course phy 121
A pendulum requires 2 seconds to complete a cycle, which consists of a complete back-and-forth oscillation (extreme point to equilibrium to opposite extreme point back to equilibrium and finally to the original extreme point). As long as the amplitude of the motion (the amplitude is the distance from the equilibrium position to the extreme point) is small compared to the length of the pendulum, the time required for a cycle is independent of the amplitude.
How long does it take to get from one extreme point to the other, how long from an extreme point to equilibrium, and how long to go from extreme point to equilibrium to opposite extreme point and back to equilibrium?
answer/question/discussion:
I will assume that all of the times are equal, that the pendulum does not change velocities ( although, in reality it seems it would as it would slow down as it reaches an extreme point in order to stop and turn back, it also would lose energy along the way and slow) during this time.
The pendulum continuously changes velocity. It's the period of the swing that doesn't change.
The time for one whole cycle is 2 secs. Extreme point to extreme point and back.
The time for ½ cycle: from one extreme point to the other would be 1 sec.
The time for ¼ cycle: from one extreme point to the equilibrium would be 1/5 sec.
The time for ¾ cycle: from one extreme point, through equilibrium, to other extreme point and back to equilibrium would be ¾ of a sec.
What reasonable assumption did you make to arrive at your answers?
answer/question/discussion:
As I said before. I assumed that no energy was lost and that there was no change in velocity, that all of the time intervals are equal.
You need only assume that the times from equilibrium to extreme point and from extreme point to equilibrium are equal, and that these times do not vary with the pendulum's decreasing amplitude. There is no reason to assume that this constancy of rhythm is in any way related to constant velocity or to conservation of the pendulum's energy; it is in fact obvious that neither of these conditions holds.
At this stage of the course we aren't in a position to explain the constant rhythm; we simply observe that if we set up the pendulum correctly this appears, within our limits of experimental error, to be the case.
The actual explanation is pretty deep. In fact, we can't explain this phenomenon within the mathematical constraints of this course. The explanation involves a lot of calculus and differential equations, at a pretty advanced level. Also when the calculus is refined for a non-ideal pendulum like this one, a pendulum that does lose energy, we find that there are indeed tiny variations in the period and between the quarter-cycles. This is not something we can observe without sophisticated and highly accurate electronic timers, so the variations aren't relevant to this course, but they do exist.