Mass Distribution and Frequencies of a Vertical Spring

A vertical spring in a gravitational field has a density increase from top to bottom. The density variation, which is often not noticeable, depends on the ratio of the spring weight to the quantity $kz0$⁠, the spring constant $k$ times the unstretched length $z0$⁠. Two springs prepared by prestretch-ing 78 and 23 turn portions of a “Slinky” showed density increases by factors of about 10 and 4, respectively. The detailed density distribution of the smaller spring agrees very well with the results of an analysis based on Hooke's law behavior. The heavier spring shows non-Hookean effects. The lowest few frequencies can be measured by driving the spring in resonance by hand, and for the lighter spring these also agree well with calculated values. The predicted nonuniform distribution of nodes is observable. This simple apparatus may provide stimulating experiments for intermediate laboratories.