The Pendulum II

Return to Contents

This experiment is like the previous one with one new feature. We test the idea of reciprocity by introducing an "obstruction" in the path of the pendulum along the vertical line through the pivot.

When the pendulum swings in the clockwise direction through the vertical line, the rod "bends" at the obstruction and the part below the obstruction continues to swing. Assuming this process consumes little or no energy, we can predict that the rod would still rise (if it could) to its original height.

At the bottom of its swing, it has transformed a certain amount of energy of position into kinetic energy, and, rising, will lose all of that kinetic energy by rising (albeit along a different route) to the original height. Reasoning of this sort, so easy if we view things in terms of reciprocity, is again the justification for introducing the notion of gravitational potential energy. It informs us that beneath it all, something simple is going on. Noone knew what that simple thing was until Einstein introduced the geometric view of gravity and argued that in space-time, objects simply moved into their future along "straightest" paths in a 4-dimensional contunuum which was warped (or curved) by the presence of evolving matter. But that explanation is for another book.

Exploration: Pendulum interrupted

In any case, this is how you do the experiment. Grab the dark blue dot (the obstruction):

by left-clicking on it as you pull it downward to the desired height below 0. While you do this, the cursor becomes a pointing finger. When you are finished, click the left mouse button, and the cursor turns back to an arrow.

There are a few things to consider here. What if the bob cannot rise to its initial height (the obstruction is more than half way to the bottom of the "rod")? The bob starts with more energy than it can expend through reciprocity. Can you guess what will happen?

We augment this experiment with a new sort of position-velocity portrait. The black curve represents the height in meters, and the blue curve represents the velocity in meters per second. Our convention is that velocity is counted negative when the bob is moving clockwise and positive when it is moving counterclockwise. This is not the usual position-velocity portrait because velocity is measured along the arc and position is height (not position along the arc).

Question 1: This is the sort of question one often considers in Calculus. At the moment the bob makes the transition from clockwise to counterclockwise motion, what is true of the velocity? Place the obstruction out of the way, and then look at the portraits, and observe what is simultaneously true of the position portrait at such points. The relationship is not as simple as may appear. You may see that in this way. Set the initial speed to -10, and set the initial height to 0 with length 6.

Question 2: Repeat the "unstable equilibrium" experiment with the obstruction out of the way. What is true of the portraits as you get energies closer and closer to the desired energy?