group Small Group Activity

30 min.

Mass is not Conserved
Theoretical Mechanics 2021 (2 years)

energy conservation mass conservation collision

Groups are asked to analyze the following standard problem:

Two identical lumps of clay of (rest) mass m collide head on, with each moving at 3/5 the speed of light. What is the mass of the resulting lump of clay?

group Small Group Activity

10 min.

Survivor Outer Space: A kinesthetic approach to (re)viewing center-of-mass
Central Forces 2022 (2 years) A group of students, tethered together, are floating freely in outer space. Their task is to devise a method to reach a food cache some distance from their group.

face Lecture

5 min.

Central Forces Introduction: Lecture Notes
Central Forces 2022

face Lecture

10 min.

Systems of Particles Lecture Notes
Central Forces 2022 (2 years)

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 Homework

Scattering
Central Forces 2022 (2 years)

Consider a very light particle of mass \(\mu\) scattering from a very heavy, stationary particle of mass \(M\). The force between the two particles is a repulsive Coulomb force \(\frac{k}{r^2}\). The impact parameter \(b\) in a scattering problem is defined to be the distance which would be the closest approach if there were no interaction (See Figure). The initial velocity (far from the scattering event) of the mass \(\mu\) is \(\vec v_0\). Answer the following questions about this situation in terms of \(k\), \(M\), \(\mu\), \(\vec v_0\), and \(b\). ()It is not necessarily wise to answer these questions in order.)

  1. What is the initial angular momentum of the system?
  2. What is the initial total energy of the system?
  3. What is the distance of closest approach \(r_{\rm{min}}\) with the interaction?
  4. Sketch the effective potential.
  5. What is the angular momentum at \(r_{\rm{min}}\)?
  6. What is the total energy of the system at \(r_{\rm{min}}\)?
  7. What is the radial component of the velocity at \(r_{\rm{min}}\)?
  8. What is the tangential component of the velocity at \(r_{\rm{min}}\)?
  9. What is the value of the effective potential at \(r_{\rm{min}}\)?
  10. For what values of the initial total energy are there bound orbits?
  11. Using your results above, write a short essay describing this type of scattering problem, at a level appropriate to share with another Paradigm student.

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.

group Small Group Activity

120 min.

Box Sliding Down Frictionless Wedge
Theoretical Mechanics 2021 (2 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.

assignment Homework

Entropy of mixing
Entropy Equilibrium Sackur-Tetrode Thermal and Statistical Physics 2020

Suppose that a system of \(N\) atoms of type \(A\) is placed in diffusive contact with a system of \(N\) atoms of type \(B\) at the same temperature and volume.

  1. Show that after diffusive equilibrium is reached the total entropy is increased by \(2Nk\ln 2\). The entropy increase \(2Nk\ln 2\) is known as the entropy of mixing.

  2. If the atoms are identical (\(A=B\)), show that there is no increase in entropy when diffusive contact is established. The difference has been called the Gibbs paradox.

  3. Since the Helmholtz free energy is lower for the mixed \(AB\) than for the separated \(A\) and \(B\), it should be possible to extract work from the mixing process. Construct a process that could extract work as the two gasses are mixed at fixed temperature. You will probably need to use walls that are permeable to one gas but not the other.

Note

This course has not yet covered work, but it was covered in Energy and Entropy, so you may need to stretch your memory to finish part (c).

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.

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.