Period of Oscillation of Underdamped Cart attached to Wall by Spring

Problem Definition
Let $C$ be underdamped.

Let $C$ be pulled aside to $x = x_0$ and released from stationary at time $t = 0$.

Then the period of oscillation of $C$ can be expressed as:
 * $T = \dfrac {2 \pi} {\sqrt {\dfrac k m - \dfrac {c^2} {4 m^2} } }$

Proof
Let:
 * $a^2 = \dfrac k m$
 * $2 b = \dfrac c m$

From Position of Cart attached to Wall by Spring under Damping: Underdamped: $x = x_0$ at $t = 0$:
 * $x = \dfrac {x_0} \alpha e^{-b t} \left({\alpha \cos \alpha t + b \sin \alpha t}\right)$

where $\alpha = \sqrt {a^2 - b^2}$.

Let $T$ be the period of oscillation of $C$.

Then: