A nice simulator for RLC and RC and LC circuits.

Worth fooling around with this one.

Derived in lecture on 11/15

Here is the precise combination in terms of \(R\), \(L\), and \(C\).

Discharge curve, a combination of real exponential \(e^{-\alpha t}\times\) complex exponentials \(e^{\pm i\omega t}\)

Phase \(\phi=kx-\omega t\)

Phase \(\phi=0\) corresponds to the first peak of the \(E_y\) field; \(\phi=2\pi\) corresponds to second peak, and so on. This means that the equation \(kx-\omega t=0\) is the equation of motion for the first peak of the wave, the "wave top" of the wave. Another way to view this is that

$$x=\frac{\omega}{k}t$$

That factor \(\frac{\omega}{k}\) must be the same as \(c\), the speed of the light wave.

Fourier transforms, another huge tool

This concludes the derivation of the wave equation for electromagnetic waves, specifically for \(E_y\left(t,x\right)\). The constants \(\epsilon_0\text{ and }\mu_0\) are related therefore to the speed of light, \(c\).

a.k.a. the vacuum solutions

Understatement. It is a lifetime's study, and that would only be scratching the surface.

i.e., only a vacuum is needed.

Even through a vacuum

Here is a nice blurb in YouTube about interference patterns. It is a HUGE tool for physicists and engineers. https://youtu.be/D7aftTF--5w?si=hABIsE089aUZpVT0

A nice diagram

Einstein's great quest, but he died before he reached it, and, in fact, no one has reached it yet.

Initial condition: Charged up, i.e., \(q\left(0\right)=q_0\)

Nice exercise

I.e., initial load in the capacitor

plug-in exercise

Good exercise

It is interesting to normalize the total energy equation to 1 by dividing both sides by U -- should result in a version of the Pythagorean Identity \(\sin^2\left(x\right)+\cos^2\left(x\right)=1\)

Joule heating, for which the energy dissipation rate is \(P=I^2 R\), and which is also known as "\(I^2 R\) heating."

idealization

Similar to a spring system at equilibrium, \(x=0\), where kinetic energy

$$\frac{1}{2}m\dot{x}^2$$

is maximum.

If you think of the charge \(q\left(t\right)\) as a coordinate, e.g., like \(x\left(t\right)\), then \(U_C\) is like the potential energy of a spring oscillator,

$$U=\frac{1}{2}kx^2$$

but for which the spring constant is \(1/C\)

Resembles another famous system,

$$E=\frac{1}{2}kx^2+\frac{1}{2}m\dot{x}^2$$

the harmonic oscillator.

Quadratic in current \(I\), therefore current \(I\) will oscillate.

We will review this application in lecture

Note that the diagram here and Fig. 12.3, greatly exaggerate the length of the line element \(d\vec{\ell}\)

The diagram in Webcourses is much more accurate.

A big "if" with a ton of trig involved.

Since \(x\gg\ell\), the approximation as \(\Delta\ell\) is good, and so is the approximation \(r_p\approx x\)

Homework problem

We will tackle this section and 12.1 Biot Savart Law on Monday, Nov. 6 Biot-Savart is ROUGH.

Needs a diagram!!!!!! A 3-D blackboard is needed here.

heinous diagram

Because \(\vec{F}\text{ is always }\perp \vec{v}\)

Operational or experimental definition, but not a theoretical definition!

What the observer sees: the compass needle aligns to the field. Lecture, 10/23

I am not sure why this problem is here.

Good. Be sure to verify this answer.

Technically, this makes the problem insoluble, since it is not specified that \(Q_1\) and \(Q_2\) are static, fixed in place, or not. IF you consider then to be fixed in place, it is an easy problem we have already studied. So proceed on that basis: I may review this problem in lecture on Wednesday, or in Discusssions.

A good workout with the nitty gritty of EPE, KE etc.

Significant question. The potential energy curve rules the dynamical world. * Here is a potential barrier, * Here is a potential well.

There is a huge difference between them, and the implications extend from quantum electrodynamics to black holes.

Finish working this one out, and try to follow the logic of the textbook authors. It is slightly different than my description, so understanding both will help you.

"energy at infinity" is all kinetic. The charge Q effectively has no interaction with q.

As noted in lecture, 10/2

A good problem to work out.

This diagram and description will be on part 1 of the midterm, one or two multiple choice concept questions

Also a very important result in Newton's Law of Universal Gravitation.

$$k=\frac{1}{4\pi\epsilon_0}$$ so $$\epsilon_0 = \frac{1}{4\pi k}$$

Not a coincidence. The Coulomb interaction is formally an infinite range interaction, directly because it is inverse \(r^2\)

Preparing to integrate!

