Activities
- The effect of angular momentum, the force constant, and the reduced mass on the shape of the effective potential function;
- Practicing exploring parameter space for a function;
- Developing intuition about how the orbit shape depends on these parameters.
A student is invited to “act out” motion corresponding to a plot of effective potential vs. distance. The student plays the role of the “Earth” while the instructor plays the “Sun”.
In this unit, you will explore the classical mechanics of central forces, especially gravitational orbits like the earth going around the sun.
Motivating Questions
- What shapes can the orbits have?
- What are Kepler's laws and why are they true?
- What is an effective potential diagram and how can it be used to predict the shape of an orbit?
Key Activities/Problems
- Problem: Undo Formulas for Center of Mass (Geometry)
- Activity: Acting Out Effective Potentials (In class, only)
- Activity: Effective Potentials
- Problem: Hockey
- Problem: Scattering
Unit Learning Outcomes
At the end of this unit, you should be able to:
- List the properties that define a central force system.
- Calculate a reduced mass for a two-body system and describe why it is important.
- Use the solution (algebraic or geometric) to a reduced mass system to describe the motion of the original system.
- Describe the role that conservation of energy and angular momentum play in a central force system. In particular, where do these properties appear in the solutions of the equations of motion?
- Use an effective potential diagram to predict the possible orbits in a central force system: which orbits are bound or unbound? which are closed or open? where will the turning points be?
Equation Sheet for This Unit