# 3.1: Getting’ the Swing

Difficulty Level: At Grade Created by: CK-12

This activity is intended to supplement Calculus, Chapter 2, Lesson 4.

## Part 1 – Warm-up

In \begin{align*}y1\end{align*}, enter \begin{align*}\cos(x)\end{align*}. Press \begin{align*}F3\end{align*} and select 7: ZoomTrig. Use the graph to answer the following questions.

1. What is the range?

2. What is the amplitude? \begin{align*}A=\end{align*}

3. What is the period? \begin{align*}T=\end{align*}

Now change your calculator mode to split screen. Press 3 and select TOP-BOTTOM for Split Screen. For Split 1 App, select \begin{align*}Y=\end{align*} Editor. For Split 2 App, select Graph. In \begin{align*}y2\end{align*}, enter an equation in the form \begin{align*}y = A \cdot \cos(B \cdot x) + C\end{align*}, where \begin{align*}A\end{align*}, \begin{align*}B\end{align*}, and \begin{align*}C\end{align*} are integers. Press \begin{align*}2 + \alpha\end{align*} to swap applications to see the graph screen update. Press \begin{align*}2 + \alpha\end{align*} again to go back to the \begin{align*}Y=\end{align*} Editor to modify your equation. To answer the following questions, modify the corresponding variable to observe the changes each variable has to the equation.

4. Describe the effect of increasing \begin{align*}A\end{align*}.

5. Describe the effect of increasing \begin{align*}C\end{align*}.

6. Describe the effect of increasing \begin{align*}B\end{align*}.

7. What is the relationship between \begin{align*}B\end{align*} and the period, \begin{align*}T\end{align*}?

8. If a positive \begin{align*}D\end{align*} shifts the graph to the right \begin{align*}D\end{align*} units, what is the general sinusoidal equation for which this is true?

## Part 2 – Collect & Analyze Data

You will collect data of a pendulum swinging. Using the skills reviewed in the warm-up, write a cosine function that models the data collected. Estimate the amplitude and period and phase shift, \begin{align*}D\end{align*}, to the nearest tenth. If a motion detector is not available, use the lists time, distance, and velocity from your teacher and graph a function to model that data. To collect data, complete the following steps:

• Using an I/O cable, connect the motion detector to the graphing calculator.
• On the HOME screen, run the Ranger program. Select 1:Setup/Sample…. Use the settings that appear to the right and press ENTER.
• Position the motion detector so that it is facing the pendulum, swing the pendulum, and press ENTER to begin collecting data.

• If your data doesn’t look sinusoidal, press ENTER and select 3:Repeat Sample to repeat the trial. Then press ENTER to begin collecting data again.
• Model the distance-time data with a function. Derive the velocity and acceleration equations. Select 7:Quit when you are finished.

Record your position, velocity and acceleration equations for your experiment data here:

\begin{align*}y=\\ \\ \\ v=\\ \\ \\ a=\end{align*}

Confirm your position and velocity equations by graphing them. To confirm your position equation, enter your equation in \begin{align*}y1\end{align*} and select to show Plot 1 as shown to the right. To plot the velocity-time graph, use \begin{align*}L1\end{align*} for time and \begin{align*}L3\end{align*} for velocity. For the acceleration-time graph, use \begin{align*}L1\end{align*} for time and \begin{align*}L4\end{align*} for acceleration.

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