Undo Formulas for Reduced Mass (Geometry)

    • assignment Undo Formulas for Reduced Mass (Algebra)

      assignment Homework

      Undo Formulas for Reduced Mass (Algebra)
      Central Forces 2023 (2 years) For systems of particles, we used the formulas \begin{align} \vec{R}_{cm}&=\frac{1}{M}\left(m_1\vec{r}_1+m_2\vec{r}_2\right) \nonumber\\ \vec{r}&=\vec{r}_2-\vec{r}_1 \label{cm} \end{align} to switch from a rectangular coordinate system that is unrelated to the system to coordinates adapted to the center-of-mass. After you have solved the equations of motion in the center-of-mass coordinates, you may want to transform back to the original coordinate system. Find the inverse transformation, i.e. solve for: \begin{align} \vec{r}_1&=\\ \vec{r}_2&= \end{align} Hint: The system of equations (\ref{cm}) is linear, i.e. each variable is to the first power, even though the variables are vectors. In this case, you can use all of the methods you learned for solving systems of equations while keeping the variables vector valued, i.e. you can safely ignore the fact that the \(\vec{r}\)s are vectors while you are doing the algebra.
    • assignment Reduced Mass

      assignment Homework

      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.

    • face Systems of Particles Lecture Notes

      face Lecture

      10 min.

      Systems of Particles Lecture Notes
      Central Forces 2023 (3 years)
    • assignment Frequency

      assignment Homework

      Quantum Mechanics Time Evolution Spin Precession Expectation Value Bohr Frequency Quantum Fundamentals 2022 (2 years) Consider a two-state quantum system with a Hamiltonian \begin{equation} \hat{H}\doteq \begin{pmatrix} E_1&0\\ 0&E_2 \end{pmatrix} \end{equation} Another physical observable \(M\) is described by the operator \begin{equation} \hat{M}\doteq \begin{pmatrix} 0&c\\ c&0 \end{pmatrix} \end{equation} where \(c\) is real and positive. Let the initial state of the system be \(\left|{\psi(0)}\right\rangle =\left|{m_1}\right\rangle \), where \(\left|{m_1}\right\rangle \) is the eigenstate corresponding to the larger of the two possible eigenvalues of \(\hat{M}\). What is the frequency of oscillation of the expectation value of \(M\)? This frequency is the Bohr frequency.
    • group Gravitational Potential Energy

      group Small Group Activity

      60 min.

      Gravitational Potential Energy

      Mechanics Gravitational Potential Energy Zero of Potential Introductory Physics

      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 Carbon monoxide poisoning

      assignment Homework

      Carbon monoxide poisoning
      Equilibrium Absorbtion Thermal and Statistical Physics 2020

      In carbon monoxide poisoning the CO replaces the \(\textsf{O}_{2}\) adsorbed on hemoglobin (\(\text{Hb}\)) molecules in the blood. To show the effect, consider a model for which each adsorption site on a heme may be vacant or may be occupied either with energy \(\varepsilon_A\) by one molecule \(\textsf{O}_{2}\) or with energy \(\varepsilon_B\) by one molecule CO. Let \(N\) fixed heme sites be in equilibrium with \(\textsf{O}_{2}\) and CO in the gas phases at concentrations such that the activities are \(\lambda(\text{O}_2) = 1\times 10^{-5}\) and \(\lambda(\text{CO}) = 1\times 10^{-7}\), all at body temperature \(37^\circ\text{C}\). Neglect any spin multiplicity factors.

      1. First consider the system in the absence of CO. Evaluate \(\varepsilon_A\) such that 90 percent of the \(\text{Hb}\) sites are occupied by \(\textsf{O}_{2}\). Express the answer in eV per \(\textsf{O}_{2}\).

      2. Now admit the CO under the specified conditions. Fine \(\varepsilon_B\) such that only 10% of the Hb sites are occupied by \(\textsf{O}_{2}\).

    • group Box Sliding Down Frictionless Wedge

      group Small Group Activity

      120 min.

      Box Sliding Down Frictionless Wedge
      Theoretical Mechanics (4 years)

      Lagrangian Mechanics Generalized Coordinates Special Cases

      Students solve for the equations of motion of a box sliding down (frictionlessly) a wedge, which itself slides on a horizontal surface, in order to answer the question "how much time does it take for the box to slide a distance \(d\) down the wedge?". This activities highlights finding kinetic energies when the coordinate system is not orthonormal and checking special cases, functional behavior, and dimensions.
    • face Chemical potential and Gibbs distribution

      face Lecture

      120 min.

      Chemical potential and Gibbs distribution
      Thermal and Statistical Physics 2020

      chemical potential Gibbs distribution grand canonical ensemble statistical mechanics

      These notes from the fifth week of Thermal and Statistical Physics cover the grand canonical ensemble. They include several small group activities.
    • accessibility_new The Distance Formula (Star Trek)

      accessibility_new Kinesthetic

      30 min.

      The Distance Formula (Star Trek)
      Static Fields 2022 (5 years)

      distance formula coordinate systems dot product vector addition

      Ring Cycle Sequence

      A short improvisational role-playing skit based on the Star Trek series in which students explore the definition and notation for position vectors, the importance of choosing an origin, and the geometric nature of the distance formula. \[\vert\vec{r}-\vec{r}^\prime\vert=\sqrt{(x-x^\prime)^2+(y-y^\prime)^2-(z-z^\prime)^2}\]
    • group Optical depth of atmosphere

      group Small Group Activity

      30 min.

      Optical depth of atmosphere
      Contemporary Challenges 2022 (4 years) In this activity students estimate the optical depth of the atmosphere at the infrared wavelength where carbon dioxide has peak absorption.
  • 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 \(m_2>m_1\).

  • Media & Figures
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