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Figure 6.6.1 created by Rebecka Tumblin.
Section 6.6 Relationship to Circular Motion
Consider a particle in uniform circular motion with angular velocity \(\omega(t) \text{.}\) The \(x\)-component of the particle’s position when the particle has angular position \(\theta(t)\) and radius \(r=A \) can be written using trigonometric relations as
\begin{equation*}
x = A\cos(\phi)
\end{equation*}
This is equivalent to the position function \(x(t) = A \cos(\omega_s t + \phi_o) \) for an oscillating block on a spring with initial condition \(\phi_o = 0 \) at \(t=0 \text{.}\) Recall the angular velocity is the rate of change of angular position
\begin{equation*}
\omega(t) = \frac{d\theta}{dt}
\end{equation*}
This equation is separable. A small change in angle can be written,
\begin{equation*}
d\theta = \omega(t) dt
\end{equation*}
If the particle is in uniform circular motion, then \(\omega(t) = \omega \) is constant in time. Integrating both sides of the relationship
\begin{equation*}
\int_{\theta_o}^{\theta} d \theta = \omega \int_{0}^{t} dt \rightarrow \theta(t) = \omega t + \theta_o
\end{equation*}
As time increases, the particle’s \(x\)-component can be written as
\begin{equation*}
x(t) = A \cos(\theta(t) = A \cos(\omega t + \theta_o)
\end{equation*}
As the particle moves in a circle, its projection onto the \(x\)-axis oscillates between \(x = \pm A \text{.}\) Thus, uniform circular motion is simple harmonic motion when projected onto one spatial axis.
This can also help you understand the meaning of the initial phase. The phase in oscillatory motion is given by \(\phi = \omega_s t +\phi_o\) and is analogous to the angle of a particle in circular motion \(\theta = \omega t +\theta_o\text{.}\) The initial phase \(\phi_o \) in oscillatory motion is analogous to the initial angle \(\theta_o\) when \(t = 0 \text{.}\) Setting \(t = 0 \) in the position function for a simple harmonic oscillator, you can find the initial \(x\)-position
\begin{equation*}
x_0 = A \cos(\theta_o)
\end{equation*}
Thus different values of the phase constant represent different starting points on the circle and correspond to different initial conditions of the motion.
References References
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*need to find a gif image that can be referenced* Image from Kathy’s website is hosted on imgur. Not sure how to reference? Maybe get Jadon to create a gif? We could also reach out to (http://www1.phys.vt.edu/~takeuchi/Tools/) to see if we can use the image gif.
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Simple harmonic motion animation by Chetvorno on Wikimedia Commons published under Creative Commons CC0 1.0 Universal Public Domain Dedication.
