This is important later when we consider angular momentum in an astronomical setting, where the axis of rotation is not necessarily the center of the orbit. In fact, an astronomical object like 'Oumuamua might not even be on a bound orbit, but an unbound orbit -- it enters the solar system, interacts gravitationally with the Sun and then exits the solar system.

Same as a Joule of energy, rather mysterious!

Good definition.

This is a vocabulary term, "lever arm."

Skip this example. It is basically a trig workout

Idealized force structure

Perfect description

Yes, tidal friction exists, tides due to our moon and to the Sun, but this skips over the main question: can gravity exert a torque on a planet? Take Venus, a planet nearly the size of Earth but no moon to cause significant tidal friction. Answer: no. The gravitational field of the Sun is spherically symmetric, no angular dependence, no vorticity.

I do not consider this so reasonable. It is better to say that, in general, for every spatial dimension u, there is a corresponding momentum \(p_u\). In spherical coordinates that could be latitude \(\theta\) and longitude \(\phi\) angles as well as distance from the center of coordinates, \(r\). If the energy states are not dependent on coordinate u, then \(p_u\) is a conserved quantity. For an ice skater, her spin angle can be seen as a longitude angle \(\phi\); he angular momentum, \(p_{\phi}\) is conserved if the ice is smooth enough.

But the author's use of analogy here is OK.

Decent "loose" definition

In fact, energy is a deep mystery, like entropy and time. There is no standard Joule on display inside a glass case in Paris at the famous Rotonde du Assiette de Crevettes or anything like that. Prof. Richard Feynman taught that energy can be calculated in various ways but he was stumped as to what it is.

It is important to realize that in physics today, we have no knowledge of what energy is. ...there are formulas for calculating some numerical quantity, and when we add it all together it [is] always the same number. [Feynman Lectures]

Huh?

57 gram tennis ball. That is a ton of g's of acceleration.

$$a=\frac{F}{m}=\frac{661\,N}{0.057 \, kg}=11596 \frac{m}{s^2}=1183 \times g$$

So about 1200 g's!!!

They say that half the Ph.D.s on the planet are at NASA working out spaceflight trajectories using this most general form, $$\vec{F}_{net}=\frac{\Delta \vec{p}}{\Delta t}$$ because, e.g., a rocket boosting a spacecraft into orbit is always losing mass through the rocket motor -- the "flames" blazing out of the rocket, hundreds of kg per sec of fuel oxidized and violently expelled. So \(\frac{\Delta m}{\Delta t}\) is not negligible and must be included in calculations... not easy

$$F_{net}=ma$$

$$F_{net}=m \frac{\Delta v}{\Delta t}$$

$$F_{net}=\frac{m \Delta v}{\Delta t}$$

and for objects with constant mass, $$m \Delta v = \Delta \left(mv\right)$$ so $$F_{net}=\frac{\Delta p}{\Delta t}$$

Sir Isaac Newton used the term "quantity of motion," and it is his second definition on page 1 of the Principia.

Poor comparison. A better comparison is with another player of similar though different mass and with velocity \(v=8.00 \frac{m}{s}\) and antiparallel to the first player's velocity.

Excellent.

A lovely paragraph, right on the money

Kepler's first law of planetary motion, before Newton, had discovered this conic section idea, but Newton SHOWED or PROVED as in geometric analysis that conic sections were required, no coincidence.

terrestrial and celestial, which at the time of Newton, were not universally considered as a unified system with universal laws governing both realms. In fact, the verification of Newton's law of universal gravitation simplified our view of the physical universe: one book to describe them all, as Galileo foretold.

1. "gravity"
2. "is able to supply"
3. "the necessary centripetal force."

$$1 \longrightarrow 2 \longrightarrow 3$$

$$F \left[ 1 \right] \,= \left[ 2 \right] \frac{mv^2}{r} \left[ 3 \right]$$

:D

We will bypass this section, but it is interesting to read about Coriolis force in regard to hurricanes and weather systems. Cf., Fig. 5

Good.

Basic

Good free body diagram to work out.

Frictionless?

Similar to the Bumblebee tilt test.

This is an important concept in how we use F = ma. The m and the a are measurable quantities (m) or derived from measurable quantities (a).

The F, however, is a slot that can be filled with tension from a rope or gravitational attraction, electrical repulsion or electrical attraction, a strong nuclear attraction or a week nuclear interaction.

Similar to my derivation, 2/15

Another good example.

Good example.

Nice definition of radians system for measuring angles.

Bypass this entire section

No calculations from this section! But plenty of conceptual questions are possible.

