Consider now the mass-spring oscillator:
Newton's second law of motion:
Hooke's law for ideal springs:
Newton's third law of motion:
We have thus derived a second-order differential equation governing the motion of the mass and spring. (Note that is both the position of the mass and compression of the spring at time .)
Taking the Laplace transform of both sides of this differential equation gives
Solving for gives
denoting the modulus and angle of the pole residue , respectively.