We will now see reciprocity
(conservation of energy) with a vengeance. Our instrument is a pendulum, that
is, a heavy weight suspended from a pivot by a stiff, but essentially weightless
rod. Such pendula, when designed to minimize friction, swing back and forth
with remarkable regularity. In fact, as you may know, they are used to regulate
the mechanism of clocks.
Why do they swing in a regular
way? If they are raised to a certain height, they fall until they reach the
lowest point they can attain. As you saw in earlier labs, they have at this
point acquired the maximum speed. If they started with zero speed, this is the
speed v such that 
(kinetic energy, or energy of motion) is equal to the loss of gravitational
energy due to the decrease in height. From that point, the energy of motion
is slowly transformed to gravitational potential energy as the pendulum rises,
until it is all gone. This happens when it rises again to the height from which
it started!
So this is a nice example
of reciprocity. Of course, in a real pendulum, there is a decrease in energy
due to loss of heat through the rubbing together of parts, friction with the
air, etc. and this brings the pendulum eventually to a stop. But here we banish
friction from the picture, and can study the theoretical behavior.
There are several factors
that you may control in this model. You may set the length of the rod, the initial
height of the bob, and the initial velocity of the bob. You do these things
by entering the values in the textfields:
Finally, to observe the motion, press: 
Then press P (or Esc) to stop it.
The initial height is measured
from the zero position in meters, and can vary from -8 to 8. The length of the
pendulum can be 8 at most. Now the velocity needs perhaps a little explanation.
It is measured in meters per second, and gives the speed along the arc of swing
of the pendulum. Positive values for velocity give speed in the counter-clockwise
sense. Negative values for velocity give speed in the clockwise sense. Thus,
if the length of a pendulum is 8, the full circular arc of swing is 
(or roughly 50) meters. A speed of 1 meter per second is relatively small with
reference to this circumference.
Question 1: Does the time of swing (say from some initial position with 0 speed all the way back to that same position) depend on the length of the rod? This time of swing is called the period of oscillation of the pendulum. Do you think it depends on the weight of the bob?
Question 2: You might like to compile some numbers for time of swing versus period of oscillation, then go to the calculator through the contents page and plot them. To plot a point (x,y) type in the command line:
point [x,y];
or
point [x,y] using color red;
or
point [x,y] using width 2, color blue;
Can you guess the relationship between length and period of oscillation?
Question 3: Choose a length for the pendulum, and set the height equal to the negative of that length, so the bob rests at the lowest possible point. Set initial speed to 0. What happens when you "swing" the pendulum? Now find the speed for which the bob would rise to the top of the circle of swing. We are using the value g = 9.81 meters per second per second for gravitational acceleration. Thus calculate how much kinetic energy it would need initially to rise through exactly 2 radii. Use the formula:
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But you must be patient! The rise to the topmost point is not observable. It is an unstable phenomenon (like a pencil standing on its point) in the sense that any small deviation from the precise speed will cause some "neighboring" phenomenon to be observed. What are the two "neighboring" observable phenomena? How long do you think the bob would take to get to the unstable equilibrium if it could do it?
