Question

Our goal in this question is to model the behaviour of a mass on a spring with and without friction. In both cases, we’ll seek the solution to a differential equation of the form: a* (d^2x /dt2) + b (dx /dt)+ cx = 0 where x = x(t) is a function of time t and a, b, and c are constant real numbers.

(a) Show that the differential equation is satisfied by x(t) = e^rt*(C1 cos(ωt) + C2 sin(ωt)) where r = − b /2a and ω = (√ 4ac − b^2/ 2a) . (Note, this equation for x(t) is the general solution to the DE when b^2 − 4ac < 0 so that ω is a non-zero real number.) Now, consider a mass m attached to a spring with spring constant k (which quantifies the stiffness of the spring). The spring exerts a force on the mass proportional to the extension/compression of the spring from equilibrium. Let us denote the extension of the spring at a time t by x(t). Ignoring friction, Newton’s second law gives the differential equation m* (d^2x/ dt^2) + kx = 0

(b) Determine the general solution to this DE using information given in the part (a) of this question.

(c) Choose initial conditions (that do not give x(t) = 0) and set m = 4 + 1 and k = 9 + 1 where 4 and 9 are the seventh and eighth digits of your student number. Sketch the corresponding solution for values of t ≥ 0. 2

(d) In your own words, describe the motion of the mass with respect to time. The above model of a mass on a spring is plausible but not very realistic: in practice, friction will act to decelerate the mass. Including a frictional force with magnitude proportional to velocity (i.e., dx/dt), Newton’s second law becomes m* (d2^x/ dt^2) + q*(dx/ dt) + kx = 0 where q is the damping coefficient representing friction.

(e) Determine the general solution to this DE using information given in part (a) of this question.

(f) Using the same initial conditions and values for m and k from part (c), set q = √ 4mk − 1 (so that q^2 − 4mk < 0) and sketch the corresponding solution for values of t ≥ 0. (g) In your own words, describe the motion of the mass with respect to time. Explain what is different in comparison to what you found when there was no friction (i.e., parts (b) - (d)).

Answer 1