Question

A spring having spring constant $k$ is cut into two parts in the ratio $1:3$. If these parts are connected in parallel then the equivalent spring constant for the combination will be:

• A. $\frac{4k}{3}$
• B. $k$
• C. $\frac{16k}{3}$
• D. $\frac{k}{4}$

Answer 1

The correct option is C $\frac{16k}{3}$

Let the natural length of original spring be $l$.

After the spring is cut in ratio $1:3$, the lengths will be $\frac{l}{4}\text{and}\frac{3l}{4}$

We know that $k\propto \frac{1}{l}$

$\Rightarrow kl=\text{constant}....\left(i\right)$

After cutting the spring let the new spring constant be $k_1\text{and}{k}_2$ for the individual parts respectively.

From Eq.$\left(i\right)$,

$k_1\left(\frac{l}{4}\right)=k_2\left(\frac{3l}{4}\right)=kl$

$\Rightarrow \frac{k_{1}l}{4}=kl$

or, $k_1=4k$

Similarly,

$\frac{3k_{2}l}{4}=kl$

$\text{Or,}{k}_2=\frac{4k}{3}$

If the springs are now connected in parallel,


$\Rightarrow k_{\text{eq}}=k_1+k_2$

$\therefore k_{\text{eq}}=4k+\frac{4k}{3}=\frac{16k}{3}$

$\begin{array}{c}\text{Why this question}?\\ \underset{–}{\text{Tip}}\text{: When the springs are connected in parallel,}\\ \text{then extension in both springs will be}\\ \text{equal i.e,}{x}_1=x_2\text{however force in}\\ \text{spring will vary. On the other hand when}\\ \text{connected in series the extension in both springs}\\ \text{will be different when acted by an external load,}\\ \text{i.e.}{x}_1\ne x_2\text{.}\end{array}$