Simple Harmonic Motion of Mass-Spring Systems Lots of things vibrate or oscillate. A vibrating tuning fork, a moving child’s playground swing, and the loudspeaker in a radio are all examples of physical vibrations. There are also electrical and acoustical vibrations, such as radio signals and the sound you get when blowing across the top of an open bottle. A simple system that vibrates in SHM is a mass hanging from a spring with a mass ms and a spring constant k. The force applied by an ideal spring is proportional to how much it is stretched or compressed, given by Hooke’s law:

F kx

Given this force behavior, the up and down motion of the mass is called simple harmonic and the position can be modeled with y Asin t  (1) In this equation, y is the instantaneous vertical displacement from the equilibrium position, A is the amplitude (maximum displacement from the equilibrium position of the mass) of the motion, f is the frequency of the oscillation, t is the time, and  is a phase constant. If the mass of the spring is ignored the period of oscillation is given by: m 2 (2) k where m is the mass of the spring and k is the spring constant. If the mass of the spring ms is not ignored, the period of oscillation can be estimated as:

mm /3 2 s (3) k If there are damping forces like air resistance acting then the position and frequency are given by:

b 2  t kb x Ae2m cos( t   ),     (4) mm2

The objective of the lab is to test these equations.

Equipment: Lab Jack, Table clamp, Long Rod, short rod, Lab Pro, Motion detector & cables, right angle clamp, spring, wire basket, mass hanger, 50 g, 100 g and 200 g masses (500 grams total each)

Lab Report: Open a Word document and an Excel data sheet for your group lab report. Use the Logger Pro file in the Physics 41, Lab 1 folder. Your lab report will need to include a brief summary. Have the lab title, number, date, lab section, lab partner names and summary on the first page. Include sample logger pro files showing data. Make it clear and pretty.

Experimental Procedure

Part 1: Finding the Spring Constant 1. Weigh the mass of the spring first. Set the equilibrium position of the spring – hanger system by zeroing the motion detector. Do this by allowing the mass to hang free and at rest. The zero is under the experiment menu in Logger Pro. Collect data to confirm that the motion detector is reading the bottom of the hanger as the zero point. 2. Add 100 grams to mass hanger and let the spring system come to rest. Measure the distance to the hanger by collecting data. THE MASS IN NOT OSCILLATING!!!! Use the statistics analysis to find the mean value of the distance. Copy and paste a sample logger pro graph (only 1) into your report. Repeat for 100 g or so increments for a total of 5 data points. Calculate the value of k for each weight (do NOT include the mass of the hanger!) and displacement. USE EXCEL FORMULAS to do the calculations. Average the values of k and find the average deviation between them. This will be the uncertainty in your k. Plot the applied force (the added weights – not the hanger!) vs the displacement and do a linear fit to find the slope of the line. This will give you the force constant, k, for the spring. Copy and paste the F vs x graph into your lab along with the linear fit information. Write the value of k in standard form: kk Part 2: Finding the Period of Oscillation 1. Now remove all but 100 g from the hanger. Lift the mass upward about 5 cm and release it. The mass should oscillate along a vertical line only. Click to collect data. Examine the graphs. The pattern you are observing is characteristic of simple harmonic motion. Do a sinusoidal curve fit for the graph. Copy and paste the graph and function into your lab report. Find the frequency of the mass spring system from the coefficient of t in the sin function. 2. Repeat with 4 more mass combinations on the hanger. For each weight, find the frequency from the sin function. Enter this into your table. 3. Calculate period of spring-hanger system for each weight using the k you derived above from the slope of the linear fit and equations 2 and 3. USE EXCEL FORMULAS to do the calculations. Make sure you include the mass of the hanger in your ‘hanging mass’ m! For equation (3) include only 1/3 of the mass of the spring! Find the percent error between the experimental values you found for the period in step 1 and the periods calculated with equations 2 and 3 for each mass. Average your percent errors for both theoretical periods. 4. Which equation gave the best agreement with experimental values? How does the agreement change with increasing mass? Or does it?

Part 3: Damped Oscillations 1. Put a mass on the hanger that gives you a very nice sinusoidal plot. Now add a record to produce a damping force. Set the system oscillating and collect a beautiful plot. Does the plot agree with equation 4? Write a brief summary of your results including your final , percent errors, and answers to questions in the lab. Have the lab title, number, date, lab section, lab partner names and summary on the first page. Make it clear and pretty and you have labeled everything and have reasonable significant figures and units, etc. Print out your data sheet, plots and graphs and attach. You may have to use landscape for your data table.