## Student handout: 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.
• computer Mathematica Activity schedule 30 min. build Computers with Mathematica and the cfeffpotential.nb Mathematica notebook, A handout for each student description Student handout (PDF)
What students learn
• The effect of angular momentum, the force constant, and the reduced mass on the shape of the effective potential function;
• Practicing exploring parameter space for a function;
• Developing intuition about how the orbit shape depends on these parameters.

You have four different sliders that control the values of four parameters $k$, $\ell$, $\mu$, and $E$.

1. What is the physical/geometric meaning of each parameter $k$, $\ell$, $\mu$, $E$?
2. How does each parameter $k$, $\ell$, $\mu$, $E$ affect the plot?
3. Which term in the effective potential ($-k/r$ or $\ell^2/(2\mu r^2))$ dominates for small values of r? For large values of r? Explain in terms of both the equation and the graph.
4. What are the classical turning points? Under what conditions will the particle be bound? Unbound?
5. How do your answers for the last question change (if at all) if you consider a repulsive potential? Hint: Figure out what you must change in this notebook and investigate.

• 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 Effective Potentials: Graphical Version

assignment Homework

##### Effective Potentials: Graphical Version
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.

• group Gaussian Parameters

group Small Group Activity

10 min.

##### Gaussian Parameters
Periodic Systems 2022

Fourier Transforms and Wave Packets

Students use an applet to explore the role of the parameters $N$, $x_o$, and $\sigma$ in the shape of a Gaussian \begin{equation} f(x)=Ne^{-\frac{(x-x_0)^2}{2\sigma^2}} \end{equation}
• 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.
• assignment Theta Parameters

assignment Homework

##### Theta Parameters
Static Fields 2023 (6 years)

The function $\theta(x)$ (the Heaviside or unit step function) is a defined as: \begin{equation} \theta(x) =\begin{cases} 1 & \textrm{for}\; x>0 \\ 0 & \textrm{for}\; x<0 \end{cases} \end{equation} This function is discontinuous at $x=0$ and is generally taken to have a value of $\theta(0)=1/2$.

Make sketches of the following functions, by hand, on axes with the same scale and domain. Briefly describe, using good scientific writing that includes both words and equations, the role that the number two plays in the shape of each graph: \begin{align} y &= \theta (x)\\ y &= 2+\theta (x)\\ y &= \theta(2+x)\\ y &= 2\theta (x)\\ y &= \theta (2x) \end{align}

• group Equipotential Surfaces

group Small Group Activity

120 min.

##### Equipotential Surfaces

Students are prompted to consider the scalar superposition of the electric potential due to multiple point charges. First a single point charge is discussed, then four positive charges, then an electric quadrupole. Students draw the equipotential curves in the plane of the charges, while also considering the 3D nature of equipotentials.
• 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: \begin{equation} U(r)= -\frac{k}{r} e^{-\frac{r}{\alpha}} \end{equation}

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 Phase transformations

face Lecture

120 min.

##### Phase transformations
Thermal and Statistical Physics 2020

These lecture notes from the ninth week of Thermal and Statistical Physics cover phase transformations, the Clausius-Clapeyron relation, mean field theory and more. They include a number of small group activities.
• 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.

• 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?

Author Information
Corinne Manogue
Learning Outcomes