Find Force Law: Spiral Orbit

    • assignment Differential Form of Gauss's Law

      assignment Homework

      Differential Form of Gauss's Law
      Static Fields 2022 (4 years)

      For an infinitesimally thin cylindrical shell of radius \(b\) with uniform surface charge density \(\sigma\), the electric field is zero for \(s<b\) and \(\vec{E}= \frac{\sigma b}{\epsilon_0 s}\, \hat s\) for \(s > b\). Use the differential form of Gauss' Law to find the charge density everywhere in space.

    • assignment Find Force Law: Logarithmic Spiral Orbit

      assignment Homework

      Find Force Law: Logarithmic Spiral Orbit
      Central Forces 2022 (2 years)

      In science fiction movies, characters often talk about a spaceship “spiralling in” right before it hits the planet. But all orbits in a \(1/r^2\) force are conic sections, not spirals. This spiralling in happens because the spaceship hits atmosphere and the drag from the atmosphere changes the shape of the orbit. But, in an alternate universe, we might have other force laws.

      Find the force law for a mass \(\mu\), under the influence of a central-force field, that moves in a logarithmic spiral orbit given by \(r = ke^{\alpha \phi}\), where \(k\) and \(\alpha\) are constants.

    • accessibility_new Using Arms to Visualize Transformations of Complex Two-Component Vectors (MathBits)

      accessibility_new Kinesthetic

      30 min.

      Using Arms to Visualize Transformations of Complex Two-Component Vectors (MathBits)
      Quantum Fundamentals 2021

      arms complex numbers phase rotation reflection math

      Arms Sequence for Complex Numbers and Quantum States

      Students, working in pairs, represent two component complex vectors with their left arms. Through a short series of instructor led prompts, students move their left arms to show how various linear transformations affect each complex component.
    • assignment Isolength and Isoforce Stretchability

      assignment Homework

      Isolength and Isoforce Stretchability
      Energy and Entropy 2021 (2 years)

      In class, you measured the isolength stretchability and the isoforce stretchability of your systems in the PDM. We found that for some systems these were very different, while for others they were identical.

      Show with algebra (NOT experiment) that the ratio of isolength stretchability to isoforce stretchability is the same for both the left-hand side of the system and the right-hand side of the system. i.e.: \begin{align} \frac{\left(\frac{\partial {x_L}}{\partial {F_L}}\right)_{x_R}}{\left(\frac{\partial {x_L}}{\partial {F_L}}\right)_{F_R}} &= \frac{\left(\frac{\partial {x_R}}{\partial {F_R}}\right)_{x_L}}{\left(\frac{\partial {x_R}}{\partial {F_R}}\right)_{F_L}} \label{eq:ratios} \end{align}

      Hint
      You will need to make use of the cyclic chain rule: \begin{align} \left(\frac{\partial {A}}{\partial {B}}\right)_{C} = -\left(\frac{\partial {A}}{\partial {C}}\right)_{B}\left(\frac{\partial {C}}{\partial {B}}\right)_{A} \end{align}
      Hint
      You will also need the ordinary chain rule: \begin{align} \left(\frac{\partial {A}}{\partial {B}}\right)_{D} = \left(\frac{\partial {A}}{\partial {C}}\right)_{D}\left(\frac{\partial {C}}{\partial {B}}\right)_{D} \end{align}

    • keyboard Kinetic energy

      keyboard Computational Activity

      120 min.

      Kinetic energy
      Computational Physics Lab II 2022 (2 years)

      finite difference hamiltonian quantum mechanics particle in a box eigenfunctions

      Students implement a finite-difference approximation for the kinetic energy operator as a matrix, and then use numpy to solve for eigenvalues and eigenstates, which they visualize.
    • assignment Energy, Entropy, and Probabilities

      assignment Homework

      Energy, Entropy, and Probabilities
      Energy Entropy Probabilities Thermodynamic identity

      The goal of this problem is to show that once we have maximized the entropy and found the microstate probabilities in terms of a Lagrange multiplier \(\beta\), we can prove that \(\beta=\frac1{kT}\) based on the statistical definitions of energy and entropy and the thermodynamic definition of temperature embodied in the thermodynamic identity.

