Question

A light spring of spring constant $k$ is cut into two equal halves. Each half is connected in parallel then net spring constant of the combination is:

• A. $\frac{k}{4}$
• B. $\frac{k}{2}$
• C. $2k$
• D. $4k$
• E. $k$

Answer 1

Answer: (D)

$4k$

Spring constant is inversely proportional to length of the spring. i.e $k\propto \frac{1}{l}=c\frac{1}{l}$ where $c$ is proportional constant.
so when it is cut into two equal half, spring constant each half, $k^{\prime }=c\frac{1}{l/2}=\frac{2c}{l}$
thus, $\frac{k^{\prime }}{k}=2\Rightarrow k^{\prime }=2k$ (Basically there are now half as many coils to take up the force via displacement, thus spring constant increases)
As the two half are connected in parallel so the constant of the combination is $k_{net}=2k+2k=4k$