Consider two (ideal) springs, with resting lengths *L*_{1} and *L*_{2}, and respective spring constants *k*_{1} and *k*_{2}. 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*=*L*_{1}+*L*_{2}?

Suppose we have spring one stretched by an amount *x*_{1}, and spring two stretched by an amount *x*_{2}. Then *x*=*x*_{1}+*x*_{2}. 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*=*k*_{1}*x*_{1}=*k*_{2}*x*_{2}

So, we now have two equations relating *x*_{1} and *x*_{2}; 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*=*L*_{1}+*L*_{2} and spring constant

.

This latter can be written as

;

and for *n* springs in series, we have

.

### Like this:

Like Loading...

*Related*

Tags: Friday Physics, physics, Spring Constant, Springs

This entry was posted on December 17, 2010 at 12:02 am and is filed under Math/Science. You can follow any responses to this entry through the RSS 2.0 feed.
You can leave a response, or trackback from your own site.

## Leave a Reply