Motion of Cart attached to Wall by Spring with no Damping

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 $\mathbf x$.

Suppose the damping force $c$ is zero.

Then the motion of $C$ can be described by the second order ODE:

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

Proof

From Motion of Cart attached to Wall by Spring under Damping:

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

The result follows by setting $c = 0$.

$\blacksquare$

Sources

• 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $1$: The Nature of Differential Equations: $\S 1$: Introduction: $(5)$