Question

A spring of force constant $k$ is cut into lengths of ratio $1:2:3$. They are connected in series and the new force constant is $k^{\prime }$. Then they are connected in parallel and force constant is $k"$ . Then $k^{\prime }:k^{\prime \prime }$ is :

• A. 1 : 9
• B. 1 : 11
• C. 1 : 14
• D. 1 : 16

Answer 1

Answer: (B)

Spring constant $\propto \frac{1}{\text{ length}}$
$k\propto \frac{1}{l}$
i.e, $k_1=6k$
$k_2=3k$
$k_3=2k$
In series
$\frac{1}{k^{\prime }}=\frac{1}{6k}+\frac{1}{3k}+\frac{1}{2k}$
$\frac{1}{k^{\prime }}=\frac{6}{6k}$
$k=k$
$k^{\prime \prime }=6k+3k+2k$
$k^{\prime \prime }=11k$
$\frac{k^{\prime }}{k^{\prime \prime }}=\frac{1}{11}\text{ i.e }{k}^{\prime }:k^{\prime \prime }=1:11$