I think my students and I finally got some things right this semester. Too bad it only took twenty-five years, but I’ll take it anyway. Rather than wallow in self pity, I’ll just get right to it.
My students and I have finally mainstreamed special relativity as the starting point in introductory calculus-based physics. There are no equations, only significant conceptual understanding that, when you get right down to it, is far simpler than almost all students expect it both could and should be. Sometimes we get as far as the Lorentz transformation, but usually we don’t and that’s fine. As long as students understand that all…ALL…of the perceived “weirdness” of special relativity comes from the realization that light’s speed is invariant, we’re good. This leads directly to both time dilation and length contraction. Understanding that nothing “really” happens to moving clocks or moving rods is far more valuable than manipulating countless equations.
My students and I have deprecated pencil and paper in favor of LaTeX. We still use whiteboards, but for anything that is to be turned in the expectation is now that it will be done using LaTeX. The Overleaf environment negates the need for a local TeX/LaTeX installagion and works on every device with which I’ve tested it and allows students to build a library of problems and solutions. Of course they also use my mandi package, which was designed specifically for student use in mind. Note that I almost always use a version that is more recent than that on CTAN. Anyway, at the end of the semester students leave with a folder/portfolio of problems and solutions they’ve written. That’s something tangible I never really had as a student.
My students and I have deprecated traditional tests, quizzes, and such in favor of an approach I rather publicly stole from a Caltech course taught by Kip Thorne. The idea is to let students choose what to do in order to demonstrate learning and understanding. I mean, I’m continually told that “ALL physics students will ultimately be judged by their ability, or lack thereof, to work textbook problems” (note that I don’t necessarily agree with this but it is indeed a strong status quo opinion) so why not just eliminate the traditional tests and get down to the nitty gritty. I present a list of problems from each chapter, and not trivial problems either, from which students choose the ones they feel most accurately convey and demonstrate their achievement. I’m not entirely happy with this in that it implicitly assumes that the textbook problems are the ultimate endpoint, and they are most certainly not. Therefore, next time I will experiment further directing students to my list of problems and questions on this blog that, hopefully, go deeper than many textbook problems.
My students and I have almost deprecated handheld calculators in favor of Python. My ultimate goal is to completely depricate them and have ALL calculation done with Python/VPython scripts. That way, students can build a library of scripts for various purposes. However, writing scripts requires learning Python and students are hesitant to dig in at first and frequently end up resorting to calculators. I must try harder to reinforce the utility of Python in subsequent semesters over and above computational problems from the textbook.
In my opinion, I still lack the ability to sufficiently motivate students to engage in their education outside of the classroom environment I strive to create for them. However, just this morning I read the blog entry in this tweet and began reflecting on my own practices. Maybe I should loosen up even more (I run a comparitively relaxed classroom as it is) and just let students be the people they are and see what happens. It’s something I need to think about.
I also want to experiment with having students present problems and solutions to the class as part of their assessments. This seems like an efficient way to implement oral examination into the course without it being a logistical nightmare.
As always, feedback is welcomed.
This question might serve as a final exam for an introductory physics course. It could serve that purpose for my own courses, but it may not be appropriate for your courses so don’t worry if that’s the case. If you do not include system schemas in your course then this question won’t make any sense to you. I recommend this paper by Lou Turner in The Physics Teacher for familiarizing yourself with system schemas.
(a) Consider four entities in the same region of space, each of which interacts with the other three through some interation, the exact nature of which is not important for our purposes. Draw a schema showing four entities and all of their mutual interactions. Label the entities as 1, 2, 3, and 4.
(b) Draw a system boundary around the system consisting of entities 2 and 4. Explicitly label this sytem.
(c) For this system, write the momentum principle, the energy principle, and the angular momentum principle as specifically as you can given what you know about the system and the four entities. You may feel free to make any necessary (thermodynamic) assumptions for certain approximations to be made as long as you explicitly state them.
