20130129

Online reading assignment: motions and cycles (SLO campus)

Astronomy 210, spring semester 2013
Cuesta College, San Luis Obispo, CA

Students have a weekly online reading assignment (hosted by SurveyMonkey.com), where they answer questions based on reading their textbook, material covered in previous lectures, opinion questions, and/or asking (anonymous) questions or making (anonymous) comments. Full credit is given for completing the online reading assignment before next week's lecture, regardless if whether their answers are correct/incorrect. Selected results/questions/comments are addressed by the instructor at the start of the following lecture.

The following questions were asked on reading textbook chapters and previewing presentationsEarth's rotation/precession/revolution/tilt, and the moon's motions and cycles.

Selected/edited responses are given below.

Describe something you found interesting from the assigned textbook reading or presentation preview, and explain why this was personally interesting for you.
"There are some 'circumpolar' constellations that never rise or set. That makes them stand out more than the others."

"That the seasons are not caused by the variation in the distance between Earth and the sun, but by the changes in the amount of solar energy the Earth's hemispheres receive."

"It was helpful to picture precession as Earth's axis wobbling like a top, causing the 'North Star' to change over time."

"The slide with the moon going through all of its phases very interesting. It was sort of fun to quickly see the different phases and recognize them."

"That the sky appears to rotate westward, but its really an eastward rotation of Earth. I've never quite paid attention to the sky, but using the starwheel I was able to recognize how the sky changes with times with Earth's rotation."

"That certain parts of the world can't see certain constellations, for example people in Australia can't see the Big Dipper."

"Sun-sign astrology because of how the sun lines up with different astrology signs depending on what time of the year it is."
Describe something you found confusing from the assigned textbook reading or presentation preview, and explain why this was personally confusing for you.
"In general, learning about astronomy is confusing for me. It is very difficult for me to picture something that is so far out of reach. I am easily becoming lost in all the motions and cycles of the stars, constellations, the sun and the moon."

"I found Earth's precession kind of hard to follow. I'm having a hard time wrapping my brain around how the 'North Star' changes over time."

"What I found confusing for me was the tilt and revolution. I don't understand when the northern hemisphere is facing away from the sun the northern hemisphere will see the sun low in the sky and it will be our winter. Does the northern hemisphere face away from the sun the entire winter?"
Ask the instructor an anonymous question, or make a comment. Selected questions/comments may be discussed in class.
"What's your favorite type of beer?" (Firestone Walker Brewing Company's Velvet Merlin.)

"Will all of the material on the quiz be included on the study guides?" (Yes.)

"What direction are the constellations actually moving? I know it depends on your perspective, but which way do they actually go? Do I move the constellations clockwise in the direction of time or counterclockwise? Thanks" (If you move the starwheel in the direction of advancing clock time, this should be the same direction that moves constellations counterclockwise around Polaris.)

"Can we please have a little more practice with the starwheel as more people get their own? It was a little difficult for me sharing with everyone and didn't really get a hands-on chance." (Yes--looking at the class' responses to the homework starwheel question, we will set aside more starwheel practice questions in class.)

"As a physics professional, why did you choose to teach astronomy?" (Start with an introductory physics textbook, but pull out all of the fascinating and awe-inspiring material. All that stuff...is astronomy.)

"omg pdawg u r so legit and #SWAGGED OUT very respectable u da man." (Yeah, uh, thanks.)

"What is your astrological sign? Do you believe in it?" (In order to answer your question, I will need to consult my Mattel™ Magic 8-Ball®.)

"Why P-dog? I know it sounds cool, but did you pull it out of a hat?" (My teaching assistants at UC-Davis starting referring to me as 'P-diddle' and 'P-money,' etc. to the class, so I asked them to call me 'P-dog' (after 'O-Dog,' from the movie Menace II Society.)

"P-dog, do you have faith in humanity? Where will we go once we have depleted what is left of Earth's natural resources? Do you believe that we are in one of the most critical times in human history? Did you know that Stephen Hawking was born 300 years after Galileo's death? Time to cook dinner."

