23 June 2013

July 3rd is the Shortest Day of the Year

21 June is the longest day and 3 Jul is the shortest.

Many people consider the first day of summer to be the longest day of the year (if you count just the daylight time).  But did you know that 3 July can be considered the shortest day of the year (and this year it is the 54rd a?

How long is a day?

Ask someone how long a day is and you will probably get the answer, "24 hours". However, ask someone at the US Naval Observatory (and at WWVB) and you will get "9,192,631,770x60x60x24 cycles of the standard cesium-133 transition". Ask a farmer and you will get "From sunup to sunset". We have any different definitions. For this post, I am going to use the synodic definition, the time it takes the Sun to go from due south (local solar noon) to the next time it is due south for an Earthbound observer

It starts with Kepler.

Thanks to Johannes Kepler, we know that the Earth orbits the Sun in an eliptical orbit, and that the Earth is moving fastest when it is closest to the Sun (perihelion) and slowest when it when it is farthest (aphelion).

This year, aphelion occurs on 4 Jul at 12:24 EDT, which will be the shortest day of the year.

Let's figure it out.

Since on the above diagram, the Earth is also spinning counter-clockwise, the diagram below shows (with the added lines) one complete rotation of the Earth for both the day of aphelion (3-4) and the day of perihelion (1-2).

Notice that it takes a little more than one rotation of the Earth to have the Sun directly due south again. Also notice that the Earth has to rotate a little more at perihelion than at aphelion to be facing the Sun 1 synodic day later.  So a synodic day is shortest at aphelion.

But how much shorter is it?

While the above argument is enough to show that 4 Jul 2016 is the shortest (synodic) day of the year, it does not say by how much.  Is it minutes or milliseconds? To answer this, we will need some numbers and the more precise form of Kepler's Law used above.  Kepler figured out that orbiting bodies sweep out equal areas in equal time. And by looking at the diagram above, you can see that the angle swept out by the Earth in one sidereal day is equal to the extra angle the Earth has to rotate to get the Sun due south for an Earthbound observer.

Since the area of a circular sector is
this gives the relationship between for angles swept in equal time.

The Earth-Sun distance is 152,098,232 kilometres (or 1.01671388 AU) at aphelion and is 147,098,290 kilometres (or 0.98329134 AU).  This gives

Microseconds or Hours?

So, the Earth rotates around the Sun just a little bit less on 6 July than on 4 January. Since the Earth rotates about 1 degree (actually 360/364 degrees), it rotates about 0.033 degrees less on 5 July for 1 synodic day than on 4 January. How long does it take the Earth to rotate that angle? Since it takes 24 hours to rotate 360 degrees, it takes about 8 seconds to rotate that 0.033 degrees. So the synodic day on 6 July is about 8 seconds shorter than the synodic day on 4 January.

Classroom use.

Unless you are teaching summer school, this is not much use in the classroom.  However, if you flip the seasons, you can show that the day of perihelion is the longest day of the year (and it is close the the "shortest" day of the year, 22 Dec). Since the first day back from the Christmas break is close to 4-5 Jan, I use this for a gradual welcome-back class.  In a regular physics class, we just do the qualitative approach with students playing the parts of the Sun and the Earth.

The discussion starts with what causes the seasons.  It helps to have a globe handy at this point. We then discuss Kepler's laws. I get one student to play the part of the Sun and one the Earth.  Depending on time, I might have the class discuss how the Earth should rotate (the Sun rises in the east). If not, I will just get the Earth rotating in the correct way.  The class decides when the Earth has rotated one complete rotation relative to the room and then notices that it is the Earth (the student) is not facing the Sun. That 5 minute activity sets up the above discussion.

A related post (http://canisiusphysics.blogspot.com/2013/03/if-spring-starts-tuesday-why-does.html)

11 June 2013

A Week in the Life of an AP Physics Reader

What happens between the time the AP Physics Exam is given and when scores are posted? While I can't walk you through all the steps, I can tell you how they are graded.

