Check out the article - The Effective Mass of an Oscillating Spring -
AJP Vol.38 Jan 1970, pp98-100, where the authors show that this value
ranges between 0.333 and 0.405
Now, a challenge to the mathematically abled is to do a similar calculation
with a non-uniform spring as being discussed in this thread. Use a measured
linear density function for the spring to find a new range of values. Will
this range then depend upon which end of the spring is at the top?
A lab experiment I have students perform is to measure the period as a function
of the total mass attached to the spring. Then I have them do a least squares
fit to the 'usual' equation, but with an unknown fraction of the mass of the
spring to add to the attached mass. From the least squares result for slope and
intercept, I now obtain the two unknowns of spring constant and fraction of
spring mass to use. Someday if I get the urge to take lots of data myself,
I may try to get the results published.
Tom Walkiewicz (814)732-2468
Dept. Physics walkiewicz@edinboro.edu
Edinboro University
Edinboro, PA 16444