Online reading assignment: total internal reflection

Physics 205B, spring semester 2013
Cuesta College, San Luis Obispo, CA

Students have a weekly online reading assignment (hosted by SurveyMonkey.com), where they answer questions based on reading their textbook, material covered in previous lectures, opinion questions, and/or asking (anonymous) questions or making (anonymous) comments. Full credit is given for completing the online reading assignment before next week's lecture, regardless if whether their answers are correct/incorrect. Selected results/questions/comments are addressed by the instructor at the start of the following lecture.

The following questions were asked on reading textbook chapters and previewing presentations on total internal reflection.

Selected/edited responses are given below.

Describe something you found interesting from the assigned textbook reading or presentation preview, and explain why this was personally interesting for you.
"I did not know that if you shine light on a diamond that no light will escape the back of the diamond. And rhinestones are actually coated in metal powder to force the reflected light back upwards."

"Light reflecting off the surface of air--I had never really thought of it that way when looking toward the sky from the bottom of a pool."

"That there's a major advantage to using prisms instead of mirrors for many applications. I was always fascinated with how light waves making up images can be bounced off sets of mirrors and not lose the image--for example, binoculars and a periscope!"

"Fiber optic cables used for long distance broadband data transmission. I remember reading somewhere that if someone wanted to cause trouble, he could try cutting the cables that run from North America to Europe under the Atlantic Ocean."
Describe something you found confusing from the assigned textbook reading or presentation preview, and explain why this was personally confusing for you.
"I found it a little bit difficult to keep the rules for total internal reflection straight."

"The three possible cases if light starts in a higher refractive index material--I still get all the different angles mixed up in my head! Total internal reflection and the critical angle would be nice to see drawn out."

This may be a silly thing to be confused about, but even though I read the text; I am still not sure how messages are sent through fiber optics. I always thought that the 'remote cameras on wires' were just regular, small, digital cameras (with the sensor near the end) with wire for sending electrical signals back to a screen. Is the actual wire a fiber optic cable that transmits light from the end of the wire to a sensor at the receiving end?"
Ask the instructor an anonymous question, or make a comment. Selected questions/comments may be discussed in class.
"For a ray of light incident on a stack of three substances, why does the second substance not matter? That doesn't make sense to me." (All three substances do matter when tracing the path of light, but if you only need to find the final transmitted angle in the third substance, then the second substance's n2·sin(θ2) term can be substituted out of the two sets of equations:

n1·sin(θ1) = n2·sin(θ2),
n2·sin(θ2) = n3·sin(θ3),

leading to n1·sin(θ1) = n3·sin(θ3), which can be solved for θ3.)


Will you never be lecturing in class anymore on the slides?" (Not quite; I'll still discuss difficult concepts and do worked-out examples in class. That said, introducing new terminology, and looking at 'cool' applications of physics will be covered during homework rather than in class.")

"Can you describe the phenomenon of a rainbow? It's very interesting but the textbook doesn't describe it quite enough. What is the cause for the light ray to reflect twice inside a raindrop?" (Total internal reflection, where the light ray is greater than the critical angle as it tries to exit the raindrop.)

"A diamond is supposedly the hardest substance naturally create, with all the tight-knit carbon atoms, how does it refract light so well?" (None of the electrons that bond the carbon atoms together have energy level spacings that would absorb visible light photons (thus making it transparent instead of opaque), but they will readily scatter those photons, impeding their passage through the material.)

Online reading assignment: motions and cycles (NC campus)

Astronomy 210, spring semester 2013
Cuesta College, San Luis Obispo, CA

Students have a weekly online reading assignment (hosted by SurveyMonkey.com), where they answer questions based on reading their textbook, material covered in previous lectures, opinion questions, and/or asking (anonymous) questions or making (anonymous) comments. Full credit is given for completing the online reading assignment before next week's lecture, regardless if whether their answers are correct/incorrect. Selected results/questions/comments are addressed by the instructor at the start of the following lecture.

The following questions were asked on reading textbook chapters and previewing presentations on Earth's rotation/precession/revolution/tilt, and the moon's motions and cycles.

Selected/edited responses are given below.

Describe something you found interesting from the assigned textbook reading or presentation preview, and explain why this was personally interesting for you.
"The online presentation was very helpful because there were a lot of pictures."

"The whole reason for the seasons--I didn't know that revolution causes the seasons to change along with the tilt."

"That there are four motions in the sky: rotation, precession, revolution and tilt."

