Question

A spring has stiffness k_s. You cut the spring in half. What is the stiffness of the half-spring?

(a) 2k_s,

(b) k_s,

(c) k_s/2

Short answer

Option (a)

The stiffness of the half-spring is2k_s.

Step-by-step solution

Given data

A spring of stiffness k_s is cut in half.

Stiffness of spring

The spring constants of springs (k₁and k₂) of the same material but of different lengths l₁andl₂are related as,

${\mathbf{k}}_{\mathbf{1}}{\mathbf{l}}_{\mathbf{1}}\mathbf{=}{\mathbf{k}}_{\mathbf{2}}{\mathbf{l}}_{\mathbf{2}}$ (I)

Determining the change in spring constant

Given information:

${\mathrm{k}}_1={\mathrm{k}}_{\mathrm{s}}$

Since the length is halved, that can be expressed as,

$$\begin{gathered} {\mathrm{I}}_2=\frac{{\mathrm{I}}_1}{2} \\ 2{\mathrm{I}}_2={\mathrm{I}}_1 \end{gathered}$$

Substitute the values in the above equation, and we get,

$$\begin{gathered} {\mathrm{k}}_{\mathrm{s}}\times 2{\mathrm{I}}_2={\mathrm{k}}_2{\mathrm{I}}_2 \\ {\mathrm{k}}_2={\mathrm{k}}_{\mathrm{s}} \end{gathered}$$

The new spring constant is $2{\mathrm{k}}_{\mathrm{s}}$.

Thus, option (a) is correct.