When we cut the spring into ratio of length \(1: 2: 3,\) we

get three springs of lengths \(\frac{l}{6}, \frac{2 l}{6}\) and \(\frac{3 l}{6}\) with force

constant,

\(\therefore k_{1}=\frac{k l}{l_{1}}=\frac{k l}{l / 6}=6 k\)

\({k_{2}=\frac{k l}{l_{2}}=\frac{k l}{2 l / 6}=3 k}\)

\({k_{3}=\frac{k l}{l_{3}}=\frac{k l}{3 l / 6}=2 k}\)

When connected in series,

\(\frac{1}{k^{\prime}}=\frac{1}{6 k}+\frac{1}{3 k}+\frac{1}{2 k}=\frac{1+2+3}{6 k}=\frac{1}{k}\)

\(\therefore \quad \overline{k^{\prime}}=k\)

When connected in parallel,

\({k^{\prime \prime}=6 k+3 k+2 k=11 k}\)

\({\frac{k^{\prime}}{k^{\prime \prime}}=\frac{k}{11 k}=\frac{1}{11}}\)