Laplace transforms:Mass-Spring Oscillator

Problem Statement:

An ideal mass m sliding on a frictionless surface, attached via an ideal spring k to a rigid wall. The spring is at rest when the mass is centered at x=0. Find the equation of motion that the spring mass follows.

Solution:

We first begin by setting up a few equations from Newton's laws.

By Newton's first law:

$\mathbf {F} =m\mathbf {a} \Rightarrow \mathbf {f} _{m}(t)=m{\ddot {x}}$