Hanging by a Slinky In this activity, we will study the motion of a mass on a spring (in this case, a washer on a slinky). We will do this by investigating the height of the washer overtime. A motion detector will be used to record the height of the washer with respect to time. We will also study what function represents this motion. CCSS Alignment: Interpreting Functions (F-IF.4) For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Interpreting Functions (F-IF.7) Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Building Functions (F-BF.3) Identify the effect on the graph of replacing by, and for specific values of (both positive and negative); find the value of given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Equipment: Trigonometric Functions (F-TF.5) Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. Table/desk Table Clamp Slinky Tape Washer Stopwatch Tape Measure/Yard Stick Motion Detector with CBL program Graphing Calculators
Experiment: 1. Begin by placing one end of the slinky in the table clamp you may need to tape the slinky together so it does not fall off of the clamp (see figure 1 below) and then fix the table clamp to the table (see figure 2 below). Note: Do not overstretch the slinky! Figure 1 Figure 2 2. Tape the washer to the free end of the slinky. Let the washer hang freely on the slinky. Measure and record the distance from the washer to the ground in the table below. 3. Place the motion detector facing upwards directly under the washer (see figure 3 on the right). 4. In a moment, you will be stretching the slinky down to a desired height above the motion detector and letting it go. Record this height to make sure you release the washer from this height for each trial in the table below. 5. Stretch the slinky to this height, let go, and record the time it takes for the washer to return back to this position in the table below (this will be consider one full oscillation). Height of washer at rest Release Height Time for One Oscillation
6. Predict how you think the motion of the washer will behave after you release the slinky from the specified height? 7. Using the height of the washer when the slinky is at rest for the zero point, sketch a time versus height graph for the washer. 8. Now, we will use the motion detector to graph the motion of the washer with respect to time. On the CBL program, set the options as 1: MOTION, 2: DISTANCE-RT, and 1: AUTO SCALE. 9. Stretch the slinky from the same release point, start the motion detector, and let go. 10. Sketch the graph you observe on the CBL program. Analysis: 11. How does the graph compare to your predictions in parts and? What is the same? What is different? 12. Does your graph appear to represent a periodic function? Why or why not? 13. At what point in the washer s motion represents the maximum on your graph? At what point represents the minimum?
14. Around what y-value is the graph centered? 15. What is the period of the function? 16. What is the amplitude of the graph? 17. How are these values related to the values in your table? 18. Determine a function that represents the time vs. height graph of the washer. How did you come up with this equation? 19. Using the graph, create a table that provides x-values of the graph and their corresponding y- values. x y 20. Input the data from the table into you graphing calculator. Use SinReg option to fit a sine regression equation to the data. Write this equation below.
21. Using this equation that is of the form determine the following in regards to this experiment: a) What does represent? b) What does represent? c) What does represent? d) What does represent? 22. How does this sine regression equation relate to your function from part? 23. How does the sine regression equation relate the parent function? Conclusions: 24. Describe the motion of the washer. When was it the highest? When was it the lowest? Did the washer undergo acceleration or did it move with a constant velocity? Explain. 25. Is the washer ever at rest during its time of motion? Does it move faster at some points than others? Explain. 26. In general, how does each constant in the function affect the graph of the function? 27. Does the graph given by the CBL program represent a perfect sine function? How does the graph behave overtime? Why do you think this is? This activity was adapted from Swing Low by Charlene Beckmann, an activity relating the sine function to the motion of a pendulum.