Moving Mass

Figure D.1 depicts a free mass driven by an external force along
an ideal frictionless surface in one dimension. Figure D.2
shows the *electrical equivalent circuit* for this scenario in
which the external force is represented by a voltage source emitting
*volts*, and the mass is modeled by an *inductor*
having the value
*Henrys*.

From Newton's second law of motion `` '', we have

Taking the unilateral Laplace transform and applying the differentiation theorem twice yields

Thus, given

- Laplace transform of the driving force ,
- initial mass position, and
- initial mass velocity,

If the applied external force is zero, then, by linearity of the Laplace transform, so is , and we readily obtain

Since is the unilateral Laplace transform of the Heaviside unit-step function

and since is the transform of , we find that the position of the mass is given for all time by

Thus, for example, a nonzero initial position and zero initial velocity results in for all ; that is, the mass ``just sits there''.

To summarize, this simple example illustrated use the Laplace transform to solve for the motion of a simple physical system (an ideal mass) in response to initial conditions (no external driving forces). The system was described by a differential equation which was converted to an algebraic equation by the Laplace transform.

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University