# Position of Cart attached to Wall by Spring

## 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 with zero friction.

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

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

Then the horizontal position of $C$ at time $t$ can be expressed as:

- $x = C_1 \cos \alpha t + C_2 \sin \alpha t$

where:

- $C_1$ and $C_2$ depend upon the conditions of $C$ at time $t = 0$
- $\alpha = \sqrt {\dfrac k m}$

### $x = x_0$ at time $t = 0$

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

Then the horizontal position of $C$ at time $t$ can be expressed as:

- $x = x_0 \cos \alpha t$

## Proof

From Motion of Cart attached to Wall by Spring, the horizontal position of $C$ is given as:

- $\dfrac {\d^2 \mathbf x} {\d t^2} + \dfrac k m \mathbf x = 0$

As both $k$ and $m$ are positive, $\dfrac k m$ can be expressed as:

- $\dfrac k m = \alpha^2$

for some $\alpha \in \R_{>0}$.

Hence:

- $\dfrac {\d^2 x} {\d t^2} + \alpha^2 x = 0$

and so from Linear Second Order ODE: $y'' + k^2 y = 0$:

- $x = C_1 \cos \alpha t + C_2 \sin \alpha t$

Hence the result.

$\blacksquare$

## Sources

- 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $\S 3.20$: Vibrations in Mechanical Systems: $(4)$