Answer
The units for the spring constant $k$ are: $\frac{kg}{s^2}$
Work Step by Step
The units for force are newtons which in base units are $\frac{kg~m}{s^2}$
In dimensions, we can represent this as: $[M][L][T]^{-2}$, where $M$ is mass, $[L]$ is length, and $[T]$ is time.
The dimensions for the distance $x$ are $[L]$
Let $D$ be the dimensions for the spring constant $k$. Using the equation $F = kx$, we can find the dimensions for the spring constant $k$:
$[M][L][T]^{-2} = D~([L])$
$D = [M][T]^{-2}$
The dimensions for the spring constant $k$ are $[M][T]^{-2}$, which means that the units are: $\frac{kg}{s^2}$
