A mass m = 750 g is connected to a spring with spring constant k = 1.5 N/m. At t = 0 the mass is set into simple harmonic motion (no damping) with the initial conditions represented by the point P in the phase space diagram at the right. **(This phase space diagram has nothing on it, that is my job to fill in. The point p is located at (x, x dot) = (0.4,0).
a. Using the given information, sketch an accurate phase space plot for the oscillator. Explain your reasoning and show all work.
The Attempt at a Solution
I know from studying simple harmonic motion in the past that the solution comes of the form x'' = -w2x, giving the solution x(t) = Acos(wt) + Bsin(wt)
Since I know that at t = 0, it is released at x = 0.4, so x(0) = 0.4. This gives A = 0.4. Next, I know that at t = 0, x dot (dx/dt) = 0. This gives B = 0.
Therefore, our equations become: x(t) = 0.4cos(wt) and x'(t) = 0.4wsin(wt).
Next, I know that the total energy (call it E) is equal to .5mv2 + .5kx2. Since V = x'(t), I can square my x'(t) equation to get the kinetic energy and square my x(t) equation to get the potential energy term. This gives me:
E = (.5m)(.16cos2(wt)) + (.5k)(.16w2sin2(wt))
However, I am unsure as of how to make this into a phase space diagram. Any help would be appreciated.[/sup]