"How astrology is linked to astronomy and how the zodiac signs are evenly spaced around earth. I thought that they were randomly everywhere."

"The phases of the moon were very interesting to me because ever since I was a little kid I was fascinated by the moon and I can't wait to learn about why it looks the way it does."

"That revolution is the motion of Earth around the sun, it takes one year to complete. Also that the sun directly lines up on a different zodiac constellation on certain times of the year."

"I don't really know where I got this from but I always thought the seasons changed because Earth got farther away from the sun. I knew Earth tilted but I thought it moved farther away as well for the seasons to change."

"Australia doesn't see the Big Dipper! I thought everyone could see that or that we all saw the same stars just different times throughout the year."

"I didn't know that over time the sun made a higher arc in the summer and a lower one in the winter because of the way Earth is tilted."
Describe something you found confusing from the assigned textbook reading or presentation preview, and explain why this was personally confusing for you.
"There wasn't anything to be confused on, like at all."

"I didn't really find anything confusing with the reading or the slides. I do have a bit of trouble with the starwheel. I just need a little more practice."

"I found precession confusing and I need a little more clarification on what it exactly does. The spinning top reference helps me understand it a little more, but I could use a bit more explaining on that. But I still don't get exactly what it means and why it's important."

"All the rotation stuff and timing things were confusing because I didn't realize all the different components that went into the seasons, moon phases, etc."

"I need a better visual in order to fully understand all of the Earth's simultaneous movements."

"Telling the difference between a waxing and waning moon, since I think they look similar."
Ask the instructor an anonymous question, or make a comment. Selected questions/comments may be discussed in class.
"I really like how you let us preview the presentations before class."

"I really like the way that this online reading assignment is an easy way to be held accountable for our reading. Not often an instructor knows how to develop a presentation and asks for specific feedback after every assignment. I'm excited to see what the rest of the semester is like." (So am I.)

"Need a little more explaining of material in class." (This is why we have these online reading assignments--so I will be able to expand upon whatever it is you need more explaining on.)

"How often do you swing dance at the Madonna Inn?" (Nearly each and every week. Why aren't you there?)

"How are you an astronomy teacher if you never took an astronomy class?" (Start with an introductory physics textbook, but pull out all the interesting and awe-inspiring material. All that stuff goes into an astronomy textbook. The dry, boring material that's left behind--yeah, that's what we call physics.)

"Your cat is one good-looking feline." (She has her own Facebook page.)

20130127

Online reading assignment: reflection and refraction

Physics 205B, spring semester 2013
Cuesta College, San Luis Obispo, CA

Students have a weekly online reading assignment (hosted by SurveyMonkey.com), where they answer questions based on reading their textbook, material covered in previous lectures, opinion questions, and/or asking (anonymous) questions or making (anonymous) comments. Full credit is given for completing the online reading assignment before next week's lecture, regardless if whether their answers are correct/incorrect. Selected results/questions/comments are addressed by the instructor at the start of the following lecture.

The following questions were asked on reading textbook chapters and previewing presentations on reflection and refraction.

Selected/edited responses are given below.

Describe something you found interesting from the assigned textbook reading or presentation preview, and explain why this was personally interesting for you.
"This one is one of my favorite topics, light reflection/refraction is cool."

"Refraction, because I would expect the light ray to stay in the same direction rather than changing."

"How reflection and refraction relate to geometry, because I am love seeing how geometry relates to everyday life."

"Diffuse reflection actually helps us to see better because more of the light waves are coming off of the reflective surface at once instead of being reflected in an angle equal to the angle of incidence."
Describe something you found confusing from the assigned textbook reading or presentation preview, and explain why this was personally confusing for you.
"I unfortunately have not received my textbook yet."

"Figuring out the difference between specular reflection and diffuse reflection. The textbook even says there is not a sharp distinction between them but I was wondering if there is an easy way to identify each."

"Nothing too confusing yet."

"Snell's law will probably become clear after applying it."
Ask the instructor an anonymous question, or make a comment. Selected questions/comments may be discussed in class.
"Light slows down when it passes through glass, but will it speed up once it reaches air again, or will it stay at the slower speed?" (It will speed up once it is again in air, as the speed depends only the medium it is currently traveling through.)

Could we possibly have an example of Snell's law done in class? A worked-out problem?" (Be careful of what you ask for.)

