### Graphs of Position, Velocity, and Acceleration

In simple harmonic motion (SHM), the general equations for position, velocity, and acceleration are:

x(t) = A cos(ωt + φ)

v(t) =
 dx dt
= - A ω sin(ωt + φ)
a(t) =
 d2x dt2
= - A ω2 cos(ωt + φ)

The phase angle φ is determined by the initial position and initial velocity.

The constant multiplying the sine or cosine represents the maximum displacement. Thus:

xmax = A      vmax = Aω      amax = Aω2

Graphing the position, velocity, and acceleration reveal general features of SHM:

• Maximum speed when the mass is at the equilibrium position (x = 0)
• Acceleration is opposite in direction, and proportional to, the displacement
• x, v, and a have the same frequency - they just differ by phases of 90o.

The first set of graphs is for ω = 1 rad/s. The second set is for ω = 0.6 rad/s. This change of ω is accomplished either by decreasing the spring constant or by increasing the mass. Which change was made in this case?

1. Decreased the spring constant
2. Increased the mass
3. Either, you can't tell the difference