A valuable model for figuring out how we're going to save the Earth
Let's start by visualizing the energy flow associated with driving a gasoline-powered car. We will use a box and arrow diagram, where boxes represent where energy can accumulate, and arrows show energy flow.
The energy clearly starts in the form of gasoline in the tank. Where does it go?
Actually ask this of students.
Visualize the energy as an indestructable, incompressible liquid.
“Energy is conserved”
The heat can look like
Hot exhaust gas
The radiator (its job is to dissipate heat)
Friction heating in the drive train
The work contribute to
Rubber tires heated by deformation
Wind, which ultimately ends up as heating the atmosphere
The most important factors for a coarse-grain model of highway driving:
The 75:25 split between “heat” and “work”
The trail of wind behind a car
What might we have missed? Where else might energy have gone?
We ignored the kinetic energy of the car, and the energy dissipated as heat in the brakes. On the interstate this is appropriate, but for city driving the dominant “work” may be in accelerating the car to 30 mph, and with that energy then converted into heat by the brakes.
In rectangular coordinates, the natural unit vectors are \(\{\boldsymbol{\hat x},\boldsymbol{\hat y}\}\), which point in the direction of increasing \(x\) and \(y\), respectively. Similarly, in polar coordinates the natural unit vectors are
\(\boldsymbol{\hat r}\), which points in the direction of increasing \(r\), and \(\boldsymbol{\hat\phi}\), which points in the direction of increasing \(\phi\).
The unit tangent vector to a parametric curve is the unit vector tangent to the curve which points in the direction of increasing parameter. The principal unit normal vector to a parametric curve is the unit vector perpendicular to the curve “in the direction of bending”, which is the direction of the derivative of the unit tangent vector.
Consider the parametric curve
\(\boldsymbol{\vec r} = 3\cos\phi\,\boldsymbol{\hat x} + 3\sin\phi\,\boldsymbol{\hat y}\)
with \(\phi\in[0,2\pi]\). Calculate the unit tangent vector \(\boldsymbol{\hat T}\) and the principal unit normal vector \(\boldsymbol{\hat N}\) for this curve in terms of \(\boldsymbol{\hat x}\) and \(\boldsymbol{\hat y}\).
Consider a circle of radius \(3\) centered at the origin. Determine the unit tangent vector \(\boldsymbol{\hat T}\) and the principal unit normal vector \(\boldsymbol{\hat N}\) for this curve in terms of \(\boldsymbol{\hat r}\) and \(\boldsymbol{\hat\phi}\).
Compare your answers.
Instructor's Guide
Main ideas
Geometric introduction of \(\boldsymbol{\hat r}\) and \(\boldsymbol{\hat\phi}\).
Geometric introduction of unit tangent and normal vectors.
Prerequisites
The position vector \(\vec{r}\).
The derivative of the position vector is tangent to the curve.
Warmup
See the prerequisites. It is possible to briefly introduce these ideas
immediately preceding this activity.
Props
whiteboards and pens
Wrapup
Emphasize that \(\boldsymbol{\hat r}\) and \(\boldsymbol{\hat\phi}\) do not live at the origin! Encourage
students to use the figure provided, which may help alleviate this confusion.
Point out to the students that \(\boldsymbol{\hat r}\) and \(\boldsymbol{\hat\phi}\) are defined everywhere
(except at the origin), whereas \(\boldsymbol{\hat{T}}\) and \(\boldsymbol{\hat{N}}\) are properties of the curve.
It is only on circles that these two notions coincide; \(\boldsymbol{\hat r}\) and \(\boldsymbol{\hat\phi}\) are
adapted to round problems, and circles are round! Symmetry is important.
Emphasize that \(\{\boldsymbol{\hat r},\boldsymbol{\hat\phi}\}\) can be used as a basis (except at the
origin). Point out to the students that their answer to the last problem
gives them a formula expressing \(\boldsymbol{\hat r}\) and \(\boldsymbol{\hat\phi}\) in terms of \(\boldsymbol{\hat{x}}\) and
\(\boldsymbol{\hat{y}}\). When comparing these basis vectors, they should all be drawn with
their tails at the same point.
Details
We have had success helping students master the idea of “direction of
bending” by describing the curve as part of a pickle jar; the principal unit
normal vector points at the pickles!
In the Classroom
The easiest way to find \(\boldsymbol{\hat{N}}\) is to use the dot product to find vectors
orthogonal to \(\boldsymbol{\hat{T}}\), then normalize. Students must then use the “direction
of bending” criterion to choose between the two possible orientations.
Finding \(\boldsymbol{\hat{N}}\) in this way requires the student to give names to the its
unknown components. This is a nontrivial skill; many students will have
trouble with this.
It may be important to draw some examples. Despite that, students still feel wary of embracing \(\boldsymbol{\hat{r}}\) and \(\boldsymbol{\hat{\phi}}\). People will feel more comfortable over the next few classes but emphasize the geometry: the circle is still the circle and the unit tangent remains the same regardless.
Some students are natural geometors and will realize what the desired vectors are. This is terrific. Certainly, it is worthwhile to explicitly demonstrate this, but the point is the geometry can often do the work for you. This will convince students of the value of smart coordinates.
Subsidiary ideas
Dividing any vector by its length yields a unit vector.
Using the dot product to find vectors perpendicular to a given vector.
Homework
Some students will not be comfortable unless they work out the components of
\(\boldsymbol{\hat r}\) and \(\boldsymbol{\hat\phi}\) with respect to \(\boldsymbol{\hat{x}}\) and \(\boldsymbol{\hat{y}}\). Let them.
Enrichment
What units does a unit vector have? Do \(\boldsymbol{\hat r}\) and \(\boldsymbol{\hat\phi}\) have the same
units?
