Lesson 8
Limitations of Newtonian Mechanics, the Plane Pendulum

1) A "phase diagram" for a nonlinear system provides a visual or graphical way of considering the behavior in time of the system. The figure to the right (Figure 4.10 from the text) shows a "phase diagram" for a plane pendulum. On the phase diagram's horizontal axis is q(t), the angular displacement of the pendulum bob (measured from the vertical) as a function of time. On the phase diagram's vertical axis is dq/dt, the time derivative of q, divided by a normalization constant ([g/l]^1/2).

In your own words, try to briefly explain this diagram. What kinds of motion of the plane pendulum correspond to the different curves/shapes on the phase diagram? The red letters placed on the diagram may help you refer to specific spots on the diagram.

Also, consider a plane pendulum that is subject to a damping force. Suppose the pendulum begins with enough energy to be able to make two complete revolutions around the pivot point before it loses enough energy that it cannot complete another revolution. After the two full revolutions, it just oscillates back and forth with an ever-decreasing oscillation amplitude. Briefly describe how the phase diagram would look for this case. (Again, you might find it helpful to refer to the red letters to describe approximate locations.)

2) Consider a plane pendulum like the one shown in the figure. The plane pendulum just consists of a mass m held a fixed distance l from a pivot point by a rod of negligible mass. The mass m can swing only in a vertical plane.

For small initial angles of displacement q_o, the plane pendulum is like the simple pendula we've seen before, and it obeys simple harmonic motion (SHM), with the period of oscillation given by T = 2p(l/g)^1/2. For larger angles, the period deviates from this expression.

Estimate about how big the initial displacement angle q_o would have to be for the SHM expression for the period to be off by 20%. (That is, at about what angle would the actual period and the SHM expression for the period differ by 20%?)

3) Newtonian Mechanics has some fundamental limitations. These can surface when

1. very small scales are involved
2. very fast speeds are involved
3. large numbers of particles are involved
4. more than one but NOT ALL of the above
5. all of the above

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