Activity: Effective Potentials

Central Forces 2021
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.
• Media
• assignment Theta Parameters

assignment Homework

Theta Parameters
Static Fields 2022 (4 years)

The function $\theta(x)$ (the Heaviside or unit step function) is a defined as: $$\theta(x) =\begin{cases} 1 & \textrm{for}\; x>0 \\ 0 & \textrm{for}\; x<0 \end{cases}$$ 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 2021

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 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 Reduced Mass

assignment Homework

Reduced Mass
Central Forces 2021

Using your favorite graphing package, make a plot of the reduced mass $\mu$ as a function of $m_1$ and $m_2$. What about the shape of this graph tells you something about the physical world that you would like to remember. You should be able to find at least three things.

• assignment Hockey

assignment Homework

Hockey
Central Forces 2021

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 Approximating Functions with Power Series

computer Computer Simulation

30 min.

Approximating Functions with Power Series
Static Fields 2022 (7 years)

Power Series Sequence (E&M)

Students use prepared Sage code or a prepared Mathematica notebook to plot $\sin\theta$ simultaneously with several terms of a power series expansion to judge how well the approximation fits. Students can alter the worksheet to change the number of terms in the expansion and even to change the function that is being considered. Students should have already calculated the coefficients for the power series expansion in a previous activity, Calculating Coefficients for a Power Series.
• group Visualization of Divergence

group Small Group Activity

30 min.

Visualization of Divergence
Vector Calculus II 2022 (7 years) Students predict from graphs of simple 2-d vector fields whether the divergence is positive, negative, or zero in various regions of the domain using the definition of the divergence of a vector field at a point: The divergence of a vector field at a point is flux per unit volume through an infinitesimal box surrounding that point. Optionally, students can use a Mathematica notebook to verify their predictions.
• group Electrostatic Potential Due to a Pair of Charges (with Series)

group Small Group Activity

60 min.

Electrostatic Potential Due to a Pair of Charges (with Series)
Static Fields 2022 (4 years)

Power Series Sequence (E&M)

Ring Cycle Sequence

Students work in small groups to use the superposition principle $V(\vec{r}) = \frac{1}{4\pi\epsilon_0}\sum_i \frac{q_i}{\vert\vec{r}-\vec{r}_i\vert}$ to find the electrostatic potential $V$ everywhere in space due to a pair of charges (either identical charges or a dipole). Different groups are assigned different arrangements of charges and different regions of space to consider: either on the axis of the charges or in the plane equidistant from the two charges, for either small or large values of the relevant geometric variable. Each group is asked to find a power series expansion for the electrostatic potential, valid in their group's assigned region of space. The whole class wrap-up discussion then compares and contrasts the results and discuss the symmetries of the two cases.
• format_list_numbered Quantum Ring Sequence

format_list_numbered Sequence

Quantum Ring Sequence
Students calculate probabilities and expectation values for a quantum mechanical particle confined to a circular ring in bra/ket, matrix, and wave function representations and compare the different calculation methods. Several different graphical representations of the time dependence for both states with special symmetry and arbitrary states are explored in a Mathematica notebook. Compared to the analogous particle-in-a-box, this quantum system has a new feature---degenerate energy eigenstates.

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.

Instructor's Guide

Prerequisite Knowledge

• Students should know that the effective potential is used to reduce the 2-D central force problem to a 1-D problem.
• Students should know that the shape of the orbit need not be elliptical - all conic sections are solutions to the equations of motion (depending on the values of the various parameters).

Introduction

We usually start this activity with a general discussion about sets of solutions to an equation and the professional sense-making activity of exploring parameter space to build an understanding of the types of solutions that exist in the set.

Student Conversations

• In exploring the parameter space of solutions, students should be encouraged to identify limiting and special cases (e.g. when is the orbit is circular, what is the shape of the function for large and small separations, etc).
• Some students point out (correctly) that the force constant $k$ ought to depend on the reduced mass $\mu$. This is a result of the gravitational force being the typical example of a central force; reminding students that other central forces exist (e.g, the Coulomb force) clears up any concerns.

Author Information
Corinne Manogue
Learning Outcomes