Activity: 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.
  • Media
    • activity_media/cfeffpotential_5svwhe0.nb

Download and run this Mathematica notebook or this Geogebra applet.

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

Answer the following questions:

  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.
  • assignment Effective Potentials: Graphical Version

    assignment Homework

    Effective Potentials: Graphical Version
    Central Forces 2023 (2 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.

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    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}

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    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?

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    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?

  • assignment Scattering

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    Scattering
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    Consider a very light particle of mass \(\mu\) scattering from a very heavy, stationary particle of mass \(M\). The force between the two particles is a repulsive Coulomb force \(\frac{k}{r^2}\). The impact parameter \(b\) in a scattering problem is defined to be the distance which would be the closest approach if there were no interaction (See Figure). The initial velocity (far from the scattering event) of the mass \(\mu\) is \(\vec v_0\). Answer the following questions about this situation in terms of \(k\), \(M\), \(\mu\), \(\vec v_0\), and \(b\). ()It is not necessarily wise to answer these questions in order.)

    1. What is the initial angular momentum of the system?
    2. What is the initial total energy of the system?
    3. What is the distance of closest approach \(r_{\rm{min}}\) with the interaction?
    4. Sketch the effective potential.
    5. What is the angular momentum at \(r_{\rm{min}}\)?
    6. What is the total energy of the system at \(r_{\rm{min}}\)?
    7. What is the radial component of the velocity at \(r_{\rm{min}}\)?
    8. What is the tangential component of the velocity at \(r_{\rm{min}}\)?
    9. What is the value of the effective potential at \(r_{\rm{min}}\)?
    10. For what values of the initial total energy are there bound orbits?
    11. Using your results above, write a short essay describing this type of scattering problem, at a level appropriate to share with another Paradigm student.

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  • assignment Reduced Mass

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    Reduced Mass
    Central Forces 2023 (3 years)

    Using your favorite graphing package, make a plot of the reduced mass \begin{equation} \mu=\frac{m_1\, m_2}{m_1+m_2} \end{equation} 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. Hint: Think limiting cases.


Author Information
Corinne Manogue
Learning Outcomes