Another problem that does not come with solution. : )

No solution offered for this problem, I see.

Hmmm. Like the tennis ball classwork problem.

This is a brain burner

An excellent study problem.

WIll be interesting to calculate.

"...give him the heater"

Newton was the first one to work out this idea of projectile becoming a spacecraft in orbit, depending on initial conditions \(\left(x\left( 0\right),\, y\left( 0 \right)\right)\) and \(\vec{v}\left( 0 \right)\).

Would make a nice multiple choice item.

I like this combo problem.

Similar to HW 3, but would be simpler to check.

I like this problem.

"analyze"

However, the author does not complete the square, as we did!

Hang time

I never use this equation. YUKK

This example is like driving a car off a cliff, like in a movie.

https://youtu.be/iUy_61hkGPM

Do not try this at home.

This angle, upon which the author concentrates, is not so important to me.

INCORRECT! The entire curve can be parametrized by \(x\) as we discussed in Session 3 and 4

They are not really independent except mathematically. They are actually coupled. For instance, the horizontal range, \(x_{max}\), is controlled by the size of \(v_y\left(0\right)\)

In the fourth equation here,

$$v_y^2=v^2_{0y}-2g\left(y-y_0\right),$$

if you have eyes to see, you can see the remnants of potential energy and kinetic energy terms.

Note that the author uses a slightly different notation than I use:

1. author, \(-g=-9.8\frac{m}{s^2}\)
2. Dr. Brueckner, \(g=-9.8\frac{m}{s^2}\)

Keep this in mind as you study.

standard assumption for basic physics like PHY2048

...plus the fact that there is no such thing as horizontal gravity!

We discussed this in session 7, concerning uniform circular motion. The direction of the acceleration vector \(\vec{a}\left( t\right)\) is the direction of the \(\Delta \vec{v}\left( t\right)\) vector.

I.e., draw the sketch then let the numbers do the talking

Simply the vector itself, divided by the magnitude of the vector. $$\hat{u}=\frac{\vec{U}}{U}$$

This will be a frequent task in PHY2048

We will discuss this problem on Thursday, among other things.

We will occasionally work in polar coordinates.

Important geometric tool

...except this is what a pilot would use, a distance and a heading.

easy to calculate components from each subdiagram

tail to tail

another important vocabulary term

Important vocabulary term.

Who does this? Measure distance to a fishing spot from your tent? Really!

I just use an arrow, \(\vec{A}\), or boldface B, but not both.

Ummm, no.

:)

This is why we do not emphasize formula sheets and memorization.

Cf., e.g, the index of refraction for BK-7 borosilicate crown glass at various wavelengths.

This is news to me, but accurate. Note that this number of hydrogen atoms is much smaller than \(1.00\, mole\). Curious

This is a polite way of saying, "All science is either physics or stamp collecting," which Prof. Ernest Rutherford said way back in his days of discovering the nuclear structure of matter. (Ernest Rutherford)

This is how most scientists work:

• from easily remembered basic principles and examples,
• then extending to new concepts and previously unseen problems.

(Enrico Fermi)

This mental process will be helpful

1. during our exams, which, by definition, will have previously unseen problems,
2. during future courses deeper into your UCF curriculum, and
3. throughout all of your professional work as a professional user of physics knowledge.

...helpful IF you want to excel. (Albert Einstein)

Professor Galileo wrote, "Philosophy is written in this grand book — I mean the universe — which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it it written. It is written in the language of mathematics, and its characters are triangles, circles and other geometrical figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering about in a dark labyrinth.” [(Assayer, VI)]

That is our challenge, the mountain we climb: to understand the grand book of nature.

(https://bibdig.museogalileo.it/Teca/Viewer?an=300984&lang=en)

Good example of what makes an image virtual. I always describe it this way: if you place a photographic film at the image location (behind the mirror, in this case) and no light rays actually hit the film, which is true in this case, then it is a virtual image.

Yuge

important

We observed this with the green laser pointer.

$$\frac{\lambda}{2},\:\frac{3\lambda}{2},\text{ etc.}$$

Green laser pointer

In our sketch, wavelet sources \(S_1\) and \(S_2\).

