## Activity: Central Forces Introduction: Lecture Notes

Central Forces 2022

In this course, we will examine a mathematically tractable and physically useful problem - that of two bodies interacting with each other through a central force, i.e. a force that has two characteristics:

Definition of a Central Force:

1. A central force depends only on the separation distance between the two bodies,
2. A central force points along the line connecting the two bodies.

The most common examples of this type of force are those that have $\frac{1}{r^2}$ behavior, specifically the Newtonian gravitational force between two point (or spherically symmetric) masses and the Coulomb force between two point (or spherically symmetric) electric charges. Clearly both of these examples are idealizations - neither ideal point masses or charges nor perfectly spherically symmetric mass or charge distributions exist in nature, except perhaps for elementary particles such as electrons. However, deviations from ideal behavior are often small and can be neglected to within a reasonable approximation. (Power series to the rescue!) Also, notice the difference in length scale: the archetypal gravitational example is planetary motion - at astronomical length scales; the archetypal Coulomb example is the hydrogen atom - at atomic length scales.

The two solutions to the central force problem - classical behavior exemplified by the gravitational interaction and quantum behavior exemplified by the Coulomb interaction - are quite different from each other. By studying these two cases together in the same course, we will be able to explore the strong similarities and the important differences between classical and quantum physics.

Two of the unifying themes of this topic are the conservation laws:

• Conservation of Energy
• Conservation of Angular Momentum
The classical and quantum systems we will explore both have versions of these conservation laws, but they come up in the mathematical formalisms in different ways. You should have covered energy and angular momentum in your introductory physics course, at least in simple, classical mechanics cases. Now is a great time to review the definitions of energy and angular momentum, how they enter into dynamical equations (Newton's laws and kinetic energy, for example), and the conservation laws.

In the classical mechanics case, we will obtain the equations of motion in three equivalent ways,

• using Newton's second law,
• using Lagrangian mechanics,
• using energy conservation.
so that you will be able to compare and contrast the methods. The third approach is slightly more sophisticated in that it exploits more of the symmetries from the beginning.

We will also consider forces that depend on the distance between the two bodies in ways other than $\frac{1}{r^2}$ and explore the kinds of motion they produce.

• group Box Sliding Down Frictionless Wedge

group Small Group Activity

120 min.

##### Box Sliding Down Frictionless Wedge
Theoretical Mechanics 2021 (2 years)

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_ind Magnetic Moment \& Stern-Gerlach Experiments

assignment_ind Small White Board Question

30 min.

##### Magnetic Moment & Stern-Gerlach Experiments
Quantum Fundamentals 2022 (2 years)

Students consider the relation (1) between the angular momentum and magnetic moment for a current loop and (2) the force on a magnetic moment in an inhomogeneous magnetic field. Students make a (classical) prediction of the outcome of a Stern-Gerlach experiment.
• group Mass is not Conserved

group Small Group Activity

30 min.

##### Mass is not Conserved
Theoretical Mechanics 2021 (2 years)

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?

• face Ideal Gas

face Lecture

120 min.

##### Ideal Gas
Thermal and Statistical Physics 2020

These notes from week 6 of Thermal and Statistical Physics cover the ideal gas from a grand canonical standpoint starting with the solutions to a particle in a three-dimensional box. They include a number of small group activities.
• computer Effective Potentials

computer Mathematica Activity

30 min.

##### Effective Potentials
Central Forces 2022 (2 years) Students use a pre-written Mathematica notebook or a Geogebra applet to explore how the shape of the effective potential function changes as the various parameters (angular momentum, force constant, reduced mass) are varied.
• 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.
• group Survivor Outer Space: A kinesthetic approach to (re)viewing center-of-mass

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.
• groups Air Hockey

groups Whole Class Activity

10 min.

##### Air Hockey
Central Forces 2022 (2 years)

Students observe the motion of a puck tethered to the center of the airtable. Then they plot the potential energy for the puck on their small whiteboards. A class discussion follows based on what students have written on their whiteboards.
• face Equipartition theorem

face Lecture

30 min.

##### Equipartition theorem
Contemporary Challenges 2022 (3 years)

This lecture introduces the equipartition theorem.
• face Thermal radiation and Planck distribution

face Lecture

120 min.

##### Thermal radiation and Planck distribution
Thermal and Statistical Physics 2020

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.

Learning Outcomes