Chapter 5, Problem 27RE

Suppose the mass m in the spring/mass system in Problem 25 slides over a dry surface whose coefficient of sliding friction is μ > 0. If the retarding force of kinetic friction has the constant magnitude f_k = μmg, where mg is the weight of the mass, and acts opposite to the direction of motion, then it is known as coulomb friction. By using the signum function

$\mathrm{sgn}(x^{\prime })=\{\begin{array}{cc}\begin{array}{c}-1,\\ 1,\end{array}& \begin{array}{c}{x}^{\prime }<0(\text{motion}\text{to}\text{left})\\ x^{\prime }>0(\text{motion}\text{to}\text{left})\end{array}\end{array}$

determine a piecewise-defined differential equation for the displacement x(t) of the damped sliding mass.

25. 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 5.R.2(a) the mass is shown at the equilibrium position x 5 0, that is, the spring is neither stretched nor compressed. As shown in Figure 5.R.2(b), the displacement x(t) of the mass to the right of the equilibrium position is positive and negative to the left. Determine a differential equation for the displacement x(t) of the freely sliding mass. Discuss the difference between the derivation of this DE and the analysis leading to (1) of Section 5.1.

Figure 5.R.2 Sliding spring/mass system in Problem 25