|MadSci Network: Physics|
I will follow the derivation of the answer to this problem that I found in the book Physics by John Cutnell and Kenneth Johnson (John Wiley & Sons, 5th edition, 2001) [Yes, the publishing date is 2001. I have an advance copy of the book!!]. On page 275 there is an example which exactly mimics your question. It is in the chapter about simple harmonic motion and elasticity. The question deals with a 10-coil spring that is cut in half to make 2 separate 5-coil springs.
Here's the answer: Suppose we apply a force to the 10-coil spring that
compresses it 1 centimeter. So each coil is compressed 0.1 cm. If we
compress the 5-coil spring 1 cm then each coil is compressed 0.2 cm. The
famous ideal-spring equation is
F = -kx
where F is the applied force, k is the spring "constant", and x is the compressed distance. So the force needed to compress a single coil by 0.2 cm is twice as large as to compress it 0.1 cm. Therefore the spring constant (k) of the 5-coil spring must be twice that of the 10-coil spring, because each individual coil of the 5-coil spring is compressed twice as far as each individual coil of the 10-coil spring for the same overall compression distance.
In general the spring constant of a spring is inversely proportional to the number of coils in the spring, all else being equal.
John Link, MadSci Physicist
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