Understanding the Effective Spring Constant When Cutting a Spring in Half

[DLinkage]

I should probably know this being a college grad w/ 4 years of mechanics, but can anyone give me a clear explanation why the effective spring constant of a spring doubles when you cut the spring in half? I understand physically smaller springs are stiffer but is there a proof someone can give? thanks

 

[Doc Al]

Consider this: Stretch the spring a distance X. The tension is the same throughout the spring. Now consider the spring as consisting of two springs (each half as big) hooked together. They exert the same force, but each is only stretched X/2. So the spring constant of each must be twice the original. Make sense?

 

[DLinkage]

So is this to say that if I manufacture springs originally with a length of 1 meter and spring constant k, then if i cut the thing into n equal lengths, the spring constant of each length will be nk?

Obviously I would assume this is only a very basic high level description of a spring and the actual constant is a function ofthe material, cross sectional area, and numbers of turns per unit lenght?