"Looking forward to another one of your classes!" (Be careful of what you ask for.)

"I have no questions." (Thus you have made a comment.)

"I'm just trying to reserve my PIN. I'm a dork." (No comment.)

20130126

Presentation: interference

And he has a butler.
This is your neighbor. You know, the guy who plays his stereo system way too loud. (Video link: "Maxell Tape: Blown Away (1979).")
Butler: "The usual sir?"
Blown Away Guy: "Please."
(Tape player starts blaring Richard Wagner's Walkürenritt ("Ride of the Valkyries").)
Narrator: "Even after 500 plays, our high-fidelity tape still delivers...high fidelity."
Nobody who plays cassette tapes over a two-channel sound system deserves to crank up the volume.
Maybe we can do something about that, next time we just happen to be in his apartment (invited or not), with some minor adjustments to his stereo system wiring.

Last semester we discussed the behavior of sound waves, and so far this semester have been extending those concepts to model the behavior of electromagnetic radiation. Here we specifically look at the superposition of two waves in general, first sound, then later extending these concepts to visible light in a subsequent presentation.

First, defining a few terms.

Here we have two speakers, which are our sources of two sound waves. Since they are plugged into the same frequency source, they will generate sound waves of the same frequency f (which is depends only on the source), same speed v (which depends only on the medium), and thus the same wavelength λ (which depends on both f and v). If the speakers are wired the same way--red and black wires to red and black plugs--then they will oscillate in phase, with both speaker cones moving forward and backwards in unison.

However, if the speakers are wired with opposite polarities--here, the speaker on the left is wired with black and red wires to red and black plugs--then they will oscillated out of phase, with one speaker cone moving backwards while the other is moving forwards, and then forwards while the other is moving backwards.

When we have two in phase sound sources with speaker cones that move in unison with each other, then the waves they generate will have crests and troughs that line up with each other. The superposition of these two waves will result in constructive interference, which will be a single louder wave.

If instead we have two out of phase sources with speaker cones that move contrary to each other, then the waves they generate will have crests and troughs that line up with the other speaker's troughs and crests. The superposition of these two waves will result in destructive interference--which would ideally be silence--but more realistically would be a single wave that is much quieter. (This is what would result if you switched the speaker wire polarities for one side of your neighbor's stereo system.)

Now let's consider two in phase sound speakers, but for an observer located at a position where the distance from each speaker--the path length--is different.

Here waves from the left speaker travel approximately 0.81 m, while the path length for the waves from the right speaker is about 0.63 m. The path differencel is the (absolute value) of how much farther one wave travels than the other, so in this case ∆l = 0.81 m - 0.63 m = 0.18 m.

This is why you should sit in the 'sweet spot,' equally distant from both speakers in order to minimize any path differences that may cause destructive interference.
Even with in phase speakers we can get either constructive or destructive interference, if the waves from each speaker travel different path lengths, resulting in certain path differences ∆l. For two in phase speakers where one wave travels a half-wavelength longer than the other, the path difference is (1/2)λ, and as a result crests and troughs line up with the other speaker's troughs and crests: destructive interference.

For two in phase speakers where one wave travels a whole wavelength longer than the other, the path difference is λ, and as a result crests and troughs line up with the other speaker's crests and troughs: constructive interference.

Second, mixing up the source phases and path difference conditions for constructive and destructive interference.

Here are two cases where both source phases and path differences matter. The top example is where two sources with a half-wavelength path difference results in constructive interference. The bottom example is where two sources with a whole wavelength path difference results in destructive interference. So how can we account for cases like these?

Whether constructive or destructive interference occurs depends on both the sources (how the waves start out, whether in phase or out of phase) and the path difference ∆l (how far each wave travels farther than the other, whether a whole wavelength or a half-wavelength longer than the other). There are four different cases:
  • For two in phase sources, if each wave travels a whole wavelength longer than the other, then constructive interference occurs (this is the solid black line.)
  • For two in phase sources, if each wave travels a half-wavelength longer than the other, then destructive interference occurs (this is the dashed black line.)
  • For two out of phase sources, if each wave travels a whole wavelength longer than the other, then destructive interference occurs (this is the solid red line.)
  • For two out of phase sources, if each wave travels a half-wavelength longer than the other, then constructive interference occurs (this is the dashed red line.)
Right now these different conditions look rather intimidating, so we'll make sure to be able to practice applying these conditions to various scenarios of in phase sources and out of phase sources with different shifted positions. Remember, there are only four unique cases of different phases and path differences.

