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

x(t) = A cos(ωt + δ)

v(t)

=

dx dt

= -Aω sin(ωt + δ)

a(t)

=

d2x dt2

= -Aω2 cos(ωt + δ)

The phase angle δ 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.