Laplace transforms: Critically Damped Motion

Using the Laplace Transform to solve a spring mass system that is critically damped

Problem Statement

An 8 pound weight is attached to a spring with a spring constant k of 4 lb/ft. The spring is stretched 2 ft and rests at its equilibrium position. It is then released from rest with an initial upward velocity of 3 ft/s. The system contains a damping force of 2 times the initial velocity.

Solution

Things we know

$m=\frac{8}{32}=\frac{1}{4}slugs$

$\text{k=4}$

$\text{Damping constant C=2}$

$\text{x(0)=0 and x'(0)=-3}$

$\text{Therefore the equation representing this system is}$

$\frac{1}{4}\frac{d^{2}x}{d{t}^2}=-4x-2\frac{dx}{dt}$

$\text{Now we put the equation in standard form}$

$\frac{d^{2}x}{d{t}^2}+8\frac{dx}{dt}+16x=0$

$\text{Now that we have the equation written in standard form we need to send}$ $\text{it through the Laplace Transform}$