Finding the Radial Equation from Conservation of Energy Lecture (15 minutes)

Central Forces Notes Section 11

  • A quick review of the meaning of potential energy diagrams is followed by the solution of the radial equation of motion for the gravitational case using conservation of energy. Students encounter the idea of potential energy diagrams in an earlier Modern Physics course and in the Oscillations Paradigm course.
  • It is sometimes useful to review the method of finding the potential energy function from the force. From introductory physics students often remember that work is equal to the force times the distance rather than $W = \int{\Vec{F} \cdot \Vec{d{r}}}$
  • It is sometimes helpful to reason through what the sign on the expression for the gravitational potential energy needs to be in order to make the gravitational potential attractive. One way to do this is to ask the students to make a rough plot of the function $1 \over r$. By discussing this plot, students can reason that the sign on the expression for the gravitational potential needs to be negative.
  • In preparation for talking about the effective potential it is useful to follow up the above small white board question by asking them to plot $1 \over r$ and $1 \over {r^2}$ on the same graph making particular note of the fact that $1 \over {r^2}$ is larger at small distances and $1 \over r$ dominates at large distances.
  • Together the class writes down the energy equation $$E=\frac{1}{2} \mu \dot r^2 + \frac{1}{2} \frac{l^2}{\mu r^2} + U(r)$$ which can be solved for $\dot r$ to get $$\dot{r} = \pm \sqrt{\frac{2}{\mu}(E-U(r)) - \frac{l^2}{\mu^2 r^2}}$$

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