What is the differential equation of motion in Problem 53 if kinetic friction (but no ot...

What is the differential equation of motion in Problem 53 if kinetic friction (but no other damping forces) acts on the sliding mass? [Hint: Assume that the magnitude of the force of kinetic friction is f_k = μmg, where mg is the weight of the mass and the constant μ >0 is the coefficient of kinetic friction. Then consider two cases: x’ > 0 and x’ < 0. Interpret these cases physically.]

Reference:

Problem 53:

Suppose a mass m lying on a flat, dry, frictionless surface is attached to the free end of a spring whose constant is k. In FIGURE 3.R.2(a) the mass is shown at the equilibrium position x =0; that is, the spring is neither stretched nor compressed. As shown in Figure 3.R.2(b), the displacement x(t) of the mass to the right of the equilibrium position is positive and negative to the left. Derive a differential equation for the free horizontal (sliding) motion of the mass. Discuss the difference between the derivation of this DE and the analysis leading to (1) of Section 3.8.