Why Does a Spring's Work Calculation Include a Factor of 1/2?

[Nick G]

if W = Fd

and F = kx

then W = kx^2 ?

where does the 1/2 in PEspring = 1/2 k x^2 come from...

F not constant when applied to a spring?

i know it has something to do with the integration but my high school kids have yet to take calc

help please

 

[DylanB]

The spring force is a variable force. Therefore you must integrate kx dx from 0 to x.

If your kids haven't done any integration, its possible for you to explain it using the average force which the spring will exert when compressing or stretching the spring which will equal 1/2kx. Maybe draw a force vs. time graph of the stretching of a spring to illustrate this.