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Eigenvalue Lesson
Shodor > CSERD > Resources > Activities > Eigenvalue Lesson
  Lesson  •  Materials  •  Lesson Plan

Lesson Plan - Eigenvalues

Audience

The eigenvalue activity is designed for use in an undergraduate level course in physics or mathematics.

In an upper level undergraduate mathematics or physics course, this activity could be used to illustrate eigenvalue problems in applied mathematics.

Solutions

While using the small angle approximation, this can be solved either numerically or analytically. Students may well see that the solution to this problem can be written as


\begin{displaymath} \theta(t) = \theta(0)*cos(\sqrt{\frac{g}{l}}t) \end{displaymath}

In order for the pendulum to be at the original position when $t_{final} = 10$, the length must satisfy $\sqrt{\frac{g}{l}} t_{final} = 2 n \pi$. This will occur when


\begin{displaymath} l = g \left( \frac{t_{final}}{2 n \pi}\right)^2. \end{displaymath}

For the question of how this will be affected by the small angle approximation, students will need a numerical solution. By adjusting the length in the model, students can see which values of l will allow them to have the pendulum return to the initial position at the final time. For an initial displacement of 0.5 radians, after 1 pass the longest and second longest pendulums will be 0.1 seconds out of phase. Students may have different ways of showing this. To answer the question as posed, the student might start with the predicted values for the small angle approximation, use those in an actual solution, and show how far out of phase the pendulums will be depending on the initial displacement of the pendulums.

An important point to stress is that problems like this, while not the classical mathematical eigenvalue problem, are what physicists typically mean by eigenvalue problems. In physics, an eigenvalue problem is a problem with one or more parameters where the boundary conditions only allow a solution for specific values of those parameters. Standing waves are a classic example. The solution to


\begin{displaymath} \frac{d^2 P}{dx^2} = -P(x) \end{displaymath}

is clearly sinusoidal function, but if P is constrained, only waves with specific frequencies will both solve the equation and match the boundary conditions.

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