The Pendulum

Part 5: Summary

1. The undamped pendulum equation has the form

y'' + b sin(y) = 0,

where b is a constant. This form is similar to the linear equation

y'' + b y = 0.

To what extent do you see similarities between the solutions and trajectories of the two equations? Explain why the solutions and trajectories should be similar for small values of y.

2. To what extent do you see differences between the solutions and trajectories of the equations

y'' + b sin(y) = 0

and

y'' + b y = 0?

Explain why the solutions and trajectories should be different for larger values of y.

3. What are the equilibrium solutions of the equation

y'' + b sin(y) = 0?

Distinguish between stable and unstable equilibria.

4. For small-amplitude oscillations of the undamped pendulum, how does the period of oscillation depend on the length of the pendulum?
5. Describe the effects of including damping in the pendulum model.

Last modified: March 20, 1998