Sketched out in Session 37, https://youtu.be/BF5e89OmsQQ

We did this calculation during Session 37, at about 45:11. https://youtu.be/BF5e89OmsQQ

Yuge

Approximately the B one octave above middle C.

purple

i.e., plane waves

As we sketched plane waves in Session 37.

I.e., a real lens for which thin lens model is just an approximation

Also, adaptive optics, like the COSTAR optics placed in the Hubble Space Telescope to correct spherical aberration.

About aperture, pupil etc.

Yup, I have astigmatism, which I noticed during my first year of college. (Test diagram)

Interesting simulation of spherical aberration in a microscope system. Eyeglasses can have aspherical lenses to correct this aberation:

Here is a coma test panel, comparing stars imaged at the center of a Canon 24mm f/1.4L II lens, at intermediate distance and at the edge of the lens:

At the edge, you get a lot more coma abberation, but at f2.8, it seems minimized.

E,g., the Zeiss apochromat lens that focuses all colors nicely.

"Achieving this level of correction is not simple. It requires the use of exceptional quality optical materials: long-crown glass (fluor crown glass and calcium fluoride) and special short flint glass."

We have discussed chromatic aberration in class a few times.

Schwarzschild radius" of the Sun. We do not think it will ever get collapsed to this small of a size, because the Sun is not large enough to have a supernova detonation.

This is the classical way of thinking about black holes, and a century after Sir Isaac Newton wrote Principia, another scientist, John Michell, worked it out using non-relativity concepts.

Escape velocity:

1. from the surface of Earth, \(11,200\frac{m}{s}\)
2. from event horizon of a black hole, \(c=3\times 10^8\frac{m}{s}\)

Diameter of DNA helix is about 2 nanometers. So wavelengths on the order of a nanometer are good to use. Rosalind Franklin probably used \(\lambda = 0.15\:nm\) for her xray diffraction images.

Huge

E.g., our handheld diffraction gratings,

with 500 lines per millimeter. That is about one line every 2000 nm, so just a few times larger than the typical Roy G. Biv wavelengths.

ok

This is like the difference between Newtonian physics and quantum physics.

"only"

80 MHz is just below our FM radio band and 2 GHz is in the microwave region of the spectrum.

Used to be very highly classified technology.

Isaac Newton developed this concept.

Second flip = nice!

The best way to think of this idea is the concept of aperture: the pupil of your eye, a few millimeters, vs., e.g. the aperture of the Hubble Space Telescope (HST), about 2.4 meters. HST is about 300 times wider, about 90,000 times more area than the human eye's pupil.

The HST is about the size of a school bus.

(Hubble being transferred from the Vertical Assembly Test Area (VATA) to the High Bay at the Lockheed assembly plant in preparation for transport to the Kennedy Space Center (KSC) after final testing and verification.)

We will conclude Ch. 26.4 with this example, then bypass the remainder of Ch. 26.4

Notice that, in this diagram, the rays are almost parallel coming out of the eyepiece to the eyeball. If the rays are parallel, your eye does not have to strain to focus the image on your retina

Or just one in a simple microscope

Always a big proviso.

An understatement. When we view a star in the night sky, our retina collects very small numbers of Roy G. Biv photons per second.

One refractive interaction, an approximation to the two physical interactions.

Most of this section is bypassed. However, we discussed LC circuits extensively in terms of oscillation, e.g., in Fig. 5.

We discussed "phase" a little bit differently than this.

Bypass or skim the diagrams. We talked about oscillating circuits from the standpoint of energy, but we did not get into any detail of this section, 23.11.

Study this secrion, 23.10. We discussed inductance and RL circuits in Session 27 and other places.

Bypass. It won't be on Exam 3, although Florida residents would do well to read about the GFI technology, which many major appliances and pool pumps have.

Bypass

Bypass

We mentioned generators and motors in general terms, but all of the materials in this section we bypassed.

Bypass

We discussed this briefly in Session 26 in the context of the navy's "rail gun."

Other material in this chapter is bypassed.

Brain burner deluxe....

Brain burner!! Attempt this diagram at your own risk!! :D

Nice mental exercise.

Basic calculations.

I like this question. You can work it out with typical values of indices of refraction, \(n_{air}=1.00,\, n_{water}=1.33,\, n_{glass}=1.5\)

You will have a small table like this on the cover page of Exam 3

Similar to the headless body situation.

VIP, very important principle.