An object with basic dimension D has surface area proportional to \(D^2\) but mass proportional to \(D^3\). So quadrupling the size, like mouse size to rat size, the area presented to the wind of \(\times 16\) but it has mass \(\times 64\). Mouse is safe but the rat gets greased by higher terminal velocity.

I might actually try this. : )

Discussed 2/13. Of course, the parachute lends an enormous effective area to the skydiver, so that the new terminal speed -- the landing speed -- is slow enough to survive.

The National Advisory Committee for Aeronautics (NACA), where Hidden Figures' Katherine Johnson rose, was originated during WW I, was an intensive research facility focused on airframe optimization -- aerodynamic shaping. One famous airframe became the P51 Mustang, a prime fighter against the German Luftwaffe.

Katherine Johnson at NASA Langley

Good to remember, in a nutshell. We will not use the complicated version involving density, area etc.

This is why a concept like terminal velocity arises: the upward drag force depends on the speed, \(\propto v^2\)

You may choose as many of these to practice on as you like.

We will bypass this subtopic

I have used diagrams like this on MANY PHY2053 midterms, frequently with different lengths left and right, so that the dip angles are different. Here the dip angle is 5º left and right, and the ropes form an isosceles triangle. So if I make the ropes different lengths, the angles might be 4º and 8º.

and no pushes

This is why I always say that intro physics is SO filled with right triangles.

Still, do not memorize this.

Slightly smaller than without friction. Good, that makes sense

Yup, previous annotation is verified.

The force parallel to the slope will exist if there is some tilt angle θ. It equals \(\vec{w}\) if \(\theta=90\) but equals zip zap if \(\theta=0\). So it has to be proportional to the \(\sin\left(\theta\right)\).

NICE. These components are important when analyzing forces and putting together a free body diagram and a net force calculation, but \(\vec{w}_{\perp}\) and \(\vec{w}_{\parallel}\) are also tricky to sketch in and keep track of.

GMTA

The molecules and atoms of the material act like a tiny trampoline, flexing downward (like the table in Fig. 1(b)) microscopically but exerting plenty of Newtons \(\perp\) to the surface.

So this section 4.5 is also a vocabulary section in which to learn the nomenclature.

This kind of problem and questions could be used to torture a physics graduate student. That is, it is a very difficult problem because of the concepts through which one must matriculate to the calculations, which are quite simple in themselves.

key finding.

Whereas if YOU, an external agent wanted to accelerate her to the left, your push would have to be leftward.

Because this is in the structure of the physical universe, the mundane skateboarders demonstration on first day of lecture is a good demonstration for understanding all forces

HUGE

This is not unreasonable. The normal force, perpendicular to the surface, i.e., to the rails, rises from the intermolecular and interatomic forces in the steel alloy of the rails. They act like little trampolines, dipping just enough to support whatever weight is on it... up until the weight passes the rail's breaking point.

Same as the relationship between the propulsion force Fp from the road surface on the chopper and the friction force f, in the homework

Why we concentrated on that part of Ch. 3

As in HW 3.

an extrapolation that Galileo first made.

This was Aristotle's view. An arrow flies through the air because it presses into the air, air rushes bacwards behind the arrow and then pushes the arrow forward.

Note: the net force can be absent from the free body diagram, for clarity, but the net force could also be drawn in over to the side or overlaid on the f.b.d. if labeled clearly and not too cluttered up.

e.g., the forces of cohesion that hold the skateboard together, which is why you cannot make a skateboard of jello.

i.e., graphical or analytical methods

or the skateboarders' demonstration

I.e., non-quantum

Non-relativistic, v << c.

A short paragraph to describe a very complex controversy! The paragraph is acceptable, but does not do it justice. Well worth reading about, e.g., Galileo's Daughter, by Dava Sobel (UCF Main Library General Collection - 4th Floor QB36.G2 S65 1999 Available)

This is the ordinary assumption in elementary physics. Air flow properties are very difficult to model, so we ignore them. This is the same as saying to our speeds are not large enough to draw appreciable drag from the air, so they can be ignored... until you need high precision!

Sneak preview on syllabus is related to this analysis. Study both.

ooops should be Ry^2

Lovely diagram

The angles here are not that helpful. Main thing is to build resultant from four components Ax, Ay, Bx, By, then get the trig on the resultant R

Huge. Fig. 2 is one to keep in mind all of the semester.

Very important

Excellent sentence about analysis of a ballistic trajectory

This is known as "adding in quadrature" -- i.e., add the squares of each perpendicular side.

Oh, no! Nothing could be further from the truth! But we can provisionally accept this statement for the moment.