Potential energy of gas in gravitational field

Potential energy Heat capacity

Hockey
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?
Thermal radiation and Planck distribution

Lecture

120 min.

Thermal radiation and Planck distribution
Thermal and Statistical Physics 2020
Planck distribution blackbody radiation photon statistical mechanics
These notes from the fourth week of Thermal and Statistical Physics cover blackbody radiation and the Planck distribution. They include a number of small group activities.
Einstein condensation temperature

Homework

Einstein condensation temperature
Einstein condensation Density Thermal and Statistical Physics 2020
Einstein condensation temperature Starting from the density of free particle orbitals per unit energy range \begin{align} \mathcal{D}(\varepsilon) = \frac{V}{4\pi^2}\left(\frac{2M}{\hbar^2}\right)^{\frac32}\varepsilon^{\frac12} \end{align} show that the lowest temperature at which the total number of atoms in excited states is equal to the total number of atoms is \begin{align} T_E &= \frac1{k_B} \frac{\hbar^2}{2M} \left( \frac{N}{V} \frac{4\pi^2}{\int_0^\infty\frac{\sqrt{\xi}}{e^\xi-1}d\xi} \right)^{\frac23} T_E &= \end{align} The infinite sum may be numerically evaluated to be 2.612. Note that the number derived by integrating over the density of states, since the density of states includes all the states except the ground state.
Note: This problem is solved in the text itself. I intend to discuss Bose-Einstein condensation in class, but will not derive this result.
The puddle
Homework

The puddle
differentials Static Fields 2023 (5 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.
Boltzmann probabilities and Helmholtz

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.
Phase transformations

Lecture

120 min.

Phase transformations
Thermal and Statistical Physics 2020
phase transformation Clausius-Clapeyron mean field theory thermodynamics
These lecture notes from the ninth week of Thermal and Statistical Physics cover phase transformations, the Clausius-Clapeyron relation, mean field theory and more. They include a number of small group activities.
Fermi and Bose gases

Lecture

120 min.

Fermi and Bose gases
Thermal and Statistical Physics 2020
Fermi level fermion boson Bose gas Bose-Einstein condensate ideal gas statistical mechanics phase transition
These lecture notes from week 7 of Thermal and Statistical Physics apply the grand canonical ensemble to fermion and bosons ideal gasses. They include a few small group activities.
Differentials of Two Variables
Homework

Differentials of Two Variables
Static Fields 2023 (8 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. \(y=u\cos(3v^2-2)\)
Zapping With d 1
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. \[k=5 \tan\left(\ln{\left(\frac{V_1}{V_2}\right)}\right)\]
Quantum Reference Sheet

Lecture

5 min.

Quantum Reference Sheet
Central Forces 2023 (6 years)

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.