2 Weeks before

About 48 hours after the exam is given, the questions are posted online and shortly after that, the initial question assignments are sent to the invited Readers. Most then solve the appropriate question and develop a preliminary rubric. Some Readers may also join online discussions about these questions.

Day -3 to Day -1

Three days before the Readers arrive at the grading site (currently Kansas City for the physics exam), the leadership team gathers. The Table Leaders (the ones directly supervising the readers) and the Question Leaders (supervising the TL's) start working on the scoring rubrics. Starting with an initial one, the TL's start looking through random samples to develop a sense of how typical students  answered the questions and how well the rubric matches those responses. The TL's and QL's go through a few rounds of discussing and making small tweaks to the rubric until all concerned are satisfied that it is a fair way of assessing student knowledge of the material covered by the question. Once rubric version 0.99 has been developed, the TL's start developing the training material for the graders. As the TL looks through more and more student responses, the rubric may undergo some final adjustments, however it is locked down before the graders arrive.

Day 1

After an 8 am initial group meeting, the graders go to their assigned locations for training. The Table Leader introduces the Readers to Rubric Version 1.0. That initial discussion includes what responses get what point values and why the leadership team decided that. Common mistakes also get mentioned.

Once the TL is convinced that the Table has a good initial understanding of the rubric, the team breaks into small groups which then reviews a small set of common student responses and scores them as a group. Invariably, there is a spread of scores. The whole table then discusses the individual responses and what the correct scoring should be. The next round consists of graders scoring a sample set by themselves and then comparing scoring with a partner. Again, a group discussion follows, but usually the scoring is becoming more consistent.

Another round of partner discussions follows but with a set of more uncommon responses. The idea is that while everyone will score correctly the perfect or nearly perfect responses and the completely wrong responses, everyone needs to grade the strange and unusual responses the same way. As before, a round of individual scoring follows the partner scoring with discussions subsequent. If warranted, another round may follow either with the entire group or with individuals, but usually this two rounds of training is sufficient.

Days 2-7

Now the slog starts. Each day starts with some quick announcements, but then it is heads down and grading. This is not a time where you grade a few exams, go get a cup of coffee, exchange gossip, and then go back to grade a few more. The expectation of graders is that they are actively scoring exams from 8:00 until the mid-morning break is called. Then, precisely 15 minutes later, you are back to scoring until the lunch break. One hour later, your head is down and you are scoring until the mid-afternoon break and then until the "End of Day" call comes. At first, it seems to be brutal and demeaning to highly-training professionals. But, when you personally have to go through several thousand exams, and the group has to score all, discipline is key.

Once the rubric is "internalized", a typical Reader develops a rhythm and gets into a zone. While the day is long, the repetition seems to make it shorter.

One quirk with the Reading is that you are not expected to start scoring until the actual start time. In fact, readers who want to get an early start are teased and chided (gently). While tardiness is frowned upon and will have a detrimental effect on your review, trying to impress the bosses by starting a half-hour early in the morning will not be noticed (officially).

Once a couple of days have passed, the Chief Reader may determine that certain questions are not being read as fast as anticipated, and so some Readers may be reassigned to other questions. For those Readers, the training process starts anew, but the slog continues after at most half a day. Should a Reader be on a question that finishes early, that Reader is then put in the "Bubble Room". That Reader then bubbles in the scores that another Reader has recorded on a question that has a lot of exams to grade. The goal is that everybody finishes together.

So, why do it?

Some Readers come to one year's reading, hate the regimentation, and are never seen again. But most are eager to come back. Why? One reason is that some see this as summer camp for physics teachers with many of your favorite "bunkmates" returning year after year. Another is that you learn so much about how students can see a problem and what misconceptions they may have. It helps your teaching.

The Reading is held in a section of Kansas City that has a small but vibrant urban core, so there are many evening activities (the picture above is the home of the KC Symphony which is a couple of blocks away from our hotel). The College Board also hosts a couple of evening forums to let people know about the future direction of the AP Exams. And not to be missed is the Physics Professional/Unprofessional Night. So when the Readers are waiting outside the hotel early in the morning to catch flights home, many are already talking about the next year.