Homework for Potential Energy Diagrams

  1. A particle of mass $m$ and total energy $E$ that oscillates under the influence a potential energy \[U(x)=\left\{ \begin{matrix} \infty & x<0 \\ \alpha x & x\ge 0 \\ \end{matrix} \right.\] where $α$ is a known constant. Draw a diagram of the potential and label any quantities you define in your solution. Calculate the period of motion and discuss whether the is period amplitude-independent.
  2. Suppose the particle oscillates in a “half” harmonic potential \[U(x)=\left\{ \begin{matrix} \infty & x<0 \\ \frac{1}{2}kx^{2} & x\ge 0 \\ \end{matrix} \right.\] with the same amplitude as in the above example, and at the particular amplitude in question, the total energy in each case happens to be the same. Draw an energy/position graph that represents this information, and discuss whether the period is longer, shorter, or the same.
  3. Identify a physical example of the potential in #1, and calculate the period of the motion by means more familiar from introductory physics. Does your answer agree with the period calculated by the energy method?

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