(d) For this same system, identify any interactions that could be accounted for using interaction energy (a scalar) rather than force (a vector).
(e) Now, draw a system boundary around the system consisting of all four entities. Explicitly label this system.
(f) For this system, write the momentum principle, the energy principle, and the angular momentum principle as specifically as you can given what you know about the system and the four entities. You may feel free to make any necessary (thermodynamic) assumptions for certain approximations to be made as long as you explicitly state them.
(g) For this same system, identify any interactions that could be accounted for using interaction energy (a scalar) rather than force (a vector).
(h) For the two systems defined above, what would make one preferable over the other? Be as specific and as comprehensive as you can be.
(i) Assume nonrelativistic circumstances in both systems. What, if anything, would change in either schema if you were to transform into a new reference frame?
I will probably add to this list of parts as I think of good items to add. Feedback welcome!
What time is it anyway, and what does that question even mean?
I want to describe a classroom activity that is the culmination of our discussion of time. I’ll start with a brief description of the background leading up to this activity and then describe the activity itself.
If there is any aspect of astronomy that is directly relevant to all of our lives, it is the measurement of time, which, ultimately began with watching the sky. I begin by defining the concept of a prime mover, a term I shamelessly stole from somewhere, I think a discussion of electric circuits, because it seems like the most appropriate term for this purpose. I will happily acknowledge the source if anyone can jar my memory.
A prime mover is any celestial object whose motion we observe for the purposes of measuring time.
Any celestial object can be a prime mover, but there are two main ones we usually adopt: Sun, and the point on the celestial sphere where the ecliptic crosses the celestial equator during the month of March (aka the vernal equinox). In this elementary discussion, we neglect all mention of precession and nutation. To complicate matters, there are actually two (yes, two!) solar prime movers. The Sun we’re all familar with constitutes one of them, and in this context we call it the apparent Sun. The apparent Sun moves along the ecliptic during the year, but does so at a variable rate, moving fastest in January (near perihelion) and solwest in July (near aphelion). This variability is caused by Earth’s non-zero obliquity and non-zero orbital eccentricity. I usually don’t discuss these causes, but sometimes I do depending on how much the class wants to get into it. If they drag me there (and I secretly always hope they do!), I feel obligated to follow their lead. There’s a second Sun, though, and it’s the one by which we’ve ALL lived our entire lives, at least within the context of telling time. This Sun is called the mean Sun. It’s not a physical entity that gives off light. Unlike the apparent Sun, the mean Sun moves around the celestial equator at a uniform rate. Both the apparent and mean Sun take one year, by definition, to go around their respective celestial great circles.
Any given prime mover defines a unique timescale inherent to that prime mover’s motions.
Two consecutive passages of the prime mover over an observer’s local celestial meridian define a day on that prime mover’s timescale.
That interval can then be subdivided into twenty-four hours (Why twenty-four? Probably because it is divisible by so many small integers.) of time on that timescale. Using the apparent Sun defines a timescale called Local Apparent Solar Time (LAST). The interval between two consecutive meridian passages of the apparent Sun defines an apparent solar day and by definition, it is subdivided into twenty-four hours of apparent solar time. LAST is embodied by a sundial (or simply a stick in the ground), which uses a shadow to track the apparent Sun’s diurnal motion across the sky. Using the mean Sun defines a timescale called Local Mean Solar Time (LMST). The interval between two consecutive meridian passages of the mean Sun defines a mean solar day and by definition, it is subdivided into twenty-four hours of mean solar time. LMST is embodied by a mechanical clock designed to track the otherwise invisible mean Sun. Because the two prime movers move at different rates around the sky, an interval or mean solar time isn’t the same as an interval of apparent solar time. Sundial users know this. The discrepancy between the two is called the equation of time.