You can also use the marching band effect with a diagram like this.

Here, the left wheels move more slowly on the thick grass, while the right wheels move more rapidly when they hit the sidewalk. Net result: lawn mower turns left, line of travel cuts away from the normal (dotted line).

Left wheels move more rapidly on the sidewalk, and right wheels move more slowly on the thick grass. Net result: lawn mower turns right, line of travel close to the normal (dotted line).

Example: for many types of glass, \(n=1.5\) or so.

!

100% reflected

Very nice simulation of refraction. "More tools" can even produce total internal reflection. It is an HTML file. Download it then click it open -- runs good.

"Not steep enough to permit transmission and refraction."

Good rule of thumb.

Like molecules and crystals, with grid spacings on the order or nanometers.

E.g., using silicon, opaque to Roy G. Biv, but fairly transparent to 1200 - 7000 nm infrared.

By far, our dominant sense, for which our brains are amazingly well developed to analyze and understand.

One of the mirrors on the James Webb Space Telescope, to be launched in 2021, we hope.

Image is as far behind the mirror as the object is in front of the mirror.

I also like this problem: will the electrons in the antenna circuit of the typical radio oscillate enough to an electric field of this intensity? Enough to provide an audible, uninterrupted signal without having to compete with too much random noise in the background? Because I have noticed, while driving around Orlando, that some of our Orlando AM radio stations really fade out in various locations, especially at night.

I like this calculation because lasers are frequently used to move electrons and ions around, with the E field. So knowing, for example, the size of the molecule or metallic lattice, you can tune the laser intensity to get the molecule or ion or electron to dance exactly the way you want it to dance.

Deceptive statement, in my opinion, because saying one field is weak is completely relative to the two scales we have for \(E\) and \(B\). For the electric field the author uses \(\frac{V}{m}\) but for the magnetic field, Teslas. Perhaps a different scale should be used for magnetic field?

It would be interesting to work out a system of constants and so forth so that E and B have the same unit and in radiation are the same size.

Fairly simple plug-in, for the definition, \(I=\frac{P}{A}\)

I like this part of the calculation.

This reminds me of something that astronomers calculate: the "solar constant,", which, it turns out is not quite constant because the Sun's output varies on a daily time scale and slowly over its 11-year sunspot cycle. Currently, the observed value is about \(1360.66\frac{W}{m^2}\).

Oscillation, like the square \(x^2\) in a spring system and the "kinetic term" with a \(v^2\) in it

The electromagnetic wave equation: $$c=\lambda f$$ ...simple yet powerful.

One of the best Interactive Simulations. I completely recommend it. Download the java file, then open it. Try the "radiated field" option -- it is spectacular.

Another way to look at this quotient, \(\frac{E}{B}=c\) involves Newtons of force.

enhance

They carry energy \(E\) and momentum \(p\).

Stated as an aside, but this is a critical concept. The source of electromagnetic radiation is accelerating charge, like the northern lights

In fact, the intensities of radiation in each direction is controlled primarily by the shape and dimensions of the antenna.

Profound. Why should lightning and lodestones have this affinity, i.e., that the \(\vec{E} \perp\vec{B}\)? Lightning Magnetite

One-time exercise, to get over the strangeness of \(\epsilon_0\) and \(\mu_0\).

Discussed in Session 28.

Now we can see where the mysterious constants, \(\epsilon_0\) and \(\mu_0\) come in and combine for the speed of light, $$c^2=\frac{1}{\mu_0 \epsilon_0}$$

WAY beyond

Oscillation analogy, similar to the diagram in Session 27. https://youtu.be/T_2R9JBu89E

Nice!

I.e., tons of trig and calculus

Blackout!

Can lead to huge arcing of current

ANother significant dependence

Significant: \(\mathcal{emf}\, \propto \, \frac{\Delta I}{\Delta t}\) is going to lead to oscillation

A dodge. The average field value requires a ton of calculus and trig!

The coil gains RIGHTWARD field lines, do the coil's induced current produces leftward field lines, \(\vec{B}_{coil}\)

Images (b) and (c) show induced fields \(\vec{B}_{coil}\) consistent with Lenz' Law.

Another big understatement!

If we were to make a full general definition, accounting for shapes of circuit and of the field, then there'd be

• a ton of trig, and
• a ton of calculus

The flux must now be dealt with!