A student is invited to “act out” motion corresponding to a plot of effective potential vs. distance. The student plays the role of the “Earth” while the instructor plays the “Sun”.
Students observe the motion of a puck tethered to the center of the airtable. Then they plot the potential energy for the puck on their small whiteboards. A class discussion follows based on what students have written on their whiteboards.
Let's apply the relationship of heat, entropy, and temperature to a contemporary challenge!
We'd like to maximize the efficiency of any process that is based on heat flow as an input.
Just a few examples of heat engines.
Energy flow diagram
Energy flow diagram
The efficiency of the machine is
\begin{align}
\text{efficiency} &= \frac{W}{Q_{\text{in}}}
\\
\textit{e.g.} &=\frac{500\text{ J}}{1000\text{ J}} = 50\%
\end{align}
For a car engine, \(T_H\approx 600\text{ K}\) and \(T_C\approx 300\text{ K}\).
Remember that \(\Delta S=\frac{Q}{T}\), and \(\Delta S_{\text{tot}} \ge 0\).
Students solve for the equations of motion of a box sliding down (frictionlessly) a wedge, which itself slides on a horizontal surface, in order to answer the question "how much time does it take for the box to slide a distance \(d\) down the wedge?". This activities highlights finding kinetic energies when the coordinate system is not orthonormal and checking special cases, functional behavior, and dimensions.
In this course, we will examine a mathematically tractable and physically useful problem - that of two bodies interacting with each other through a central force, i.e. a force that has two characteristics:
Definition of a Central Force:
A central force depends only on the separation distance between the two bodies,
A central force points along the line connecting the two bodies.
The most common examples of this type of force are those that have \(\frac{1}{r^2}\) behavior, specifically the Newtonian gravitational force between two point (or spherically symmetric) masses and the Coulomb force between two point (or spherically symmetric) electric charges. Clearly both of these examples are idealizations - neither ideal point masses or charges nor perfectly spherically symmetric mass or charge distributions exist in nature, except perhaps for elementary particles such as electrons. However, deviations from ideal behavior are often small and can be neglected to within a reasonable approximation. (Power series to the rescue!)
Also, notice the difference in length scale: the archetypal gravitational example is planetary motion - at astronomical length scales; the archetypal Coulomb example is the hydrogen atom - at atomic length scales.
The two solutions to the central force problem - classical behavior exemplified by the gravitational interaction and quantum behavior exemplified by the Coulomb interaction - are quite different from each other.
By studying these two cases together in the same course, we will be able to explore the strong similarities and the important differences between classical and quantum physics.
Two of the unifying themes of this topic are the conservation laws:
Conservation of Energy
Conservation of Angular Momentum
The classical and quantum systems we will explore both have versions of these conservation laws, but they come up in the mathematical formalisms in different ways.
You should have covered energy and angular momentum in your introductory physics course, at least in simple, classical mechanics cases. Now is a great time to review the definitions of energy and angular momentum, how they enter into dynamical equations (Newton's laws and kinetic energy, for example), and the conservation laws.
In the classical mechanics case, we will obtain the equations of motion in three equivalent ways,
using Newton's second law,
using Lagrangian mechanics,
using energy conservation.
so that you will be able to compare and contrast the methods.
The Newtonian approach is the most straightforward and naive, but it
suggests changes of coordinates that inform the other methods. The Lagrangian
and energy conservation approaches are slightly more sophisticated in that they exploit more of the symmetries from the beginning.
We will also consider forces that depend on the distance between the two bodies in ways other than \(\frac{1}{r^2}\) and explore the kinds of motion they produce.
In https://paradigms.oregonstate.edu/act/2525 you learned about an experiment in which rubidium atoms are dropped from a trap into an optical two-slit experiment. During this experiment the atoms fall a total of 1.5 meters. What is the de Broglie wavelength of an atom after falling from rest 1.5 m?
\begin{align}
\lambda &= \frac{2\pi\hbar}{p}
\end{align}
Students examine a plastic "surface" graph of the gravitational potential energy of a Earth-satellite system to make connections between gravitational force and gravitational potential energy.
Students examine a plastic “surface” graph of the gravitational potential energy of an Earth-satellite system to explore the properties of gravitational potential energy for a spherically symmetric system.
Explain the consequences of energy and angular momentum conservation in a system of two particles interacting via a central force, in both classical and quantum systems.
Use effective potential diagrams to determine properties of classical orbits.
Solve for the quantum properties of a particle confined to a ring, rigid rotor, and the hydrogen atom in several different representations.
Relate the state of a quantum system (ring, rigid rotor, hydrogen atom) to graphs of a wave function.
Apply Schrödinger time dependence to central force systems (ring, rigid rotor, H atom).
Mathmatics Content Learning Objectives
Solve ordinary differential equations using power series methods.
Use eigen expansions as an orthonormal basis.
Solve the initial value problem for partial differential equations with more than one spatial variable.
Professional Learning Objectives
Communicate scientific ideas in writing and with other representations (e.g. graphs, code), using good scientific language and practices, concisely and without ambiguity.
Learn to work with and communicate productively and respectfully with peers and collaborators of different backgrounds.
Cite the information and ideas obtained from or with others in a clear and professional manner.
Communicate in a timely and professional manner with others in the work environment when things don't go to plan.
LG 2024: I added the Euler-Lagrange equation as an example of a generalized statement of Newton's 2nd Law. I'm planning on using a Lagrangian approach for the 2-body problem.
Students consider projectile motion of an object that experiences drag force that in linear with the velocity. Students consider the horizontal motion and the vertical motion separately. Students solve Newton's 2nd law as a differential equation.
A group of students, tethered together, are floating freely in outer space. Their task is to devise a method to reach a food cache some distance from their group.
