It will be graded on the completeness of the answers to the questions, the completeness and quality of the graphs, the explanation of concepts for each section, the correctness of the derivations and calculations, and the overall appearance of the lab.The answers should be complete enough for someone to understand who has not read the questions. The graphs will need a title and the axes will need to be labelled with the value being plotted and the units.At the beginning of each of the four parts, write a brief explanation of the concepts being explored.
Simple Harmonic Motion This lab is accompanied by the following video. https://youtu.be/At6U4LoZCoQ Objectives: • Review Hooke’s Law and the spring constant. • Study graphically, Simple Harmonic Motion. • Study graphically, the relationships between position, velocity, and acceleration for Simple Harmonic Motion. • Experimentally measure frequency and compare to the expected value for a known mass and spring constant. • Study graphically Newton’s Second Law as it relates to a mass and spring in Simple Harmonic Motion. • Review how kinetic energy, spring potential energy and mechanical energy relate to a mass and a spring. In this laboratory, you will watch a video of the experiment being performed and the data being taken. The data will be provided on a separate Excel document. You are asked to use either Excel, Google Sheets, or a better graphing program to analyze the data. All the graphs you submit should have a title and have the axes labelled with the variable and the units. The setup of the apparatus is shown in the following image. https://youtu.be/At6U4LoZCoQ Part A – Force from an Ideal Spring In this first part, we are going to stretch a spring and measure the stretching force vs the position of the cart. We will make the zero position for the cart the same as where the cart is when the spring is unstretched, then future measurements of the cart’s position will be the same as the amount the spring is stretched. The video will show different amounts of mass added to the hanger causing different spring stretches. Watch Part A in the video. 1. If the spring is an ideal spring, what is the relationship we expect for applied force versus spring stretch? 2. The video comments on the signs of the Force and the cart position. Why do the spring force and the cart have different signs? 3. Using the data provided in the Excel File, recreate the plot of Force versus Position. 4. Does the behavior of the graph reflect the relationship you mentioned in your answer to question 1? 5. In theory, what should the slope of this plot be equal to? (You are not meant to give a specific numerical value. This is meant to be a conceptual answer.) 6. What is the experimental value for the spring constant? 7. Suppose the zero for position of the cart were set at the location of the cart when the initial 100g mass was on it. How would this have changed the graph? 8. Would the slope of the graph have been different? Part B -- Simple Harmonic Motion In this part, you are going to look at the relationships between position, velocity, and acceleration for Simple Harmonic Motion. You will also compare the measured frequency to what is expected for this spring and mass. This time, we will make the zero position of the cart the equilibrium position, the position of the cart when it is at rest with the current hanging mass. Watch Part B in the video. 9. Write the general equations of position, velocity and acceleration that describe Simple Harmonic Motion. 10. Write the equation that relates the frequency of oscillation to the mass and spring constant. There are two data sets in the Excel file for this part. One is using a 150g hanging mass and the other is using a 250g hanging mass. Do steps 11 through 18 for each data set. 11. Using the equation you just wrote in question 10, predict the frequency of oscillation for the setup. 12. Plot the data for position, velocity, and acceleration using the same time axis for all three plots. (This is not simple in Excel or Google Sheets. Instead, you can make three separate plots that each have the same maximum and minimum values for the time axis. Then, on your lab writeup, line these graphs up under each other so that the time values on one graph line up with the time values on another graph.) 13. Explain how these curves support, or don’t support, the relationships between position, velocity, and acceleration for Simple Harmonic motion. 14. Find the period of oscillation from the graphs. To minimize uncertainty, measure the time for several periods in a row and divide by the number of periods. 15. Calculate an experimental value for the frequency. 16. How does your experimental frequency compare the expected frequency that you calculated in question 11? (Find the percent difference.) 