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Learning Introductory Physics with Activities

Paul J. Emigh, Rebecka Tumblin, Kathryn Hadley, Danielle Skinner

Section 16.4 Representing Oscillations

The general form for the position of a system undergoing simple harmonic motion is a sinusoidal function of time.
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\begin{equation*} \vec{x}(t) = \vec{x}_o + x_{\text{max}} \cos \left(\frac{2\pi t}{T} + \phi_i\right)\hat{x} \end{equation*}
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Here, the notation \(\vec{x}(t)\) indicates that the position \(\vec{x}\) is a function of time \(t\text{.}\) This position is a cosine function with Amplitude \(x_{\text{max}}\text{,}\) Oscillation Period \(T\text{,}\) initial phase \(\phi_i\text{,}\) and equilibrium position \(\vec{x}_o\text{.}\) The argument of a trigonometric function is called the phase and \(\phi_i\) tells you the initial condition for the oscillation at \(t = 0\text{.}\)
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Figure 16.4.1. Position function of an object in Simple Harmonic Motion.
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Note 16.4.2. Analogue to Circular Motion.

Recall that a particle in uniform circular motion has an angular speed \(\omega\) related to the period of motion by
\begin{equation*} \omega= \frac{2 \pi}{T} \end{equation*}
This relation holds true in simple harmonic motion. When describing oscillatory motion, \(\omega \) is called the angular frequency rather than angular velocity. The physical units of \(\omega\) are given in rad/s. Note that since radians are a dimensionless unit, often the rad is dropped for the SI units when \(\omega\) is used together with other quantities. Most of the symbolic work you will do when representing oscillations will be done in terms of the angular frequency \(\omega\text{,}\) rather than the ordinary frequency \(f\text{.}\)
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Subsubsection Activities

Activity 16.4.1. Practice with Graphing.

Identify the equilibrium position \(\vec{x}_o\) and the initial phase \(\phi_i\) in the graph above. Explain how you found them.
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Solution.
The initial phase can be found by looking at where in its cycle the cosine function is at \(t = 0\text{:}\) here, it is at a maximum, which means \(\phi_i = 0\text{.}\) The equilibrium position can be found by looking halfway between the maximum and minimum values, which here is at \(\vec{x}_o = 0\text{.}\) When possible, it is extremely common to choose both of these quantities to be zero to simplify the position equation.
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Activity 16.4.2. Exploring the Position Function.

It is often useful to be able to represent the position function in terms of the period \(T\text{,}\) the frequency \(f\) or the angular frequency \(\omega\text{.}\) Rewrite the position equation in terms of the frequency \(f\) and the angular frequency \(\omega\text{.}\)
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Answer.
You can write the position relationship as follows:
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\begin{align*} \vec{x}(t) \amp = x_{\text{max}} \cos \left(\frac{2\pi t}{T} + \phi_i\right)\hat{x}\\ \amp = x_{\text{max}} \cos(2 \pi f t+ \phi_i)\hat{x}\\ \amp = x_{\text{max}} \cos(\omega t+ \phi_i)\hat{x} \end{align*}
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often \(\vec{x}(t) = x_{\text{max}} \cos(\omega t + \phi_i)\hat{x}\) is most useful for solving problems. This is the form of the position function you will use going forward when studying simple harmonic motion.
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References References

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