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,
${k^{\prime \prime}=6 k+3 k+2 k=11 k}$
${\frac{k^{\prime}}{k^{\prime \prime}}=\frac{k}{11 k}=\frac{1}{11}}$
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Figure: $Image$
| List $I$ (Physical Quantity) | List $II$ (Dimensional Formula) |
| $(A)$ Pressure gradient | $(I)$ $\left[ M ^0 L ^2 T ^{-2}\right]$ |
| $(B)$ Energy density | $(II)$ $\left[ M ^1 L ^{-1} T ^{-2}\right]$ |
| $(C)$ Electric Field | $(III)$ $\left[ M ^1 L ^{-2} T ^{-2}\right]$ |
| $(D)$ Latent heat | $(IV)$ $\left[ M ^1 L ^1 T ^{-3} A ^{-1}\right]$ |
Choose the correct answer from the options given below: