A spring has length $L$ and force constant K . It is cut into two springs of length $L_1$ and $L_2$ such…

A spring has length $L$ and force constant K . It is cut into two springs of length $L_1$ and $L_2$ such that $\mathrm{L}_1=\mathrm{NL}_2$ ( N is an integer). The force constant of spring of length $L_1$ is
  1. $(\mathrm{N}+1) \mathrm{K}$
  2. $\frac{\mathrm{K}}{\mathrm{N}}(1+\mathrm{N})$
  3. K
  4. $\frac{\mathrm{K}}{\mathrm{N}+1}$

Solution

Given, $\mathrm{L}_1=\mathrm{NL}_2...(i)$
Spring constant, $\mathrm{k} \propto \frac{1}{\mathrm{~L}}$ $\therefore \quad \mathrm{k}_2=\mathrm{Nk}_1$
Before cutting, the springs are connected in series, so $\begin{aligned} \therefore \quad \frac{1}{\mathrm{~K}} & =\frac{1}{\mathrm{k}_1}+\frac{1}{\mathrm{k}_2} \\ \frac{1}{\mathrm{~K}} & =\frac{1}{\mathrm{k}_1}+\frac{1}{\mathrm{Nk}_1} \\ \frac{1}{\mathrm{~K}} & =\frac{\mathrm{N}+1}{\mathrm{Nk}_1} \\ \therefore \quad \mathrm{k}_1 & =\frac{\mathrm{K}}{\mathrm{~N}}(1+\mathrm{N}) \end{aligned}$

Asked in: MHT CET 2024 (15 May Shift 2)

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