation in an unfamiliar context. But only the details are different, at least algebraically. Geometrically, the fundamental difference is that hyperbolas have two branches, separated by distinguished asymptotes at 45 degrees. Remind students that the speed of light is suppos… text/html 2011-07-04T15:45:00-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:rflec:rflecintervals?rev=1309819500 Spacetime Intervals (Lecture: 5 minutes) See (ss)5.3. Reflections This short lecture provides a physical interpretation of the fact that ”(squared) distances” can now be positive, negative or zero, in terms of space, time, and light, respectively. text/html 2011-07-04T15:44:58-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:rflec:rflecmagnetism?rev=1309819498 Magnetism via Special Relativity (Lecture: 15 minutes) See(ss)11.1 of the text. Reflections Students should already be familiar with the electric and magnetic fields due to an infinite, straight, charged, current-carrying wire. The argument presented in the text is another straightforward application of hyperbolic addition formulas. However, it's a good idea to practice this material before presenting it, to ensure that the signs work out as desired. text/html 2011-07-27T10:17:38-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:rflec:rflecmass?rev=1311787058 Rest Mass and Kinetic Energy (Lecture: 10 minutes) previous lecture See (ss)9.2 and (the middle of) (ss)9.4 of the text. Reflections This is a one good place to show that the relativistic formula for energy reduces for speeds much less than the speed of light to the rest mass plus the Newtonian kinetic energy, plus higher order terms. An alternative is to wait until discussing conservation of energy, as in the text. text/html 2015-08-15T13:23:02-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:rflec:rflecmassless?rev=1439670182 Massless Particles (Lecture: 5 minutes) See (the end of) (ss)9.4 of the text. Reflections This mini-lecture is really a continuation of the preceding SWBQ, and represents the limiting case of the hyperbolic triangle relating energy, momentum, and rest mass when the hypotenuse becomes lightlike. text/html 2011-07-27T10:34:07-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:rflec:rflecnconserve?rev=1311788047 Newtonion Conservation Laws (Lecture: 10 minutes) See the first half of (ss)9.3 of the text. Reflections This material should be familiar to the students, but sets the stage for the next lecture on relativistic conservation laws. It is important to emphasize that Newtonian momentum conservation requires the additional assumption that mass is conserved, but that these two conservation laws together then imply conservation of (kinetic) energy without additional assumptions. text/html 2011-07-04T15:44:49-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:rflec:rflecoverview?rev=1309819489 Overview (Lecture: 5 minutes) Reflections This is the time to make the case that both rotational motion and special relativity are fundamentally about comparing observations between different reference frames. This is also a good time to pique student interest by pointing out that this course will likely affect students' understanding of such basic concepts as “East”, “down”, and “time”. text/html 2011-12-08T10:18:15-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:rflec:rflecparable1?rev=1323368295 Parable I (Lecture: 10 minutes) Based on the first two paragraphs of (ss)5.1 of the text. Reflections This is the first of two short lectures based on the surveyor's parable, introducing the parable in its Euclidean setting, where its consequences are rather obvious. The second lecture then discusses the implications for special relativity, setting the stage for the use of spacetime diagrams. text/html 2011-12-08T10:18:02-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:rflec:rflecparable2?rev=1323368282 Parable II (Lecture: 10 minutes) See (ss)5.1 of the text. Reflections This is the second of two short lectures based on the surveyor's parable. The first lecture introduced the parable in its Euclidean setting, where its consequences are rather obvious. This lecture discusses the implications for special relativity, setting the stage for the use of spacetime diagrams. text/html 2011-07-06T13:47:23-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:rflec:rflecpole?rev=1309985243 Resolving Paradoxes with Spacetime Diagrams (Lecture: 10 minutes) See (ss)8 of the text, especially (ss)8.1 and (ss)8.2. Reflections Students should have just had the opportunity to resolve special relativity paradoxes using spacetime diagrams (in this activity). This lecture can be part of the wrapup for that activity, or serve as a good review at the next class meeting. text/html 2011-07-04T15:44:39-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:rflec:rflecpostulates?rev=1309819479 Postulates of Special Relativity (Lecture: 15 minutes) See (ss)2.1 and (ss)2.2 of the text. Reflections The key