Question

If a spring of spring constant $k$ is divided into $m$ equal parts, and all are connected in parallel, then the equivalent spring constant of the combination is

• A. $mk$
• B. $k$
• C. $m^{2}k$
• D. $\frac{m^2}{k}$

Answer 1

The correct option is C $m^{2}k$

Let us assume that the length of given spring be $l$
Then, on dividing it into $m$ equal parts, the length of each new spring would be $\frac{l}{m}$
Let spring constant of the new springs be $k^{{}^{\prime }}$ each
Then, we have $k^{{}^{\prime }}\times \frac{l}{m}=kl\Rightarrow k^{{}^{\prime }}=mk$
Since, all are connected in parallel, then equivalent spring constant would be
$k_{eq}=(mk+mk+mk\cdot \cdot \cdot \cdot \cdot +mk)\text{m times}mk$
$\Rightarrow k_{eq}=m\left(mk\right)=m^{2}k$