Solve this in English showing all steps used

When the mass hangs in equilibrium, the upward force exerted by the spring (spring force, $F_s$) balances the downward gravitational force ($F_g$) acting on the mass.

The spring force is given by Hooke's Law: $F_s = kx$, where $k$ is the spring constant and $x$ is the extension.
The gravitational force is: $F_g = mg$, where $m$ is the mass and $g$ is the acceleration due to gravity.
Given:
$m = 0,5 \text{ kg}$
$x = 10 \text{ cm} = 0,1 \text{ m}$ (converting cm to m)
$g = 10 \text{ m/s}^2$

The spring constant is $\boxed{\text{50 N/m}}$.

The period of oscillation ($T$) for a mass $m$ attached to a spring with constant $k$ is given by:

$m = 0,5 \text{ kg}$
$k = 50 \text{ N/m}$ (from part a)

Using $\pi \approx 3,14159$:

The period of oscillation is $\boxed{\text{0,628 s}}$.