point of this lecture is that, since the speed of light appears explicitly in Maxwell's equations, the only way for those equations to be valid in all “good” reference frames is for the speed of light to be independent of reference frame. text/html 2011-07-04T15:44:36-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:rflec:rflecproper?rev=1309819476 Proper Time (Lecture: 20 minutes) See (ss)9.1 of the text. Reflections Make sure to draw a geometric representation of these algebraic identities, namely a right triangle whose legs are dx and c dt, and whose hypotenuse is . text/html 2012-05-17T20:16:40-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:rflec:rflecrconserve?rev=1337311000 Relativistic Conservation Laws (Lecture: 20 minutes) See the second half of (ss)9.3 of the text. Reflections Having set the stage in the previous lecture for Newtonian conservation laws, it is straightforward to repeat the derivation in the relativistic case. However, the Einstein addition law leads to quite different conclusions --- and the derivation becomes an exercise in the addition formulas for the hyperbolic trig functions. text/html 2011-07-04T15:44:25-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:rflec:rflecshortest?rev=1309819465 Straight Line is Longest (Lecture: 10 minutes) See the discussion at the end of (ss)8.3 of the text. Reflections This topic is optional, but is yet another surprising property of special relativity. The discussion in the text, in the context of the twin paradox, provides an intuitive justification --- and this topic could alternatively be covered as part of an early discussion of the twin paradox. text/html 2011-07-04T15:44:22-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:rflec:rflecspacetime?rev=1309819462 Spacetime (Lecture: 5 minutes) See (ss)2.2 of the text. Reflections Having digressed to discuss paradoxes, this is a good time to remind the students that the constancy of the speed of light is fundamental to special relativity. Students should bear this fact in mind througout the subsequent introduction of spacetime diagrams and hyperbola trigonometry. text/html 2011-07-04T15:44:19-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:rflec:rflecsummary?rev=1309819459 Course Summary (Lecture: 10 minutes) text/html 2011-07-27T10:37:29-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:rflec:rflectensor?rev=1311788249 Tensor Description of Electromagnetism (Lecture: 40 minutes) See (ss)11.4 and (ss)11.5 of the text. Reflections On the one hand, this lecture provides new insight into the unification of electrity and magnetism. On the other hand, the treatment is not elementary, and it is hard to keep students focused. One possibility might be to summarize the key results without deriving them, referring the interested student to the text. Another might be to skip this material altogether. At the mom… text/html 2011-07-04T15:44:15-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:rflec:rflectidal?rev=1309819455 Tidal Effects (Lecture: 10 minutes) See(ss)13.2 of the text. Reflections Some students may not yet have seen an explanation of the tides; a simple analysis reveals both that there are two high and low tides per day. In general relativity, this effect is due to geodesic deviation, which cases “straight” lines to approach each other --- just as lines of longitude do. This is in fact the modern view of gravity, as seen through general relativity: Gravity curves space, and curvature cause… text/html 2011-12-08T10:17:27-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:rflec:rflectime?rev=1323368247 Time Dilation (Lecture: 5--10 minutes) See (ss)6.3 of the text. If you have access to the Mechanical Universe video, there is a good animation of time-dilation in 3-dimensional spacetime @ 11:00 minutes. Reflections The video introduces 3-dimensional spacetime diagrams for the same animations as shown previously (in a previous lecture). text/html 2011-07-04T15:44:09-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:rflec:rflectimelength?rev=1309819449 Time Dilation and Length Contraction (Lecture: 5 minutes) See (ss)2.3 of the text. Reflections A qualitative review of time dilation and length contraction makes a good summative review prior to analyzing paradoxes. Make sure to emphasize that moving objects are shorter, and moving clocks run slower. text/html 2012-07-13T21:04:14-08:00 http://sites.science.oregonstate.edu/portfolioswiki/courses:lecture:rflec:rflecuniform?rev=1342238654 Uniform Acceleration and Black Holes (Lecture: 10 minutes) See (ss)13.5 of the text. Reflections A simple argument shows that constant acceleration corresponds to moving along a hyperbola in spacetime, and that any such hyperbola has an asymptote, corresponding to a beam of light --- which therefore never quite catches up. The spacetime diagram which describe this situation are quite similar to those which describe black hole solutions in general relativity --- which is not as surprising…