20130123

Overheard: all the starwheel questions

2013-01-23_19-39-28_490
http://www.flickr.com/photos/waiferx/8409668035/
Originally uploaded by Waifer X

(Overheard after students practice all three possible types of starwheel (planisphere) questions in class.)

Instructor: "Hey, uh--do you know what Rage Comics are?" (Beat.) "But of course you do, you're college students." (Draws in Rage Comics' "All of the Things" character next to outline of possible starwheel questions.)

20130115

Presentation: optical instruments

Look at them. Just look at them. Old school optical instruments: microscopes and telescopes.


Make sure you get a chance to look through them in class--use the pocket microscopes to look at laptop and smartphone screens, and the telescopes to look at the posters across the room. (Focus the microscopes using the ridged wheels, and focus the telescopes by sliding the eyepiece tube in or out.)

First, the similarities between microscopes and telescopes.

A microscope consists of a (short) tube that holds two lenses apart from each other: an objective lens in the front, and the eyepiece in the back.

Similarly, telescope consists of a (long) tube that holds two lenses apart from each other: an objective lens in the front, and the eyepiece in the back.

Let's look at the two-lens model of a microscope, where the objective is lens 1, and the eyepiece is lens 2. The objective takes the light from object, and creates a real image 1 (how do you know that this would be a real image?). This real image 1 then becomes the object 2 for the eyepiece.

Now let's look at the two-lens model of a telescope, where the objective is lens 1, and the eyepiece is lens 2. The objective takes the light from object, and creates a real image 1 (how do you know that this would be a real image?). This real image 1 then becomes the object 2 for the eyepiece.

Second, differences between microscopes and telescopes. (You may have started to notice some of them already.)

For the microscope ray tracing, the object 1 is placed just outside of the focal point of the objective, which makes a greatly enlarged real image 1. (Which ray tracing(s) ((1)-(10)) best match(es) this?)

Then this image 1 becomes the object 2 for the eyepiece, where it is placed on the focal point of the eyepiece to maximize its angular magnification. (Which ray tracing(s) ((1)-(10)) best match(es) this?)

(Strangely enough, the "tube length" for microscopes is defined as the distance measured between the objective and eyepiece focal points. Compare this definition to the "barrel length" for telescopes, below.)

Then for the telescope ray tracing, the object 1 is extremely distant, such that its rays are essentially parallel. The objective lens then focuses these parallel light rays onto an image 1 located at its focal point. (Which ray tracing(s) ((1)-(10)) best match(es) this?)

Then this image 1 becomes the object 2 for the eyepiece, where it is placed on the focal point of the eyepiece to maximize its angular magnification. (Which ray tracing(s) ((1)-(10)) best match(es) this?)

Where are the ray tracings for microscopes and telescopes most similar? Where do they differ?

(Note how the "barrel length" for telescopes is defined as the distance measured between the objective to the eyepiece lenses, which is the same as the sum of their focal points. Compare this definition to the "tube length" for microscopes, above.)

For the microscope equation, 'L' is the distance between the objective and eyepiece lenses, and 'N' refers to the near point, which is assumed to be the nominal 25 cm value.
Notice the negative sign in the angular magnification equations for microscopes and telescopes--what does this mean for the orientation of the final image seen through the eyepiece? Did you notice this for both the microscope and telescope?

What type of focal lengths would you want for the objective lens of a microscope? Telescope? What type of focal lengths would you want for the eyepiece lens of a microscope? Telescope?

The telescope angular magnification equation does not explicitly refer to the distance between the objective lens and the eyepiece lens. How is this distance related to the focal lengths fo and fe of the objective and eyepiece?

20130114

Presentation: magnifiers

Look at this magnifying glass. Just look at it. Um, through it.

In this presentation we will look through, um, at how magnifiers magnify. (In the next presentation we'll see how these magnifying lenses are used as eyepieces in telescopes and microscopes.)

First, defining what magnifiers do, and to what.

The angular size Θ is not the actual size, it is a measure of how large an angle it subtends with your eye at the origin, and is a measure of how big something "seems" from your viewpoint.

