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Consider now the mass-spring oscillator depicted physically in Fig.D.3, and in equivalent-circuit form in Fig.D.4.
By Newton's second law of motion, the force applied to a mass equals its mass times its acceleration:
We have thus derived a second-order differential equation governing the motion of the mass and spring. (Note that in Fig.D.3 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
To simplify notation, denote the initial position and velocity by and , respectively. Solving for gives
denoting the modulus and angle of the pole residue , respectively. From §D.1, the inverse Laplace transform of is , where is the Heaviside unit step function at time 0. Then by linearity, the solution for the motion of the mass is
If the initial velocity is zero (), the above formula reduces to and the mass simply oscillates sinusoidally at frequency , starting from its initial position . If instead the initial position is , we obtain