In SHM, the general equations for position, velocity, and acceleration are:

x(t) = A cos(ωt + φ)

v(t)

=

dx dt

= -Aω sin(ωt + φ)

a(t)

=

d2x dt2

= -Aω2 cos(ωt + φ)

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

Whatever is multiplying the sine or cosine represents the maximum value of the quantity. Thus:

x_max = A

v_max = Aω

a_max = Aω²

Graphing the position, velocity, and acceleration allows us to see some of the general features of simple harmonic motion:

• The speed is maximum when the object passes through the equilibrium position (x = 0)
• The acceleration is opposite in direction, and proportional to, the displacement
• All three graphs have the same frequency - they just differ by phases of 90 degrees.

The first set of graphs is for an angular frequency ω = 1 rad/s. The second set of graphs 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 did we make in this case?

1. We decreased the spring constant (4/30) (13%)
2. We increased the mass (7/30) (23%)
3. We could have done one or the other, you can't tell the difference (19/30) (63%)