Answer
(a) The spring constant of a single spring would be $\frac{k}{2}$
(b) The spring constant of a single spring would be $2k$
Work Step by Step
(a) Let's consider two springs in series. The force on each spring is F, so each spring stretches a distance of $\frac{F}{k}$. The system of two springs stretches a total distance of $2\times \frac{F}{k} = \frac{2F}{k}$
We can find the spring constant $k'$ of a single spring:
$k' = \frac{F}{x} = \frac{F}{\frac{2F}{k}} = \frac{Fk}{2F} = \frac{k}{2}$
The spring constant of a single spring would be $\frac{k}{2}$.
(b) Let's consider the two springs in parallel. The total force on the system is $F$, so each spring pulls with a force of $\frac{F}{2}$. Then each spring stretches a distance of $\frac{F}{2k}$
We can find the spring constant $k'$ of a single spring:
$k' = \frac{F}{x} = \frac{F}{\frac{F}{2k}} = \frac{2Fk}{F} = 2k$
The spring constant of a single spring would be $2k$.