Nice example, very basic

$$0^{\circ}\leq\theta\leq 180^{\circ}$$ I.e., from parallel to antiparallel

Here we go. Calculus ahead

Also similar to how CGI makes Gollum from the "wire frame" generated by Andy Serkis' motion capture suit.<br />

Important

Its destiny is to become an element of area, e.g.,

$$dA=dx\, dy$$

or

$$dA=r\,dr\,d\theta$$

Unfortunate choice of symbol, because it confounds with the unit of E field, \(\frac{N}{C}\).

I.e., units of E field, \(\frac{N}{C}\,\,\times\) units of area \(m^2\).

Even better: a hula hoop held out the window by a friend as you drive up Alafaya to Publix for Chinese food. Even your hand out the window works a bit as an analogy. If your hand is orthogonal to the road and therefore to the velocity of the air rushing past, you feel its force of air resistance. But if you hold your palm flat, parallel to the road surface, you feel way less air resistance.

This statement, however, is limited, because the electric field fills ALL of space, so that is a continuous \(\infty\) of vectors, no matter what direction the element of area \(dA\) is oriented.

SO... we must agree upon a standard of field line density, somehow.

I.e., $$d\vec{E}\cdot\, \hat{n}\,dA$$, in which \(\hat{n}\) encodes the orientation of the infinitesimal tile of surface, \(dA\)

Hmmmmmmmmmmmm....... will there be time on Friday in lecture?

Hmmmmmmmm.....

Flux!

A ubiqitous diagram, very famous, because many molecules, even large ones, have a polar symmetry: a more positive end and a more negative end. Example: \(H_2O\).

The guts of our Quiz 1

Covered in lecture, 8/28/23, in my comments about \(\hat{r}_{21}\)

Where we get the word "electric" and "electron."

Or to your head. :D

You can easily do this experiment at home. Kinda cool.

We normally do this demonstration in lecture on the first day of the semester.

about 103 mph

This makes the example more of a pin ball machine: give the object some speed than inject it into the "frictionless" track.

For this formula, it is also interesting to think about the circular proving track in Italy, the Nardo ring. It is banked, unlike this example, but there is still a correspondence between optimum speed \(v\) and the track's turn radius, \(r\). That is because the track surface and tires have a set coefficient of friction, and \(g\) is known, so the turn radius for the speed you want can be calculated: $$r=\frac{v^2}{\mu_s g}$$ That is, 1. Decide the top speed at which you want to test, 2. then calculate the turn radius you need, 3. and therefore the amount of land you need to build your own test track.

A good workout, and you can check your answer. If you work through Example 2 (above) then you can definitely work this problem... though they are not identical.

Think of this as similar to what we observed with the skateboarder interactions, where the smaller skateboarder got noticeably more speed.

This is like a sideways version of an old-fashioned spring scale. A modern scale uses a device called a strain gauge or a piezo-electric cell, each of which generates shanges in electrical current based on strain or pressure applied to the device.

True true true. If you have a velocity graph, you can analyze it for two other important physical quantities: 1. position and distances, which we get by calculating areas. Terms for this calculation are velocity rectangle and velocity triangle. 2. accelerations, which we get by computing slope of the graph, rise over run in geometry class but \(\frac{\Delta\vec{v}}{\Delta t}\) on a velocity graph in physics class.

Sir Isaac Newton himself figured out that position, velocity and acceleration are the main physical quantities a scientist needs to study any moving physical object, and that they all relate directly to his construct, forces.

This section, Ch. 3.4, is the target section for Midterm Exam 1, for Summer B, 2023. For that reason, these problems and exercises are good good good practice if you are preparing to crush the midterm.

A gas like \(O_2\) molecular oxygen can be converted to a liquid by compressing it. AND, a tank of LOX liquid oxygen has to be kept cold and under pressure, like on the space shuttle, one of the last things loaded before launch.

This is a good reinforcement on the lectures about tidal heating in Io and Enceladus.

NASA image of several emission spectra:

1. hydrogen
2. helium
3. oxygen
4. neon
5. iron

*

Here is a YouTube of a nice standing wave demonstration a few years ago.

https://youtu.be/n2lSvcHKtBY

This example is slightly deceptive, in that it presumes that the value of \(G\) is already known. Historically, that was not the case. The first successful measurement of the value of \(G\) in the laboratory in 1798, by Prof. Henry Cavendish of Cambridge University, led to a calculation of the density and mass of Earth.

