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Learning Introductory Physics with Activities

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

Section 15.4 Graphical Trigonometry

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_o\right)\hat{x}\tag{15.4.1} \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 \(x_{\text{max}}\text{,}\) period \(T\text{,}\) initial phase \(\phi_o\text{,}\) and equilibrium position \(\vec{x}_o\text{.}\) The argument of a trigonometric function is called the phase and \(\phi_o\) tells you the initial condition for the oscillation (the position at \(t=0\)).
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Figure 15.4.1. Position function of an object in Simple Harmonic Motion.
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Activity 15.4.1. Practice with Graphing.

Identify the equilibrium position \(\vec{x}_o\) and the initial phase \(\phi_o\) 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 \(\phi_o = 0\text{.}\) 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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Definition 15.4.2. Amplitude.

The amplitude of an oscillatory function is the function’s maximum deviation from equilibrium. It can be represented generically with \(A\) (as in the figure) or with a more evocative name for a particular function, such as \(x_{\text{max}}\) for position, which can help make the units of the amplitude more obvious.
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Definition 15.4.3. Oscillation Period.

The period \(T\) represents the time it takes to complete one full oscillation.
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Definition 15.4.4. Frequency.

The frequency is the number of cycles completed during each second. Since period \(T\) represents the time it takes to complete one cycle, period and frequency have an inverse relationship.
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\begin{equation} f = \frac{1}{T}\tag{15.4.2} \end{equation}
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Note 15.4.5. 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}\tag{15.4.3} \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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Exercises Activities

1. Physical Units.

Determine the SI units of amplitude and period. How can you tell?
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Tip.
Try checking the units of equation (15.4.1).
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Answer.
The equation as a whole is a position and has dimension of length with SI units of meters. Trigonometric functions are dimensionless. Therefore, the amplitude must have dimension of length for the equation to be true. The SI units of amplitude are meters. The argument of a trigonometric function must also be dimensionless with SI units of radians. Therefore, the fraction \(\frac{2\pi t}{T}\) must be dimensionless. The only way for this to be true is if the dimension of period is time. The SI units of time are seconds.
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2. Exploring the Period, Frequency, and Angular Frequency.

It is often useful to represent oscillatory motion in terms of the period \(T\text{,}\) the frequency \(f\) or the angular frequency \(\omega\text{.}\) Write the relationship between angular frequency, period, and frequency.
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Answer.
\begin{equation} \omega = 2 \pi f = \frac{2 \pi}{T} \tag{15.4.4} \end{equation}
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3. 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 equation (15.4.1) 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_o\right)\hat{x}\\ \amp = x_{\text{max}} \cos(2 \pi f t+ \phi_o)\hat{x}\\ \amp = x_{\text{max}} \cos(\omega t+ \phi_o)\hat{x} \end{align*}
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often \(\vec{x}(t) = x_{\text{max}} \cos(\omega t + \phi_o)\hat{x}\) is most useful for solving problems. This is the form of the position function we will use going forward when studying simple harmonic motion.
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References References

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