Physics Friday 147

Consider two (ideal) springs, with resting lengths L1 and L2, and respective spring constants k1 and k2. We connect the springs in series. How, then, does the force exerted by the combined springs vary with a stretch or compression x beyond the combined length L=L1+L2?

Suppose we have spring one stretched by an amount x1, and spring two stretched by an amount x2. Then x=x1+x2. Considering the point where the springs connect, we see the force exerted by one spring must be equal in magnitude of the force exerted by the other, and thus equal to the force exerted by the springs:
F=k1x1=k2x2
So, we now have two equations relating x1 and x2; solving that system, we see

.
Thus, we see that the force exerted is
.
Thus, the pair of springs behaves like a single spring of resting length L=L1+L2 and spring constant
.
This latter can be written as
;
and for n springs in series, we have
.

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