Brain burner of the century!

Another good workout

A good workout

A good illustration.

Good workout that makes you think.

There is an error in this example.

Error. \(\left(5.0\, cm\right)=0.05\, m\)

So \(\Delta U_{elastic}\longrightarrow -0.0075\, J\)

"exact differential" is pertinent to a differential equations course and frequently seen in physical applications for the first time in a thermodynamics course.

SO... this section is the author's way of giving you a sneak preview of something almost every engineering and physics major will eventually see.

Bypass

Optional. Read or skim if you have a personal interest.

This paragraph distills an enormous amount of history, several century of thinking about quantifying motion, which led us to the idea of \(\frac{1}{2}mv^2\) kinetic energy.

Reminiscent of the impulse formula, $$d\vec{p}=\vec{F}_{net}\Delta t$$ which describes an increment of dynamical change to the momentum of the object, \(\vec{p}\)

"a large amount of charge" is nebulous, unless you compare it to something else well known. \(I = 1.00\) Ampere of current flowing for 1.00 second = 1.00 Coulombs. In terms of electrons, that would be much less than 1.00 mole of electrons in motion:

$$I=1.00\frac{C}{s}=\left(\frac{1.00\,C}{1.602\times10^{-19}C/e}\right)/{s}$$

$$I=\left(6.248\times 10^{18}e\right)/s \ll N_{A}/s$$

Meanwhile a calculator might operate with current \(I_2\) of a few micro Amps of current would be even fewer electrons per second.

$$I_2 =\left(6.248\times 10^{12}e\right)/s \ll N_{A}/s$$

So it all depends on what you consider to be "large" or "small."

A nice example

Huge calculus bypassed here, but average voltage etc. does an adequate job for now.

A good brain burner!!

A good study problem

Nice combo example

Another good example of two coupled free body diagrams. What is the force that mediates the coupling of the diagrams?

Sometimes I sketch the components of a constituent force in the free body diagram itself, and other times I split off a separate diagram of just the constituent force vector, like \(\vec{P}\) here, somewhere off to the side for clarity, and then sketch in its components.

I like this example, featuring two free body diagrams that are coupled together by interaction forces \(\vec{A}_{21}\) and \(\vec{A}_{12}\)

Optional. I never do this. But remember that the textbook author DOES use squigglies.

This is what we will normally do.

Important definition

A nice sketching exercise!

Be careful with the - signs in this section. :\

"only the charge creating it" ← those charge are also known as source charges.

Kind of idiotic but...

A good generic example that uses

1. integration twice and
2. the initital conditions.

A good brain burner

A very good exercise

A good exercise

Here is how the author uses \(g\) so be alert.

Good. And I like to denote its downwardness with a negative sign, $$g=-9.8\frac{m}{s^2}$$ and as a vector, $$\vec{g}=\left(-9.8\frac{m}{s^2}\right)\hat{j}$$

We will use calculus, not this average velocity argument, to derive \(x\left(t\right)=x\left(0\right)+v_x\left(0\right)t+\frac{1}{2}at^2\)

...like these guys!

Notice how the streamlines get crowded together on one side of the airfoil... the lift goes toward those crowded stream lines.

Good concept to remember. You can see the "remnant" of the conventional kinetic energy fmla in $$\frac{1}{2}\rho v^2$$

and the remnant of gravitational potential energy in $$\rho g h$$

The new term is pressure. When we talk about photons at the end of the semester, it gets even more interesting!

Cool effect. This is essentially dynamic lift, but working sideways!

Interesting example for all biology students

Continuity equation is important. Fig. 2 is good.

Actually has to do with the idea of flux, which we will also use in electromagnetism, e.g., a magnetic field flux, Fig. 4, Ch. 23.1

Amazing

so units of volume/second

Like a fireball rising through much cooler air. "Fire Breather" by Jon_Senior is licensed with CC BY-NC-ND 2.0. To view a copy of this license, visit https://creativecommons.org/licenses/by-nc-nd/2.0/

Nice.

...in the doctor's office

E.g., at Wawa, an extra 32 PSI above the ambient air pressure.

This is why "they" say that you should keep the blood pressure sleeve at approximately the level of your heart. "Blood pressure check" by Army Medicine is licensed with CC BY 2.0. To view a copy of this license, visit https://creativecommons.org/licenses/by/2.0/

Good workout for a study problem.

We will work out this idea in lecture.

Good force multiplier

so that there is no pressure differential due to \(\Delta z\) elevation difference, i.e., \(\rho g\Delta z\)

Brake fluid is not water. It is usually some kind of glycol ether, very high boiling point and other properties.

