Question

now suppose that we have attached not just two springs in series, but N springs. Write an equation that expresses the effective spring constant of the combination using the spring constant of the original spring k and the number of springs N

Answer 1

Answer:

 $k_{eq} = \frac{k}{N}$

Explanation:

For this exercise let's use hooke's law

         F = - k x

where x is the displacement from the equilibrium position.

        x = $- \frac{F}{k}$

if we have several springs in series, the total displacement is the sum of the displacement for each spring, F the external force applied to the springs

       x_ {total} = ∑ x_i

we substitute

       x_ {total} =  ∑ -F / ki

       F / k_ {eq} =  -F  $\sum \frac{1}{k_i}$

      $\frac{1}{k_{eq}} = \frac{1}{k_i}$ 1 / k_ {eq} =  ∑ 1 / k_i

if all the springs are the same

     k_i = k

     $\frac{1}{k_{eq}} = \frac{1}{k} \sum 1$

     $\frac{1}{k_{eq} } = \frac{N}{k}$

     $k_{eq} = \frac{k}{N}$