Fundamentally, we must observe a prime mover’s hour angle (the angle between the hour circle passing through the object extended to the celestial equator and the celestial meridian, measured along the celestial equator) and operationally turn that into something that we call “time.” We could operationally define “time kept by a prime mover” as its hour angle, but there’s a problem with that. That would mean that 00:00 (hh:mm) on that prime mover’s timescale would happen when the prime mover is on the celestial meridian. So what? Well, calendar makers like to have the date rollover at 00:00 and having this happen during the middle of daylight would complicate our daily lives. Imagine waking up on one date and coming home from work on another date. Yuck! So, let’s add a twelve hour offset to put the calendar rollover in the middle of nighttime, when humans are ostensibly the least active.
Time kept by a prime mover = prime mover’s hour angle + twelve hours
LAST and LMST have the “local” attribute because hour angle inherently depends on one’s local sky. All locations on the same north-south line have share a celestial meridian, but if you move east of west, you have a new celestial meridian.
Finally, the question “What time is it?” seems so simple, but to an astronomer it really means, “Where is the prime mover relative to the celestial meridian?”
Okay, that’s all the background (except for a few minor details); now for the actual activity. I have a stack of index cards, each of which has the name of a city and its longitude on it. Usually I only use cities in North America but I have several cards with the names and longitudes of cities on other continents for occasional use.
Each student group (a group of two or a group of four) picks a random card and writes its chosen city and longitude on its whiteboard. I also randomly pick a card and put my chosen location’s name and longitude on the class whiteboard. I ask a student to randomly pick a date, and I put the date and the corresponding value of the equation of time on the class whiteboard. Then I randomly choose whether to give a LMST or LAST, and write my choice on the whiteboard (e.g. a LAST of 13:50, a LMST of 05:14, etc.). So basically, I specify a reference location, a prime mover, a time measured by that prime mover, and the equation of time on the chosen date. The object is for each student group to get the LMST, LAST, STDT (standard time), and UT (Universal Time) at the same moment at its chosen location. Everything boils down to three basic rules:
- Given time kept by a prime mover at one location, to find the time kept by the same prime mover at a DIFFERENT location, the difference in time is the difference in longitude expressed in time units.
- Given time kept by a prime mover at one location, to find the time kept by the other prime mover at the SAME location, the difference in time is the equation of time.
- The STDT is the LMST at the nearest time zone center, so this is just a special case of the first rule. For our purposes, we don’t account for irregular time zone boundaries.
- The UT is the LMST at a longitude of zero degrees (the prime meridian), so once again this is a special case of the first rule. For our purposes, we don’t distinguish among GMT and the various flavors of UT (UTC, UT0, UT1, and UT2).
Since the various groups will have sometimes wildly different longitudes, there’s no way to know whose results are “right” except by doing the necessary calculuations. It is important to do the STDT and UT last, and in that order, becuase it will always be the case that the STDT’s for the various locations will all have the same number of minutes and will have hours differing by integer amounts. These two requirements serve as a sanity check on our calculuations. I tell students that if everyone ends up with STDTs that have the same minutes and hours differing by whole numbers, their results are probably correct. A final sanity check is that the UT must be the same for everyone; that’s one reason it’s called “universal” time.
An interesting extension of this activity is to have everyone determine, for their location, the STDT at noon, where “noon” refers to 12:00 LAST. This multistep calculuation requires setting the LAST to 12:00 (i.e. noon), applying the equation of time to get the LMST, and then applying the necessary longitude correction to get the STDT. Becuase of the equation of time, “noon” doesn’t always happen when a clock reads 12:00 and therefore the phrase “twelve noon” has no practical meaning.
Another useful variation is to specify not the LAST or LMST for the instructor’s reference location, but to cite the hour angle of the apparent Sun and let students express that as a measure of time and then proceed as usual. This is actually a special case of the second rule with a subsequent application of the first rule if necessary. If a location is on a time zone’s center, then no difference in longitude exists.
I may consider writing this activity up for AstroNotes in The Physics Teacher.
As always, feedback is welcome!