Yes, Pascal was a very interesting man, and for more reasons than are mentioned here.

This is what makes a hydraulic system so useful for tranmitting and exerting huge forces, like excavation machines.

Sometimes you see depth for scuba divers rated in atmospheres instead of meters. 1.00 atmosphere of depth = 10.3 meters

1013.25 millibars

...because the designers know how to use calculus, not rough averages

Very rough average, but acceptable

Or, more generally, \(p=\rho g z\) with \(z\) being the vertical coordinate over one’s head toward the zenith

Excellent simulations. I highly recommend them. It is especially nifty to view the diffusion simulation, with data enabled, and watch both sides of the container converge to an equilibrium temperature.

Important: buoyancy forces.

Good example.

A friend of mine in the Air Force did a lot of loading of tanks into the big C5-A Galaxy cargo plane. He said that, to save weight, the floor of the cargo bay was not that stout, but that a tank, with its huge weight spread over the big wide treads, would not damage the floor, and a woman walking in high heel shoes would puncture the floor!

E.g., in relation to hurricanes. On the Weather Channel etc., the intensity of a hurricane or tropical storm is usually described by a single number, its central pressure. Hurricane Dorian in 2019 had a central pressure of 910 millibars at its most intense on Sept. 1, 2019 over the Bahamas. Normal atmospheric pressure at sea level on a fair day is higher, 1013.25 millibars. Notice how small a pressure variation that is,

$$\frac{\Delta p}{1013.25\, mb}=\frac{103.25\, mb}{1013.25\, mb}=0.102$$

i.e., about 10%.

This is a lot of mass. \(1.000\text{ metric ton }=1000\, kg\), so this reservoir mass is \(2\times 10^9\, \text{tons}\), two billion metric tone. This is a lot of mass.

I like this experiment.

Heavier than air

Why icebergs float!

I always remember this, 1 gram per cubic centimeter.

Like air bubbles underwater, rising to the surface and expanding as they go.

a very small \(\Delta t\) interaction time.

← between the water molecules are hydrogen bonds which you can read about in any standard chemistry textbook/UNIT_3%3A_THE_STATES_OF_MATTER/10%3A_Solids_Liquids_and_Phase_Transitions/10.3%3A_Intermolecular_Forces_in_Liquids).

This rock looks like a steak...

or maybe a slice of a tree with visible tree rings.

But, okay, we will go with "this rock".

NOT. They are interacting and the interaction is stable against small spatial perturbations. I.e., it will oscillate around its equilibrium and rapidly radiate off the excess energy to settle back down to the equilibrium distance.

You can read about shearing forces in Chapter 5.3.

I.e., it can be shown using calculus and a ton of trig!

Hans Christian Ørsted in Denmark, and Michael Faraday in England made this discovery.

And, in fact, the electromagnetic forces produced by the muscle fibers in your quadriceps muscles DO overwhelm the gravitational force of the Earth, every time you go up a stairway!

Time does NOT slow down. However, time measurements by two different observers disagree if the two observers are timing the same events from different positions relative to the "very massive" body, like a black hole.

ALL forms of energy affect the curvature of spacetime

It is interesting to think about other large, nearby objects that pull gravitationally on us.

$$\text{You }\approx50 - 100\,kg \\ \updownarrow \\ \text{Earth }\approx 6\times 10^24\, kg,\, \\ @ r=6.371\times 10^6 \,m$$

vs.

$$\text{You on the sidewalk @ 350 Fifth Ave., NYC } \\ \approx 50 - 100\,kg\\ \updownarrow \\ \text{Empire State Building }\approx 3.31\times 10^8\, kg,\\ \text{w/ center of mass, elevation } h\approx 200\,m$$

How do those forces compare?

1. The cosmological abundance of elements in the universe is a huge topic in astrophysics.
2. The abundance or elements in solar system objects like planets, moons, comets, asteroids and meteorites that fall to Earth is a huge topic for planetary physicists, like UCF's Planetary Sciences Group.

This topic, action at a distance, has been a topic of scientific discussion for centuries.

Should be submicroscopic. VERY submicroscopic!

