Potential energy diagrams: a means to calculate time of travel

Energy diagram slides/notes

  • Introduce \(dt=\frac{dx}{v(x)}\). Use simple arguments, starting from constant velocity \(\Delta t=\frac{d}{v}\), to motivate the integral form. Ask students how they would find out $v$($x$) in a conservative system given the total energy.
  • Introduce harmonic oscillator potential energy \(U\left( x \right)=\frac{1}{2}kx^{2}\)

using mass-on-a spring prop. Discuss features of PE diagram: equilibrium points, kinetic energy and velocity, turning points, $etc$.

  • Set up integral to calculate period, time for one cycle and use symmetry to simplify limits. Perform integral.
  • Discuss qualitatively the effect on the period of the amplitude (distance traveled) vs. energy. Compare to non-harmonic systems (e.g. cosine potential).

The focus of the lecture is calculating travel time, given a velocity, with the harmonic oscillator as an example and variants considered qualitatively. The focus is not the equation of motion of a pendulum. The students know that their next activity will be to measure travel time for an oscillating system and that they will be able to calculate the predictions of the harmonic oscillator or any other model. Few or none have calculated the period of an oscillator this way. Some know that that the plane pendulum in not an harmonic oscillator. The class continues with a focus on measurement and modeling, and then to representing harmonic motion before it returns to the equation of motion of an oscillator from the force approach.

The level of detail determines the time. 50 minutes is comfortable, but it could easily expand. This is the first lecture of the course, so at least 10-15 minutes are taken with introductions and administrative matters.

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