Problems deal with a nuiss−spring−dashpot system having position function x(t) satisfying...

Problems deal with a nuiss−spring−dashpot system having position function x(t) satisfying Eq. (4). We write x0 = x(0) and v0 = x′(0) and recall that p = c/(2m), ω02 = k/m, and ω12 = ω02 − p2. The system is criticallydamped, overdamped, or underdamped, as specified in each problem.

(Underdamped) A body weighing 100 Ib (mass m = 3.125 slugs in fps units) is oscillating attached to a spring and a dashpot. Its first two maximum displacements of 6.73 in. and 1.46 in. are observed to occur at times 0.34 s and 1.17 s, respectively. Compute the damping con­stant (in pound-seconds per foot) and spring constant (in pounds per foot).

Differential Equations and Determinism

Given a mass m, a dashpot constant c, and a spring constant k, Theorem 2 of Section 2.1 implies that the equation

has a unique solution for t ≥ 0 satisfying given initial condi­tions x (0) = x0, x′(0)=v0. Thus the future motion of an ideal mass-spring-dashpot system is completely determined by the differential equation and the initial conditions. Of course in a real physical system it is impossible to measure the param­eters m, c, and k precisely. Problems explore the resulting uncertainty in predicting the future behavior of a physical system.