I.e., chemistry! The periods and groups of the periodic table are helpful because they encode the behavior of ELECTRONS in neutral atoms of each element. Hinrich's spiral "Programme of Atomechanics"

Instead of an average velocity relative to two points:

1. \(\left(x_1,\, y_1\right)\) @ time \(t_1\)
2. \(\left(x_2,\, y_2\right)\) @ time \(t_2\)

where the average speed is

$$\vec{\bar{v}}=\left(\frac{x_2-x_1}{t_2-t_1}\right)\hat{i}+\left(\frac{y_2-y_1}{t_2-t_1}\right)\hat{j}$$

When you take the instantaneous limit, \(\Delta t\longrightarrow 0\) , the other two Δs decrease, also, but the ratios \(\frac{\Delta x}{\Delta t}\) and \(\frac{\Delta y}{\Delta t}\) do not vanish; they converge to a finite value, the components \(v_x\) and \(v_y\)of the instantaneous velocity at time t, where \(t_1<t<t_2\),</p>

$$\vec{v}\left(t\right)=v_x \left(t\right)\hat{i}+v_y \left(t\right)\hat{j}$$

Compare to the \(v\left(t\right)\) formula with the instantaneous limit in the next paragraph.

Two different parameterizations of the path \(\Gamma\), one by increments of (approximately) equal distance, \(t\), and a second parameterization, \(\color{red}\theta\) , as it might be for an aggressive race car driver, with equal increments of time.

I.e., we parametrize the path along the x-axis.

We call this a path parameter. Cf., Parametrizations of Plane Curves in LibreText, for further reading.

Like a light-second

$$1.00\, LS= 3.00\times 10^8\, m$$

or the Earth-Sun distance, which we call the astronomical unit,

$$1.00\, AU = 499\, LS$$

etc.

For any \(\Delta\) the order of subtraction is always

$$\text{final}-\text{initial}$$

$$\text{later}-\text{earlier}$$

and if some kind of sequence exists, \(\left(u_1, u_2, u_3 \ldots\right) \) $$\Delta u=un - u{n-1}$$

Basic vocabulary

Tricky but important point.

...in motion relative to Earth or to any other object. For instance, when docking your capsule to a spacecraft on orbit, you are moving and your target, the spacecraft, is a moving target.

So your sense of steering includes a moving frame of reference. In fact, you are in a frame of reference that is in free fall!

In the usual sense, we can imagine three mutually perpendicular rigid meter sticks...

"starr-160503-4526-Triticum_aestivum-Glenn_seedheads_and_meter_stick_for_height-Hawea_Pl_Olinda-Maui" by Starr Environmental is licensed with CC BY 2.0. To view a copy of this license, visit https://creativecommons.org/licenses/by/2.0/

...but it could be a set of rigid curves, like on this sundial.

"Israel-05367 - Sundial" by archer10 (Dennis) is licensed with CC BY-SA 2.0. To view a copy of this license, visit https://creativecommons.org/licenses/by-sa/2.0/

This is a set of basic questions that apply to a LOT of the scientific enterprise.

Joule experiment, free expansion.

A little bit like a wavelet, in that the ring is the source of the reflected wave, on the right hand end of the string.

Boundary conditions, so important!! Just like initial conditions being important.

Bypass this chapter section. It is better for a more advanced class, but you may enjoy working through it.

Photons: $$\vec{p}=\hbar\, \vec{k}$$ $$E=\hbar\, \omega$$

Bypass this chapter section. It is better for a more advanced class, but you may enjoy working through it.

SHAPES OF ORBITS determined by energy and angular momentum states:

1. Center of mass, again.
2. The effective potential energy which we know from applying basic planetary angular momentum concepts. $$V_{eff}\left(r\right)=\frac{L^2}{2mr^2}+-\frac{GMm}{r}$$

You will use this technology, Gauss' Law, many times in PHY2049

the field concept is an abstraction useful here and in PHY2049

The Ice Skater Effect. Classic example of conservation of angular momentum.

This example is not even a trick shot in a circus -- i.e., why would anybody want to do this. But it is still illustrative

An interesting exercise. The gymnast slows down his spin just in time to land extended to his full height. Requires a lot of practice, probably.

We discussed this in Session 25

This is an important distinction to make for each and every moment of inertia calculation. The rotation axis

1. might or might not go through the center of mass,
2. it might be aligned along the length of a rod-shaped object or perpendicular to the long axis of the rod shaped axis,
3. it might be along the edge or a rectangular box-shaped object,
4. or it might be any weird axis you need to study,
5. and you might have an irregular object without even any symmetry. Many symmetries in these objects

Asteroid Eros, very irregular

This will be abstracted in PHY2049, when studying the topic of electrostatic potential, which is defined as a property of a charged object or an array of charged objects. Electrostatic potential is not measured in Joules. It is measured in Volts.

Good to keep in mind, even in PHY2049