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An analysis of the velocity and acceleration of a particle undergoing simple harmonic motion

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Simple harmonic motion is the periodic motion of a particle which follows a sinusoidal oscillation about an equilibrium position. A typical example is the motion of a body on a spring which experiences the elastic restoring force. In order to define the velocity and acceleration of a simple harmonic motion, we introduce the displacement $s$ with the following equation:

$s = m \cos nt$

(4)

where $m$ and $n$ are constants.

\calc{n^2*(m^2-s^2)} "Instant calculator eq(4)"

Taking into account the definition of velocity (for a more detailed discussion on velocity and acceleration see Linear Velocity and Acceleration ) and equation (#1), we can write:

$v = \frac{ds}{dt} = -mn \sin nt$

(5)

We can express $v$ from (#2) only in terms of $s$ by squaring the equation, which gives:

$v^2 = m^2n^2\sin^2 nt = m^2n^2(1-\cos^2 nt) = n^2(m^2-m^2\cos^2 nt)$

from which, by also considering (#1), we obtain the squared velocity:

$v^2 = n^2(m^2-s^2)$

(6)

By differentiating (#3) with respect to $s$ , we get:

$2v \frac{dv}{ds} = n^2(-2s)$

(7)

Considering one of the forms in which the acceleration can be expressed and using (#4) we obtain:

$a = v\frac{dv}{ds} = \frac{n^2(-2s)}{2}$

from which:

$a = -n^2s$

(8)

The same equation for the acceleration of a simple harmonic motion can also be obtained by considering an alternate expression form for the acceleration and also using equation (#2), when we have that:

$a = \frac{dv}{dt} = -mn^2 \cos nt$

(9)

Coupling (#6) with (#1) leads again to the final equation for the acceleration highlighted in (#5).

\calc{-n^2*s} "Instant calculator eq(7)"

Simple Harmonic Motion