### 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

Thus, given

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

*all*linear, time-invariant (LTI) systems. For nonlinear and/or time-varying systems, Laplace-transform analysis cannot, strictly speaking, be used at all.

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

^{D.3}Similarly, any initial velocity is integrated with respect to time, meaning that the mass moves forever at the initial velocity.

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.

**Next Section:**

Mass-Spring Oscillator Analysis

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Differentiation