      The internal energy and entropy are each defined as a weighted average over microstates: \begin{align} U &= \sum_i E_i P_i & S &= -k_B\sum_i P_i \ln P_i \end{align}: We saw in clase that the probability of each microstate can be given in terms of a Lagrange multiplier \(\beta\) as \begin{align} P_i &= \frac{e^{-\beta E_i}}{Z} & Z &= \sum_i e^{-\beta E_i} \end{align} Put these probabilities into the above weighted averages in order to relate \(U\) and \(S\) to \(\beta\). Then make use of the thermodynamic identity \begin{align} dU = TdS - pdV \end{align} to show that \(\beta = \frac1{kT}\).

    • assignment Energy, Entropy, and Probabilities

      assignment Homework

      Energy, Entropy, and Probabilities
      Thermal and Statistical Physics 2020

      The goal of this problem is to show that once we have maximized the entropy and found the microstate probabilities in terms of a Lagrange multiplier \(\beta\), we can prove that \(\beta=\frac1{kT}\) based on the statistical definitions of energy and entropy and the thermodynamic definition of temperature embodied in the thermodynamic identity.

      The internal energy and entropy are each defined as a weighted average over microstates: \begin{align} U &= \sum_i E_i P_i & S &= -k_B\sum_i P_i \ln P_i \end{align} We saw in clase that the probability of each microstate can be given in terms of a Lagrange multiplier \(\beta\) as \begin{align} P_i &= \frac{e^{-\beta E_i}}{Z} & Z &= \sum_i e^{-\beta E_i} \end{align} Put these probabilities into the above weighted averages in order to relate \(U\) and \(S\) to \(\beta\). Then make use of the thermodynamic identity \begin{align} dU = TdS - pdV \end{align} to show that \(\beta = \frac1{kT}\).

    • face Systems of Particles Lecture Notes

      face Lecture

      10 min.

      Systems of Particles Lecture Notes
      Central Forces 2022 (2 years)
    • assignment Adiabatic Compression

      assignment Homework

      Adiabatic Compression
      ideal gas internal energy engine Energy and Entropy 2020

      A diesel engine requires no spark plug. Rather, the air in the cylinder is compressed so highly that the fuel ignites spontaneously when sprayed into the cylinder.

      In this problem, you may treat air as an ideal gas, which satisfies the equation \(pV = Nk_BT\). You may also use the property of an ideal gas that the internal energy depends only on the temperature \(T\), i.e. the internal energy does not change for an isothermal process. For air at the relevant range of temperatures the heat capacity at fixed volume is given by \(C_V=\frac52Nk_B\), which means the internal energy is given by \(U=\frac52Nk_BT\).

      Note: in this problem you are expected to use only the equations given and fundamental physics laws. Looking up the formula in a textbook is not considered a solution at this level.

      1. If the air is initially at room temperature (taken as \(20^{o}C\)) and is then compressed adiabatically to \(\frac1{15}\) of the original volume, what final temperature is attained (before fuel injection)?

      2. By what factor does the pressure increase?

    • assignment Undo Formulas for Reduced Mass (Algebra)

      assignment Homework

      Undo Formulas for Reduced Mass (Algebra)
      Central Forces 2022 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.
  • Central Forces 2022 (2 years)

    In science fiction movies, characters often talk about a spaceship “spiralling in” right before it hits the planet. But all orbits in a \(1/r^2\) force are conic sections, not spirals. This spiralling in happens because the spaceship hits atmosphere and the drag from the atmosphere changes the shape of the orbit. But, in an alternate universe, we might have other force laws.

    Find the force law for a central-force field that allows a particle to move in a spiral orbit given by \(r=k\phi^2\), where \(k\) is a constant.