Question: figure 2 shows a translating mass on a frictionless surface...
Figure 2 shows a translating mass on a friction-less surface, acted on by an input force fa(t). The mass m is connected to ground via a linear damper b and spring k, arranged in series. The coordinate z2 tracks the relative position of the mass-less node between the damper and mass, shown with a small dark circle. The relative position of the mass (from rest/equilibrium) is z1. This lumped-element model could serve, for example, as a basic model of linear creep behavior of a polymer rope under load.
(a) Draw a free-body diagram for mass m clearly showing your sign conventions for the coordinates and external forces.
(b) Find the equation of motion for mass m. [Hint: You might find it easiest to consider the mass-less node to have a finite mass, say m0, and write its equation of motion, and then take the limit m0 → 0. The resulting expression can be used to complete the equation of motion for m, such that it does not depend on z2. Think about why coordinate z2 is not independent of coordinate z1.]