The angular magnification M (upper-case M, to distinguish it from linear magnification lower-case m) is a numerical factor denoting how much larger the angular size of something appears as seen through a magnifier, compared to with just the unaided eye.

Why would anyone use a magnifier with an angular magnification of less than 1?
When a converging lens with a focal length f is used a magnifier, the angular magnification is the ratio of the angular size as seen through the magnifier, compared to the angular size as seen with an unaided eye. This is also the ratio of the near point (the nominal closest distance an unaided eye can focus on, 25 cm) to the the focal length of the magnifier.

Second, the process of magnification using a magnifier.

A magnifying lens doesn't magnify...it FOCUSIFIES!
Let's start with a rather provocative statement: a magnifying lens doesn't really magnify. Here we see a close-up (but unfocused) view of a pliers, and the same pliers at the same distance using a magnifying lens. The angular size of the pliers is relatively unchanged (after accounting for extreme defocusing circle of confusion blurring)!

Consider an Ames room, which is constructed that a person moving along the back of the room (which is actually greatly skewed) will be farther away or closer to an observer's eye, causing the person's angular size to change unexpectedly. This is the main idea behind angular magnification, which is merely caused by bringing an object closer to an eye.


Bringing something closer biggifies it. BIGGIFIES.
Bringing an object closer to an eye increases its angular size, but there is a practical limit to how close an object can be brought such that the eye still can focus on it--the near point, with the nominal value being 25 cm. Any closer would still increase the angular size of the object, but the eye would no longer be able to focus clearly on it.


Now compare these two views of the pliers, without and with a magnifying lens. Note that without the magnifying lens, the view is focused at ∞, where objects on the horizon are sharply in focus, and the pliers is out of focus. With the magnifying lens, the view is still focused at ∞, but now the pliers is in focus, meaning that the virtual image produced by the magnifying lens is at ∞. (How do you know that this is a virtual image? Which ray tracing best matches this?) This is because the pliers is on the focal point of the magnifying lens, which produces a virtual image (of the same angular size) out at infinity.

So to refine our understanding of what magnifiers actually do:
  • The maximum angular size of an object as seen by an unaided (nominal) eye is when the object is placed 25 cm away.
  • For a magnifying lens with a focal length f of less than 25 cm, the object can then be brought closer (up to the focal point of the magnifying lens), such that the magnifying lens allows the eye to be able to focus (at ∞) on an object held closer than 25 cm.
  • Closer is bigger.
One could also think of the magnifying glass as increasing the accommodation ability of the eye to focus on objects nearer than 25 cm, such that objects can be brought closer in order to increase their angular size.

20130113

Presentation: two-lens systems

In the previous presentation we considered vision problems caused by defects in the curvature (and focal length) of the eye. By augmenting the eye with a second lens, here either using a contact lens...

...or glasses, we can compensate for these common vision defects.

This will require us to analyze the two lens systems of a contact lens and the eye, or glasses and the eye using a two-step approach. As a result, we will compress four years of post-graduate optometry school into this presentation, and be able to prescribe corrective optics for common vision defects.

Here we have light passing not just through one lens, but through two lenses. You know, double the trouble, twice the fun.

For any two lens system, the main idea is to just take it one lens at a time. The object 1 in front of lens 1 will produce an (intermediate) image 1, and the thin lens equation is applied to relate do1, di1, and f1.

This image 1 will then become the object 2 for lens 2, which will produce the (final) image 2, and the thin lens equation is again applied to relate do2, di2, and f2.

Keep in mind that the intermediate image 1 from the first lens is subsequently "fed" to the second lens as its object 2.

First, applying the two-step model to contacts and eyes.

You don't need to know the focal length f_2 of the eye, nor the size q_2 of the eye.
The contact lens is the first lens, and the eye is the second lens in this two lens system. The (final) image 2 produced by the eye is a real image on the retina, on back of the eye. If the eye suffers from a visual defect, then by itself it will only be able to see an object 2 located at a distance do2. If used correctly, a contact lens will take an object 1 (located at a nominal distance do1), and produce an image 1 that will be located at a distance do2 = di1 that the eye is able to focus on.

So knowing the nominal object distance do1, and the actual distance do2 an uncorrected eye can focus on, the focal length f1 of the contact lens can be solved for using only one thin lens equation.

