Oscillating Spring Lab



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Trigonometric Models

Oscillating Spring Lab Extensions to Calculus

Analyzing Your Equation:

1. Give values and explain the physical meaning of each of the following from your lab, including the units and numbers in your explanation. [For example, the horizontal shift indicates that the motion detector didn’t begin collecting data right as the spring passed through equilibrium, but 0.25 seconds before it was to reach it’s equilibrium position]. Each explanation should have to do with the actual behavior of the spring.

a. Amplitude: b. Period: c. frequency

d. Horizontal shift: e. Vertical shift:

2. Recall that the first derivative of position will yield an equation for velocity. Use the position vs. time equation you found in your lab to answer the following:

a. Find an equation for the velocity of the spring. What are the units of this velocity?

b. What is the fastest the spring moves? Explain how you know from your velocity equation.

c. Mathematically find at least two times when the spring is moving at this fastest velocity (one negative, one positive velocity). Show your work. [Hint: The velocity equation should be a cosine function. Cosine is largest (absolute value) when the argument equals 0, π, 2π, 3π, etc.]

d. Find the times you solved for above on your printed position vs. time graph. Put a triangle around these times on your printout and label. Does the spring travel the fastest when it is fully stretched, going through equilibrium, fully compressed, or all of the above? Explain.

3. Recall that the second derivative of position will yield an equation for acceleration. Use the position vs. time equation from your lab to answer the following:

a. Find an equation for the acceleration of the spring. What are the units of this acceleration?

b. What is the greatest value of the acceleration of the spring? Explain how you know from your acceleration equation.

c. Mathematically find at least two times when the spring is moving at this greatest acceleration (one negative, one positive acceleration). Show your work. [Hint: The acceleration equation should be a sine function. Think about when the sine is largest and set the argument of the function equal to the appropriate value.

d. Find the times you solved for above on your printed graph by using the TRACE feature on your calculator. Put a square around these times on your printout and label where the magnitude of the spring’s acceleration it the greatest. Does this occur when it is fully stretched, going through equilibrium, fully compressed, or all of the above? Explain.

4. Use the printouts of your position vs. time and velocity vs. time graphs to answer the following. How can you use your velocity graph to determine where the original position graph is:

a. Increasing

b. Decreasing

c. Has a relative maximum

d. Has a relative miniumum

5. Use the printouts of your position vs. time and acceleration vs. time graphs to answer the following. How can you use your acceleration graph to determine where the original position graph is:

a. Increasing or Decreasing

b. Concave Up

c. Concave Down

d. Has and inflection point (changes concavity)

6. Is it possible to use your velocity vs. time graph to predict:

a. Where the objects initial position is (position at time zero)?

b. Where the original position vs. time graph is Concave Up?

c. Where the original position vs. time graph is Concave Down?

d. Where the original position vs. time graph has an Inflection point (changes concavity)?

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