## Effective Potentials: Graphical Version

• assignment Yukawa

assignment Homework

##### Yukawa
Central Forces 2023 (3 years)

In a solid, a free electron doesn't see” a bare nuclear charge since the nucleus is surrounded by a cloud of other electrons. The nucleus will look like the Coulomb potential close-up, but be screened” from far away. A common model for such problems is described by the Yukawa or screened potential: $$U(r)= -\frac{k}{r} e^{-\frac{r}{\alpha}}$$

1. Graph the potential, with and without the exponential term. Describe how the Yukawa potential approximates the “real” situation. In particular, describe the role of the parameter $\alpha$.
2. Draw the effective potential for the two choices $\alpha=10$ and $\alpha=0.1$ with $k=1$ and $\ell=1$. For which value(s) of $\alpha$ is there the possibility of stable circular orbits?

• face Unit Learning Outcomes: Classical Mechanics Orbits

face Lecture

5 min.

##### Unit Learning Outcomes: Classical Mechanics Orbits
Central Forces 2023 This handout lists Motivating Questions, Key Activities/Problems, Unit Learning Outcomes, and an Equation Sheet for a Unit on Classical Mechanics Orbits. It can be used both to introduce the unit and, even better, for review.
• computer Effective Potentials

computer Mathematica Activity

30 min.

##### Effective Potentials
Central Forces 2023 (3 years) Students use a pre-written Mathematica notebook or a Geogebra applet to explore how the shape of the effective potential function changes as the various parameters (angular momentum, force constant, reduced mass) are varied.
• group Gravitational Potential Energy

group Small Group Activity

60 min.

##### 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.
• assignment Effective Potential Diagrams

assignment Homework

##### Effective Potential Diagrams
Central Forces 2023

See also the following more detailed problem and solution: Effective Potentials: Graphical Version

An electron is moving on a two dimension surface with a radially symmetric electrostatic potential given by the graph below:

1. Sketch the effective potential if the angular momentum is not zero.
2. Describe qualitatively, the shapes of all possible types of orbits, indicating the energy for each in your diagram.

• face Ideal Gas

face Lecture

120 min.

##### Ideal Gas
Thermal and Statistical Physics 2020

These notes from week 6 of Thermal and Statistical Physics cover the ideal gas from a grand canonical standpoint starting with the solutions to a particle in a three-dimensional box. They include a number of small group activities.
• assignment Hockey

assignment Homework

##### Hockey
Central Forces 2023 (3 years)

(Synthesis Problem: Brings together several different concepts from this unit.) Use effective potential diagrams for other than $1/r^2$ forces.

Consider the frictionless motion of a hockey puck of mass $m$ on a perfectly circular bowl-shaped ice rink with radius $a$. The central region of the bowl ($r < 0.8a$) is perfectly flat and the sides of the ice bowl smoothly rise to a height $h$ at $r = a$.

1. Draw a sketch of the potential energy for this system. Set the zero of potential energy at the top of the sides of the bowl.
2. Situation 1: the puck is initially moving radially outward from the exact center of the rink. What minimum velocity does the puck need to escape the rink?
3. Situation 2: a stationary puck, at a distance $\frac{a}{2}$ from the center of the rink, is hit in such a way that it's initial velocity $\vec v_0$ is perpendicular to its position vector as measured from the center of the rink. What is the total energy of the puck immediately after it is struck?
4. In situation 2, what is the angular momentum of the puck immediately after it is struck?
5. Draw a sketch of the effective potential for situation 2.
6. In situation 2, for what minimum value of $\vec v_0$ does the puck just escape the rink?

• computer Visualization of Quantum Probabilities for the Hydrogen Atom

computer Mathematica Activity

30 min.

##### Visualization of Quantum Probabilities for the Hydrogen Atom
Central Forces 2023 (3 years) Students use Mathematica to visualize the probability density distribution for the hydrogen atom orbitals with the option to vary the values of $n$, $\ell$, and $m$.
• group Conic Sections

group Small Group Activity

30 min.

##### Conic Sections
Central Forces 2023 (3 years) Students are asked to explore the parameters that affect orbit shape using the supplied Maple worksheet or Mathematica notebook.
• assignment Find Force Law: Logarithmic Spiral Orbit

assignment Homework

##### Find Force Law: Logarithmic Spiral Orbit
Central Forces 2023 (3 years)

(Use the equation for orbit shape.) Gain experience with unusual force laws.

In science fiction movies, characters often talk about a spaceship “spiralling in” right before it hits the planet. But all orbits in a $1/r^2$ force are conic sections, not spirals. This spiralling in happens because the spaceship hits atmosphere and the drag from the atmosphere changes the shape of the orbit. But, in an alternate universe, we might have other force laws.

Find the force law for a mass $\mu$, under the influence of a central-force field, that moves in a logarithmic spiral orbit given by $r = ke^{\alpha \phi}$, where $k$ and $\alpha$ are constants.

• Central Forces 2023 (3 years)

Consider a mass $\mu$ in the potential shown in the graph below. You give the mass a push so that its initial angular momentum is $\ell\ne 0$ for a given fixed value of $\ell$.

1. Give the definition of a central force system and briefly explain why this situation qualifies.
2. Make a sketch of the graph of the effective potential for this situation.
3. How should you push the puck to establish a circular orbit? (i.e. Characterize the initial position, direction of push, and strength of the push. You do NOT need to solve any equations.)
4. BRIEFLY discuss the possible orbit shapes that can arise from this effective potential. Include a discussion of whether the orbits are open or closed, bound or unbound, etc. Make sure that you refer to your sketch of the effective potential in your discussions, mark any points of physical significance on the sketch, and describe the range of parameters relevant to each type of orbit. Include a discussion of the role of the total energy of the orbit.