Potential energy of gas in gravitational field

  • Potential energy Heat capacity
    • assignment Hockey

      assignment Homework

      Hockey
      Central Forces 2022 (2 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?

    • assignment The puddle

      assignment Homework

      The puddle
      differentials Static Fields 2022 (4 years) The depth of a puddle in millimeters is given by \[h=\frac{1}{10} \bigl(1+\sin(\pi xy)\bigr)\] Your path through the puddle is given by \[x=3t \qquad y=4t\] and your current position is \(x=3\), \(y=4\), with \(x\) and \(y\) also in millimeters, and \(t\) in seconds.
      1. At your current position, how fast is the depth of water through which you are walking changing per unit time?
      2. At your current position, how fast is the depth of water through which you are walking changing per unit distance?
      3. FOOD FOR THOUGHT (optional)
        There is a walkway over the puddle at \(x=10\). At your current position, how fast is the depth of water through which you are walking changing per unit distance towards the walkway.
    • face Energy and Entropy review

      face Lecture

      5 min.

      Energy and Entropy review
      Thermal and Statistical Physics 2020 (3 years)

      thermodynamics statistical mechanics

      This very quick lecture reviews the content taught in Energy and Entropy, and is the first content in Thermal and Statistical Physics.
    • assignment Centrifuge

      assignment Homework

      Centrifuge
      Centrifugal potential Thermal and Statistical Physics 2020 A circular cylinder of radius \(R\) rotates about the long axis with angular velocity \(\omega\). The cylinder contains an ideal gas of atoms of mass \(M\) at temperature \(T\). Find an expression for the dependence of the concentration \(n(r)\) on the radial distance \(r\) from the axis, in terms of \(n(0)\) on the axis. Take \(\mu\) as for an ideal gas.
    • assignment Differentials of Two Variables

      assignment Homework

      Differentials of Two Variables
      Static Fields 2022 (5 years) Find the total differential of the following functions:
      1. \(y=3u^2 + 4\cos 3v\)
      2. \(y=3uv\)
      3. \(y=3u^2\cos wv\)
      4. \(y=u\cos(3v^2-2)\)
    • assignment Potential vs. Potential Energy

      assignment Homework

      Potential vs. Potential Energy
      Static Fields 2022 (4 years)

      In this course, two of the primary examples we will be using are the potential due to gravity and the potential due to an electric charge. Both of these forces vary like \(\frac{1}{r}\), so they will have many, many similarities. Most of the calculations we do for the one case will be true for the other. But there are some extremely important differences:

      1. Find the value of the electrostatic potential energy of a system consisting of a hydrogen nucleus and an electron separated by the Bohr radius. Find the value of the gravitational potential energy of the same two particles at the same radius. Use the same system of units in both cases. Compare and the contrast the two answers.
      2. Find the value of the electrostatic potential due to the nucleus of a hydrogen atom at the Bohr radius. Find the gravitational potential due to the nucleus at the same radius. Use the same system of units in both cases. Compare and contrast the two answers.
      3. Briefly discuss at least one other fundamental difference between electromagnetic and gravitational systems. Hint: Why are we bound to the earth gravitationally, but not electromagnetically?

    • face Boltzmann probabilities and Helmholtz

      face Lecture

      120 min.

      Boltzmann probabilities and Helmholtz
      Thermal and Statistical Physics 2020

      ideal gas entropy canonical ensemble Boltzmann probability Helmholtz free energy statistical mechanics

      These notes, from the third week of Thermal and Statistical Physics cover the canonical ensemble and Helmholtz free energy. They include a number of small group activities.
    • assignment Zapping With d 1

      assignment Homework

      Zapping With d 1
      Energy and Entropy 2021 (2 years)

      Find the differential of each of the following expressions; zap each of the following with \(d\):

      1. \[f=3x-5z^2+2xy\]

      2. \[g=\frac{c^{1/2}b}{a^2}\]

      3. \[h=\sin^2(\omega t)\]

      4. \[j=a^x\]

      5. \[k=5 \tan\left(\ln{\left(\frac{V_1}{V_2}\right)}\right)\]

    • face Review of Thermal Physics

      face Lecture

      30 min.

      Review of Thermal Physics
      Thermal and Statistical Physics 2020

      thermodynamics statistical mechanics

      These are notes, essentially the equation sheet, from the final review session for Thermal and Statistical Physics.
    • group de Broglie wavelength after freefall

      group Small Group Activity

      30 min.

      de Broglie wavelength after freefall
      Contemporary Challenges 2022 (3 years)

      de Broglie wavelength gravity

      In this activity students combine energy conservation with the relationship between the de Broglie wavelength and momentum to find the wavelength of atoms that have been dropped a given distance.
  • Thermal and Statistical Physics 2020 Consider a column of atoms each of mass \(M\) at temperature \(T\) in a uniform gravitational field \(g\). Find the thermal average potential energy per atom. The thermal average kinetic energy is independent of height. Find the total heat capacity per atom. The total heat capacity is the sum of contributions from the kinetic energy and from the potential energy. Take the zero of the gravitational energy at the bottom \(h=0\) of the column. Integrate from \(h=0\) to \(h=\infty\). You may assume the gas is ideal.