Effective Potentials & Orbital Motion

Students should be able to:

• Write Newton's second law as a differential equation.
• - reduced mass

1. (Yukawa) Not all central forces are Coulombic.

   In a solid, a free electron doesn't “see” a bare nuclear charge since the nucleus is surrounded by a cloud of other electrons. The nucleus will look like the Coulomb potential close-up, but be “screened” from far away. A common model for such problems is described by the Yukawa or screened potential:

   $$U(r)= -{k\over r} e^{-{r\over \alpha}}$$

   1. Graph the potential, with and without the exponential term. Describe how the Yukawa potential approximates the “real” situation. In particular, describe the role of the parameter $\alpha$.

   2. Draw the effective potential for the two choices $\alpha=10$ and $\alpha=0.1$ with $k=1$ and $\ell=1$. For which value(s) of $\alpha$ is there the possibility of stable circular orbits?

2. (NASA) How do you change the orbit of a spacecraft?

   NASA has launched a satellite into a circular orbit around the earth and wants to increase the radius slightly while maintaining a circular orbit. NASA scientists propose to fire the engines briefly, applying a small impulse to the satellite.

   One scientist says that it doesn't matter if the impulse is applied in a direction tangential to the satellite motion or perpendicular to the motion, arguing that both approaches will simply fine tune the total energy of the system.

   A second scientist disagrees and argues that one of the options would work but the other would definitely not work.

   A third scientist says that neither option would work. Which scientist would you side with, and why?

3. (Hockey)

   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?

      %(This is not meant to be a trick question. Do not worry about the puck getting caught in the corner or the shape of the sides of the bowl. You may consider the sides of the rink as angled very, very slightly outward and connected smoothly to the surface of the rink. You may also consider the puck to have no sharp corners.)

   3. Situation 2: a stationary puck, at a distance ${a\over 2}$ 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?

4. (FindForceLaw)

   %Marion and Thornton 8.14 Find the force law for a central-force field that allows a particle to move in a spiral orbit given by $r=k\phi^2$, where $k$ is a constant.

5. (OrbitsEnergy) This problem needs to be rewritten to be textbook independent.