Like Grand Coulee Dam on the Columbia River.

Mentioned in Lecture 22. https://youtu.be/ZJOsVs0kzgw

Mentioned in Lecture 22. https://youtu.be/ZJOsVs0kzgw

Nifty YouTube of sound waves Doppler shifting, about 0:40. https://youtu.be/w0AadZ1bs78

"Complicating factor," yes, but this Doppler shifting is something we understand, and it gives us basic leverage to extract more information:

1. "blue shift" meaning a λ wavelength shift to shorter than normal λ, and you can extract the speed at which the object approaches the observer;
2. "red shift" in which the observed λ wavelength is shifted to a longer λ, plus you can extract the speed at which the object is moving away from the observer.

Here is an interesting diagram of Doppler shift information uncovering currents underneath sunspots!:

Because the speed of light c is a constant, and because the electromagnetic wave equation is c = λf,

• when the wavelength increases (redshift), the frequency decreases, but
• when the wavelength decreases (blueshift), the frequency has to get bigger.

The arrow of Observer C's wavelength is slightly off, too short. It should go from the top edge of the dark red circle to the top edge of the next circle in (reddish pink).

As I always say, "What does the observer see?"

Will mention in Session 20 or 21.

Needs a third sub diagram that shows the reversed current.

Would be a restoring force, leading to oscillation, if current continues in the initial direction.

...using tons of trig and calculus!

Important diagram

We will try this demonstration in Friday's session 23.

Cf., The Hunt for Red October.

https://www.imdb.com/title/tt0099810/

Will review this in Session 23.

We viewed this force effect in Session 21, with the big horseshoe magnet and the thick copper strip between the poles of the magnet.

We will work out an example from this set up in our Session 21.

Bypass

For Exam 2, we will bypass the aurora and other examples past this point.

Like in CSI

Cosmic rays are protons and other particles blown out by supernova detonations and big galactic black holes.

I severely hate this version of the right hand rule. However, other semingly normal people might like it, so I will include it in the textbook without mass crossouts.

"relatively simple" ← nice, but do not forget the appearance/disappearance of the magnetic field depending on the motion of the observer.

This is a good set of questions to ask of the magnetic field of Earth. Theoretical image of field lines:

Actual field lines:

Very important diagram.

As reviewed in Session 20.

Always true, unlike electric field lines. There are electric monopoles, e.g., proton and electron, with porcupine and anti-porcupine field lines. There are looped \(\vec{E}\) field lines starting with a dipole array.

Ditto with \(\vec{E}\) field lines

Flux again.

Good -- similar to our idea about electric field lines.

NOT

crude indeed. As far as we can tell, electrons have no structure, so saying that it spins on an axis like a planet is very crude. We make do, saying that the electron has intrinsic spin angular momentum, \(\plusmn \frac{\hbar}{2}\)

Moving charge is the source of the magnetic field.

Any large structure built of steel, like a building of steel girders or a warship, can be magnetized by pounding the steel: pile drivers, riveters, etc. Liberty ship, keel plates

This topic should follow "Electromagnets" because we understood electromagnets a century earlier than we savvied ferromagnetism (iron atoms as tiny electromagnets).

Check for paired white-and-black spots on the Sun's magnetogram. The Sun is pretty quiet now, however.

So the magnetic pole up in arctic Canada is actually an S!!

Might be no such thing as a magnetic monopole. Magnets seem only to come as dipoles.

We can thank Michael Faraday for these developments.

WHat an understatement! Transistors on various electronics chips are on the order of a few dozen nanometers!

Check the Session 18 extension in YouTube.

The circuit's "time constant" \(\tau = RC\) is related to the half-life of the circuit. Both terms are used by scientists in exponentially growing or decaying systems.

Will be helpful when working on HW 2.

We will work out a few like this on Friday, 10/4.

idealization

So after an amount of time \(\Delta t\), all of the energy of a battery will dissipate at the circuit element, e.g., the light bulb, where stored Joules from the battery are dissipated as light and heat. At that point, the bulb goes cold.

This term traces way back to the 1800s, before we had figured out everything and unified the electromagnetic theory. So calling it a force is anachronistic, but we still use it for the sake of custom. The common symbol is an upper case script E, \(\mathcal{E}\)