17. What might be some of the sources of error? The Simple Harmonic Equation for a mass and spring was derived with only the spring force in the horizontal direction. This led to the general solution that you were asked to write in question 9. Friction plays a significant part for the Smart Cart on the mechanical track. Read up on the Damped Oscillator, section 15.7 in the text. 18. Considering friction and drag, should we expect our frequency measurement to be larger or smaller than the predicted frequency of question 11? Part C – Newton’s Second Law and the Simple Harmonic Oscillator In this part you will be looking at how Newton’s Second Law relates to Simple Harmonic Motion and the ideal spring. The spring is pulling to the left on the cart and gravity is pulling down on the hanging mass. Because of these forces, the cart accelerates back and forth, and the mass accelerates up and down. This time, the force sensor is zeroed at the cart’s equilibrium position. This means that the force sensor is not reading the full force of the spring. It is reading how much more or less the spring force is than the spring force at the cart’s equilibrium position. 19. Analyze this as a physics 4A style force problem. Name the distance the spring is stretched, y . Name the mass of the cart, cm , and the mass of the hanger, hm . Solve for the acceleration, a , as a function of spring stretch. Remember the cart measures the positive direction to the left. In other words, a positive acceleration is to the left. Note that when the system is at rest in the equilibrium position, the spring is already stretched. 20. Write an expression for how far the spring is stretch when the cart is at rest in the equilibrium position. Name this distance 0y . In the data run, the positions are measured from the rest (equilibrium) position of the cart. Name this position 1y . The distance that the spring is stretched is 0 1y y y= − . The minus sign is because a positive position for the cart decreases the spring stretch. (See the above image.) 21. Substitute your expression from question 20 into 0 1y y y= − . 22. Substitute your expression from question 21 into your expression from question 19. In our data run, we zeroed the force sensor at the cart’s equilibrium position. Name the total force to stretch the spring F ky= . Name the force to stretch the spring to the cart’s equilibrium position 0 0F ky= . Name the difference in the spring force when the cart from the equilibrium position, 1F . So that 0 1F F F= + 23. Write the expression for 1F in terms of 0y , 1y and k ? 24. Substitute your expression from question 23 into your expression from question 22. Re-write it as an expression for 1F . Watch Part C in the video. 25. Use the data for part C on the Excel spreadsheet to create a graph of acceleration and force, both with a shared time axis. 26. Use the data to create a graph of force versus acceleration. 27. Look at the two graphs that you created for question 25. Do they support the expression, in question 24, that you derived from Newton’s Second Law. Explain why they do or don’t support Newton’s Second Law. [Note: the force that was measured is the force 1F , the extra force from the cart’s displacement from the equilibrium position.] 28. Look at the graph of force vs. acceleration that you created for question 26. Does it support the expression, in question 25, that you derived from Newton’s Second Law. Explain why it does or doesn’t support Newton’s Second Law. 29. What should the slope of the force vs. acceleration ( 1 vs F a ) graph be? (You are not meant to give a specific numerical value. This is meant to be a conceptual answer.) 30. Using the graph of force versus acceleration, calculate an experimental value for the slope. 31. How does the experimental value for the slope compare to the expected value? 32. What are some possible sources of error? When the data taking program took its data, it connected the dots as they were added. Notice, in the video, that there are essentially two lines that were created. One of the lines corresponds to the cart moving in one direction and the other line corresponds to it moving the other direction. 33. Why do you think there are two separate lines when the expression of question 25 suggests one line? 34. Estimate the magnitude of the frictional force as it moves at its maximum speed through the equilibrium point. [Hint: the answer is based upon the two separate lines.] 35. What percentage is this of the maximum spring force? Part D – Mechanical Energy of a Simple Harmonic Oscillator In this last part, you will look at the kinetic energy, spring potential energy and mechanical energy of a Simple Harmonic Oscillator. There is no video for this part. You will use the data from either of the runs in part B. Refer to the image in part C. 36. What are the general equations for the kinetic energy of a moving mass, the gravitational potential energy of a