3.1: Gettin’ the Swing
This activity is intended to supplement Calculus, Chapter 2, Lesson 4.
ID: 11691
Time Required: 40-45 minutes
Activity Overview
In this activity, students will investigate sinusoidal functions, collect data from a swinging pendulum, model the data with a cosine function, and take the derivative to find the velocity and acceleration.
Topic: Model Sinusoidal Data
- Write an equation for a sinusoidal function.
- First and second derivative to determine velocity and acceleration
Teacher Preparation and Notes
- This activity utilizes a CBR or CBR 2, which records the movement of the pendulum and transmits it to the TI-89.
- Students will need to know that the derivative of position is velocity, and the derivative of velocity is acceleration. Students are also expected to know the derivative of trigonometric functions.
- The activity is designed to be a student-centered discovery/review involving data collection. If a motion sensor is not available, then use the data from the lists provided.
- In order to collect data using the Calculator-Based Ranger \begin{align*}2^{\text{TM}}\end{align*}, use the Ranger program. The Ranger program should be in the CBR device. To download it onto your calculator, connect the two, press \begin{align*}2nd LINK \end{align*}, arrow over to RECEIVE, and press ENTER.
- To download the calculator files, go to http://www.education.ti.com/calculators/downloads/US/Activities/Detail?id=11691 and select time, distance, and velocity.
Associated Materials
- Student Worksheet: Getting the Swing http://www.ck12.org/flexr/chapter/9727
- time.89l, distance.89l, velocity.89l
- Calculator-Based Ranger (CBR) \begin{align*}2^{\text{TM}}\end{align*} or Vernier GO! \begin{align*}^{\text{TM}}\end{align*} Motion
- pendulum (a ball hanging from a string)
- meter stick
- stop watch
Part 1 – Warm-up
Students begin the activity by answering questions to help them visualize and review characteristics of sinusoidal functions.
Next, students will translate the cosine function to explore the effects of varying the parameters of the general sinusoidal functions. Students will use a split screen to view the equation and graph of the function at the same time. Students should see that as \begin{align*}B\end{align*} increases the period decreases. Students should also see that this relationship involves \begin{align*}2 \pi\end{align*}.
Further Discussion
- Ask students what the derivative of \begin{align*}y = \cos(x)\end{align*} is. When the \begin{align*}A = 1\end{align*}, \begin{align*}B = 1\end{align*}, \begin{align*}C = 0\end{align*}, and the \begin{align*}D\end{align*} is set at \begin{align*}\frac{\pi}{2}\end{align*}, the graph looks likes \begin{align*}y = - \sin(x)\end{align*}.
Solutions
- range is from -1 to 1
- amplitude \begin{align*}A = 1\end{align*}
- period \begin{align*}T = 2 \pi\end{align*}
- \begin{align*}A\end{align*} makes the amplitude bigger
- \begin{align*}C\end{align*} produces a vertical shift
- \begin{align*}B\end{align*} changes the frequency
- \begin{align*}B = \frac{2\pi}{T}\end{align*}; The inverse function graphed on 3.2 confirms this.
- \begin{align*}y = A \cdot \cos(B \cdot (x - D)) + C\end{align*}
Part 2 – Collect & Analyze Data
This part of the activity begins with students plugging in the CBR \begin{align*}2^{\text{TM}}\end{align*} or GO!\begin{align*}^{\text{TM}}\end{align*} Motion with an I/O cable. (Note: the mini USB cables will not work with this program.) Students should start the Ranger program. The experiment should be set up to match the settings shown on the student worksheet. When ENTER is pressed, data is collected. If there is a wall, chair, person or some other object that “distracts” the motion detector, the data may not look as good as depicted on the right. Students can repeat the experiment by pressing ENTER and select 3:Repeat Sample. Have student repeat the data collection until they are pleased with the shape of the graph.
Be sure to use a smooth ball so that the sonic pulse is reflected off the surface well.
Students can see the connection to real life more clearly if meter sticks and stop watches are available. Students can measure the period with the stop watch. With the meter stick they can see how far the pendulum is from the motion detector when the ball is at rest. This will be their \begin{align*}C\end{align*}. Also they can measure how far they pull the pendulum. This is the amplitude. The phase shift is related to when they pressed start to collect the data.
Solutions
The graphs and equations for the experiment will vary. The following equations correlate to the data in the given lists.
\begin{align*}y & = 0.12 \cos \left( \frac{2\pi}{1.1} \cdot(x - 0.38) \right) + 0.35 \\ v & = -0.12 \left(\frac{2\pi}{1.1}\right) \sin \left(\frac{2\pi}{1.1} \cdot(x - 0.38) \right) \\ a & = -0.12 \left(\frac{2\pi}{1.1}\right)^2 \cos \left(\frac{2\pi}{1.1} \cdot(x - 0.38) \right)\end{align*}
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