Harmonic Oscillator - Acceleration

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on

Jul 24, 2020, 6:28:07 PM

Created by

on

Dec 17, 2014, 9:04:33 PM

a_{x} = - ω^{2} A cos (ω \cdot t + ϕ)

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

$(omega)"Angular Velocity"$

$(A)"Amplitude"$

$(t)"Time"$

$(phi)"Phase Angle"$

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

^ University Physics, page 425, eq. 13.13

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Harmonic Oscillator - Acceleration

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