A flat sheet will neither focus nor defocus parallel rays of light, such that it effectively has an infinite focal length f, and zero refractive power P.
The focal length of contact lenses are typically not specified, instead they are rated in terms of refractive power P "diopters," which are merely the inverse of the focal length f (measured in meters).

The sign convention for refractive power is the same as for focal lengths: positive values for converging contact lenses, and negative values for diverging contact lenses.

Why must optometrists disguise contact lens focal lengths by taking its inverse? What's up with that?
If you have a prescription from your optometrist, or the box that your contact lenses came in, check out the refractive power P ("diopters") listed there.

Second, applying this two-step model to prescribe contact lenses to correct common vision defects.

We only need to measure your actual far point and your actual near point. If the measured far point is less than the nominal value of infinity (i.e., some measurably finite value), and/or the measured near point is greater than the nominal value of 25 cm...

Really?  You pay your optometrist $100 for this?
...then you will need corrective optics, which we will solve for using the thin lens equation.

Let's do an example for a myopic Physics 205B student, whose uncorrected far point is 5.0 m. This is the farthest distance do2 = +5.0 m (positive object distances correspond to being in front of the eye) that this student's unaided eye can focus on, instead of the nominal farthest distance of +∞.

What this means is that the contact lens will take an object located (in front of it) at do1 = +∞, and produce a virtual image (in front of it!) at di1 = –5.0 m. This virtual image 1 becomes the object 2 (do2 = +5.0 m) for the eye, which is able to see 5.0 m in front of it.

Solving for the focal length f1 of the contact lens,

(1/do1) + (1/di1) = (1/f1),

(1/(+∞)) + (1/(–5.0 m)) = (1/f1),

f1 = –5.0 m,

as (1/(+∞)) = 0. Remember that contact lenses are prescribed in diopters rather than focal lengths, such that the refractive power is:

P = (1/f1) = (1/(–5.0 m)) = –0.20 m–1 or –0.20 D.

This means when you wear contacts (or glasses) to correct myopia, you are actually looking at virtual images!
Note that the negative focal length (and refractive power) value means that contact lenses (and glasses) to correct for myopia are diverging lenses, which take distant objects, and make closer, upright virtual images, which is then the object for the eye, which can see it at this closer distance. (Which ray tracing(s) best match this?)

Now let's do an example for a hyperopic Physics 205B student, whose uncorrected near point is 0.60 m. This is the nearest distance do2 = +0.60 m (positive object distances correspond to being in front of the eye) that this student's unaided eye can focus on, instead of the nominal reading distance of 0.25 m.

What this means is that the contact lens will take an object located (in front of it) at do1 = +0.25 m, and produce a virtual image (in front of it!) at di1 = –0.60 m. This virtual image 1 becomes the object 2 (do2 = +0.60 m) for the eye, which is able to see 0.60 m in front of it.

Solving for the focal length f1 of the contact lens,

(1/do1) + (1/di1) = (1/f1),

(1/(+0.25 m)) + (1/(–0.60 m)) = (1/f1),

f1 = +0.43 m,

Remember that contact lenses are prescribed in diopters rather than focal lengths, such that the refractive power is:

P = (1/f1) = (1/(+0.43 m)) = +2.3 m–1 or +2.3 D.

This means when you wear contacts (or glasses) to correct hyperopia, you are also looking at virtual images!
Note that the positive focal length (and refractive power) value means that contact lenses (and glasses) to correct for hyperopia are converging lenses, which take nearby objects, and make farther away, upright virtual images, which is then the object for the eye, which can see it at this farther distance. (Which ray tracing(s) best match this?)

Now what happens if a Physics 205B student is myopic (nearsighted, can see near, but can't see far) due to defects in curvature of the eye, who as a result of aging develops presbyopia, losing the ability to accommodate and see nearby objects? You would need to prescribe separate diverging and converging lenses to correct for myopia and presbyopia...

When I wear bifocals, it makes me feel sad, like tears are welling up at the bottom of my vision...
...or these two lenses could be combined into bifocal glasses, with the compromise that looking down (at reading distances) would have a diverging lens which only corrects for presbyopia, and looking straight (for far distances) would have a converging lens which only corrects for myopia.