Theta Parameters

    • assignment Undo Formulas for Reduced Mass (Geometry)

      assignment Homework

      Undo Formulas for Reduced Mass (Geometry)
      Central Forces 2023 (3 years)

      The figure below shows the position vector \(\vec r\) and the orbit of a “fictitious” reduced mass \(\mu\).

      1. Suppose \(m_1=m_2\), Sketch the position vectors and orbits for \(m_1\) and \(m_2\) corresponding to \(\vec{r}\). Describe a common physics example of central force motion for which \(m_1=m_2\).
      2. Repeat, for \(m_2>m_1\).

    • group Sequential Stern-Gerlach Experiments

      group Small Group Activity

      10 min.

      Sequential Stern-Gerlach Experiments
      Quantum Fundamentals 2022 (3 years)
    • 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?

    • accessibility_new Time Dilation Light Clock Skit

      accessibility_new Kinesthetic

      5 min.

      Time Dilation Light Clock Skit

      Special Relativity Time Dilation Light Clock Kinesthetic Activity

      Students act out the classic light clock scenario for deriving time dilation.
    • face Statistical Analysis of Stern-Gerlach Experiments
    • assignment Series Convergence

      assignment Homework

      Series Convergence

      Power Series Sequence (E&M)

      Static Fields 2022 (5 years)

      Recall that, if you take an infinite number of terms, the series for \(\sin z\) and the function itself \(f(z)=\sin z\) are equivalent representations of the same thing for all real numbers \(z\), (in fact, for all complex numbers \(z\)). This is not always true. More commonly, a series is only a valid, equivalent representation of a function for some more restricted values of \(z\). The technical name for this idea is convergence--the series only "converges" to the value of the function on some restricted domain, called the “interval” or “region of convergence.”

      Find the power series for the function \(f(z)=\frac{1}{1+z^2}\). Then, using the Mathematica worksheet from class (vfpowerapprox.nb) as a model, or some other computer algebra system like Sage or Maple, explore the convergence of this series. Where does your series for this new function converge? Can you tell anything about the region of convergence from the graphs of the various approximations? Print out a plot and write a brief description (a sentence or two) of the region of convergence. You may need to include a lot of terms to see the effect of the region of convergence. Keep adding terms until you see a really strong effect!

      Note: As a matter of professional ettiquette (or in some cases, as a legal copyright requirement), if you use or modify a computer program written by someone else, you should always acknowledge that fact briefly in whatever you write up. Say something like: “This calculation was based on a (name of software package) program titled (title) originally written by (author) copyright (copyright date).”

    • assignment Hockey

      assignment Homework

      Hockey
      Central Forces 2023 (3 years)

      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?

    • group Energy radiated from one oscillator

      group Small Group Activity

      30 min.

      Energy radiated from one oscillator
      Contemporary Challenges 2022 (4 years)

      blackbody radiation

      This lecture is one step in motivating the form of the Planck distribution.
    • group Electric Field Due to a Ring of Charge

      group Small Group Activity

      30 min.

      Electric Field Due to a Ring of Charge
      Static Fields 2022 (7 years)

      coulomb's law electric field charge ring symmetry integral power series superposition

      Power Series Sequence (E&M)

      Ring Cycle Sequence

      Students work in groups of three to use Coulomb's Law \[\vec{E}(\vec{r}) =\frac{1}{4\pi\epsilon_0}\int\frac{\rho(\vec{r}^{\,\prime})\left(\vec{r}-\vec{r}^{\,\prime}\right)}{\vert \vec{r}-\vec{r}^{\,\prime}\vert^3} \, d\tau^{\prime}\] to find an integral expression for the electric field, \(\vec{E}(\vec{r})\), everywhere in space, due to a ring of charge.

      In an optional extension, students find a series expansion for \(\vec{E}(\vec{r})\) either on the axis or in the plane of the ring, for either small or large values of the relevant geometric variable. Add an extra half hour or more to the time estimate for the optional extension.

    • group Wavefunctions on a Quantum Ring

      group Small Group Activity

      30 min.

      Wavefunctions on a Quantum Ring
      Central Forces 2023 (2 years)
  • Static Fields 2022 (5 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}

  • Media & Figures
    • figures/vfthetaparameterssolab.png
    • figures/vfthetaparameterssolc.png
    • figures/vfthetaparameterssolde.png