Phase space diagram for a spring in simple harmonic motion

[Yosty22]

Homework Statement

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.

Homework Equations

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]

 

[TSny]

Therefore, our equations become: x(t) = 0.4cos(wt) and x'(t) = 0.4wsin(wt).

These equations give enough information to plot points on the trajectory in phase space.

If you want to identify the shape of the trajectory, then write the energy equation in terms of the symbols x and x'. (x' is the same as "x dot".) Plug in values for all parameters and simplify to get a relation between x and x'. Try to identify the shape of the trajectory from this relation.