Undo Formulas for Reduced Mass (Algebra)

    • 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 \(m_2>m_1\).

    • face Systems of Particles Lecture Notes

      face Lecture

      10 min.

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

    • 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.
    • group Flux through a Cone

      group Small Group Activity

      30 min.

      Flux through a Cone
      Static Fields 2022 (4 years)

      Integration Sequence

      Students calculate the flux from the vector field \(\vec{F} = C\, z\, \hat{z}\) through a right cone of height \(H\) and radius \(R\) .
    • group Vector Surface and Volume Elements

      group Small Group Activity

      30 min.

      Vector Surface and Volume Elements
      Static Fields 2022 (3 years)

      Integration Sequence

      Students use known algebraic expressions for vector line elements \(d\vec{r}\) to determine all simple vector area \(d\vec{A}\) and volume elements \(d\tau\) in cylindrical and spherical coordinates.

      This activity is identical to Scalar Surface and Volume Elements except uses a vector approach to find directed surface and volume elements.

    • assignment Cone Surface

      assignment Homework

      Cone Surface
      Static Fields 2022 (5 years)

      • Find \(dA\) on the surface of an (open) cone in both cylindrical and spherical coordinates. Hint: Be smart about how you coordinatize the cone.
      • Using integration, find the surface area of an (open) cone with height \(H\) and radius \(R\). Do this problem in both cylindrical and spherical coordinates.

    • assignment Flux through a Plane

      assignment Homework

      Flux through a Plane
      Static Fields 2022 (3 years) Find the upward pointing flux of the vector field \(\boldsymbol{\vec{H}}=2z\,\boldsymbol{\hat{x}} +\frac{1}{x^2+1}\boldsymbol{\hat{y}}+(3+2z)\boldsymbol{\hat{z}}\) through the rectangle \(R\) with one edge along the \(y\) axis and the other in the \(xz\)-plane along the line \(z=x\), with \(0\le y\le2\) and \(0\le x\le3\).
    • group Heat capacity of N$_2$

      group Small Group Activity

      30 min.

      Heat capacity of N2
      Contemporary Challenges 2022 (4 years)

      equipartition quantum energy levels

      Students sketch the temperature-dependent heat capacity of molecular nitrogen. They apply the equipartition theorem and compute the temperatures at which degrees of freedom “freeze out.”
    • assignment Center of Mass for Two Uncoupled Particles

      assignment Homework

      Center of Mass for Two Uncoupled Particles
      Central Forces 2023 (3 years)

      Consider two particles of equal mass \(m\). The forces on the particles are \(\vec F_1=0\) and \(\vec F_2=F_0\hat{x}\). If the particles are initially at rest at the origin, find the position, velocity, and acceleration of the center of mass as functions of time. Solve this problem in two ways, with or without theorems about the center of mass motion. Write a short description comparing the two solutions.

  • 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.