Re: Pendulum: What is relationship of the T^2 vs. length?

Michael,

Your thinking is not "off".  In terms of simply solving an algebraic 
problem, you are correct.  

But now, step back and look at the problem a little bit differently.  
Let's assume you don't know anything about pendulums and want to perform 
an experiment in which you vary the pendulum length, L, the mass, and the 
starting position.  You also have the ability to measure the period 
accurately.  (You may do this by measuring one period, or taking the 
average of many periods).  You would find that T^2 is directly 
proportional to the pendulum length, and completely independent of the 
mass or the starting position (as long as the starting position 
wasn't "too high").

You now have a simple device that can accurately measure time, and is 
quite easy to calibrate (adjust the length) and keeps accurate time even 
as the power "runs down".  This type of understanding of mechanisms is 
what lead to the development of portable, accurate clocks, which 
revolutionized navigation in the 17th Century.

One goal of this type of physics analysis and experimentation is to 
develop an understanding of unique relationships between phenomena, and 
convert it into a mathematical model, if possible.  Even at this moment, 
researchers in nanotechnology are using this type of modeling to develop 
new sensors and devices.

Solve a problem, then step back to try to understand what the solution 
means.

Christopher M. Seaman
Technical Specialist
ALCOA Technical Center