# Period of Oscillation of Underdamped Cart attached to Wall by Spring

## Contents

## Theorem

### Problem Definition

Consider a cart $C$ of mass $m$ attached to a wall by means of a spring $S$.

Let $C$ be free to move along a straight line in a medium which applies a damping force $F_d$ whose magnitude is proportional to the speed of $C$.

Let the force constant of $S$ be $k$.

Let the constant of proportion of the damping force $F_d$ be $c$.

Let the displacement of $C$ at time $t$ from the equilibrium position be $\mathbf x$.

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:

\(\displaystyle T\) | \(=\) | \(\displaystyle \dfrac {2 \pi} {\sqrt {a^2 - b^2} }\) | Period of Oscillation of Underdamped System is Regular | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \dfrac {2 \pi} {\sqrt {\dfrac k m - \left({\dfrac c {2 m} }\right)^2} }\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \dfrac {2 \pi} {\sqrt {\dfrac k m - \dfrac {c^2} {4 m^2} } }\) |

$\blacksquare$

## Sources

- 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $\S 3.20$: Vibrations in Mechanical Systems: $(21)$