Classical Angular Momentum

Students should be able to:

1. (CentralForce) Determine whether several common forces in nature are central forces.

   Which of the following forces can be central forces? which cannot?

   1. The force on a test mass $m$ in a gravitational field $\Vec{g }$, i.e. $m\Vec g$

   2. The force on a test charge $q$ in an electric field $\Vec E$, i.e. $q\Vec E$

   3. The force on a test charge $q$ moving at velocity $\Vec{v }$ in a magnetic field $\Vec B$, i.e. $q\Vec v \times \Vec B$

2. (FreeCentralForce) A simple check on your understanding of center-of-mass motion.

   If a central force is the only force acting on a system of two masses (i.e. no external forces), what will the motion of the center of mass be?

3. (PlanarOrbit) A simple check on your understanding of classical angular momentum.}

   Show that the plane of the orbit is perpendicular to the angular momentum vector $\Vec L$.

4. (CMLandT) Explicitly show how the kinetic energy and angular momentum of a two particle system is related to the energy and angular momentum of the center of mass and reduced mass system.

   Consider a system of two particles.

   1. Show that the total kinetic energy of the system is the same as that of two “fictitious” particles: one of mass $M=m_1+m_2$ moving with the speed of the CM (center of mass) and one of mass $\mu$ (the reduced mass) moving with the speed of the relative position $\vec{r}=\vec{r}_2-\vec{r}_1$.

   2. Show that the total angular momentum of the system can be similarly decomposed into the angular momenta of these two fictitious particles.