Consider the force balance for a damped harmonic oscillator with mass m = 1 kg, a spring constant k > 0 in N/m, and a friction coefficient ξ > 0 in kg/s,

subject to some initial position x₀ and initial velocity v₀.

(a) Define the right-hand-side functions f₁ and f₂ and state the initial conditions if this is converted into a system of ordinary differential equations of the form

(b) What is the steady state for the system?

(c) Is the steady state stable or unstable? You do not need to show any work here if you can answer this from the physics. If you do not know the answer, do part (d) and then come back to this part.

(d) How does the classification of the steady state depend on the parameters ξ , m, and k? Report your result in terms of a friction coefficient ξ^∗ = ξ^∗(k,m) that governs the crossover.

(e) Sketch the phase plane for the two cases from part (d).

(f) Let’s assume that I integrated this system of equations using RK4.What is the maximum step size that I could take? Think carefully here with respect to the previous parts of the problem. You can report your answer as a fraction – you do not need to do the arithmetic.

(g) How does the maximum step size depend on the current values of x(t) and v(t)?

Transcribed Image Text:

d²x dt² = -kx - E dx dt