`a_x = - omega ^2 "A" cos( omega * "t" + phi )`

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This equation computes the acceleration, `a_x`, of a harmonic oscillator as a function of time, t.

Inputs

• `omega` - the angular velocity
• A - amplitude, the maximum displacement of the oscillator
• `phi` - the phase angle defining the starting displacement at time t = 0
• t - the time at which we compute the acceleration

Definition

If the phase angle `phi` is zero, then `x_0 = A*cos(0) = A`.  So, `phi = 0`, means the oscillator is at maximum displacement.

The acceleration as a function of time can be found by taking the simple derivative of velocity:

     `a_x = dv_x/dt = (d^2x) /dt^2 = - omega^2 * A cos(omega*t + phi)`, since

     `x = A*cos(omega*t +phi)` ^[1]

     `dx/dt = -omega*A*sin(omega*t + phi)` and `(d^2x) /dt^2 = - omega^2 * A cos(omega*t + phi)`

Reference

Young, Hugh and Freeman, Roger.  University Physics With Modern Physics.  Addison-Wesley, 2008. 12th Edition, (ISBN-13: 978-0321500625 ISBN-10: 0321500628 ) Pg 426, Equation #13.16

1. ^ University Physics, page 425, eq. 13.13