Goals: 1) To investigate position, velocity, and acceleration of an oscillating spring/mass system as a function of time, and 2) to compare the maximum values of these quantities to those calculated using the amplitude of the motion, spring constant, and mass on the spring.
Equipment:
TI-82 calculator spring ring stand
SPRING TI program mass hanger C-clamp
TI link cable 50g and 100g masses short rod
TI-CBL stopwatch right angle clamp
CBL ac adapter meter stick masking tape
sonic ranger with CBL adapter index card flag
Journal formatting: In addition to all previous journal formatting instructions, carefully incorporate the following into your report.
* Items in brackets in the instructions must be included in your lab form. Those that are also in italics must appear word-for-word. Leave plenty of space to show your work.

* Bold-face section titles and the goals of the experiment must appear word-for-word, and parts and steps of the experiment must be numbered as given.

* Except for percentage error, all calculations must be shown, beginning with formulas.
* Consistent symbols must be used, with subscripts as needed.

* All your work must proceed in a step-by-step form that is easy to read. If you make mistakes in a calculation or derivation, simply cross out and start again on the next line.

prelab: Read sections 6-4 and 11-1 Giancoli 3rd edition.
1) Write the identifying number of the spring you are using. Be sure to use the same spring throughout all parts of the experiment. If you change springs, you will have to redetermine the spring constant for the new spring and redo all calculations!
[Spring number =]
2) The spring should be hung from the short rod clamped to the ring stand. The ring stand should be clamped securely to the tabletop. Position the apparatus so that the spring hangs at least 0.10 m beyond the edge of the table. Tape the index card to the bottom of the mass hanger. Suspend the mass hanger (0.050 kg) from the spring and put the sonic ranger on the floor directly below the mass hanger. Position the rod that supports the spring so that the bottom of the weight hanger is 0.65 to 0.70 m above the ranger. Attach the sonic ranger to the CBL sonic port using the provided adapter. Plug the CBL into a wall outlet using the ac adapter.
3) Turn on the CBL. Hit the [MODE] button until the screen displays the word MULTIMETER in the lower left corner. Now press the [CH VIEW] button several times until the display says it is reading SONIC in M. This means the CBL will take a reading from the sonic ranger once per second and will display the results in meters. At this point you should hear one click per second coming from the ranger. Make sure the ranger is "locked on" to the mass hanger and not the table edge or some other object. When the mass hanger is motionless and you are satisfied that the displayed reading is correct, record the position reading.
[Equilibrium position for 0.050 kg = y₁ = ]
4) Now load the mass hanger with 0.150 kg (for a total of 0.200 kg) and read the position again. Turn off the CBL when you are finished gathering the position data.
[Equilibrium position for 0.200 kg = y₂ = ]
5) Note that y₀ would represent the position with no stretch in the spring, but you will not actually measure this position. Side-by-side on a clean page, sketch three (well-labeled y₀, y₁, y₂) figures depicting the placement of the ranger and hanging mass.
Just below each of the latter two figures, construct the force diagram for the hanging mass and write the corresponding F_net equation. Invoke Hooke's Law where appropriate.
6) Using the equations you wrote in #5, calculate the spring constant of your spring in units of N/m. (Hint: you should use all the data you have taken so far.)
[Calculated spring constant = ]
7) Using the spring constant, predict what the measured position of the spring would be if 0.250 kg were placed on the mass hanger. Then measure the position with the sonic ranger. Expect agreement to better than 5%.
[Calculated position of 0.300 kg = ]
[Measured position = ]
[Percentage error = ]
8) Measure the period of oscillation to 3 significant figures using a stopwatch.
[Number of cycles = ]
[Total time measured = ]
[Measured period of 0.300 kg = ]
9) Assuming that the period, T, of an oscillating spring depends only the spring constant, k, and the mass, m, determine the mathematical combination of k and m that gives the right units for period. This result should be correct to within a constant numerical factor, to be determined in step #8 by experiment.
[Units analysis]
10) Using the spring constant, mass, measured period, and the result of #9, calculate the numerical constant in the formula for period. What familiar number is it close to? What does the text (Giancoli 11-3) say it should be? Compare the two values.
[Calculated constant = ]
[Expected constant = ]

1) Remove 0.150 kg of mass from the weight hanger, leaving a total of 0.200 kg hanging from the spring. Use masking tape a) to hold the weights securely to the weight hanger, b) to hold the top of the weight hanger to the spring, and c) to hold the top of the spring to the support rod. The reason for this is so that the WEIGHTS WILL NOT FALL ONTO THE SONIC RANGER. Measure the position of the bottom of the weight hanger as you did previously.
[Equilibrium position for 0.200 kg mass = ]
2) Select M from the main menu of MOTION PLOTTER. Skip over the Graph Style Options menu. On the Real Time Graphing-Scaling menu, turn off the middle and bottom graphs. Select the minimum and maximum of the position graph to be 0.4 and 0.8 m. Select a time interval of 5 s. Finally, select G to bring up the graph display.
3) Set the spring into oscillation by lifting it up--but not so much as to completely relax the spring--and releasing it. Press ENTER on the keyboard to begin data collection. You should obtain a smooth, sinusoidal waveform. There may be stray points toward the end of the time interval if the spring starts to sway back-and-forth. Make a large sketch of the waveform, labeling the axes and placing numbers on them. Draw a horizontal dotted line to show the equilibrium position of the spring.
[Labeled sketch of waveform]
4) Now tape 0.150 kg more mass to the weight hanger for a total of 0.350 kg. Set the weight into oscillation as before. Obtain another graph and sketch it. Describe several ways in which this graph is different from the one sketched in step 3.
[Total Mass = 0.350 kg]
[Labeled sketch of waveform]
[Comparison]
5) In this step, you will learn to take accurate readings of the coordinates of the points on the graph. Press ENTER to obtain the Graph Follow-Up Options menu. Then select X to return to the main menu. Select P to plot a graph. Skip over the Graph Style Options menu and the scaling choices. After the graph is plotted, press ENTER to gring up the Graph Followup Options. On this menu, select E to examine data. When the graph comes up again, you will be able to use the cursor to read the coordinates of each point on the curve. Read the coordinates of three consecutive extremum points. (An extremum point is a peak or a valley.) These points span one complete cycle of the motion. Record the results with units.
[(t,y) coordinates of consecutive extremum points =]
6) Use the data above to calculate the equilibrium position. Measure the equilibrium position using the L option.
[Total mass =]
[Calculated equilibrium position =]
[Measured equilibrium position =]
[Percentage error =]
7) Use the data to calculate the amplitude of the motion.
[Calculated amplitude =]

8) Use the data to calculate the period of the motion. As a check, this value should be within 5% of the value calculated in Part I.
[Period calculated from graph =]
[Measured period form Part I =]
[Percentage error =]
9) Determine the equation for the position, y, of the spring as a function of time, t. First decide whether you want to call the curve a sine curve or a cosine curve. (Why is the choice arbitrary?) Then decide how the constants of the motion calculated in 6,7, and 8 will be incorporated into the equation. (Section 11-3 of Giancoli may help.) Finally, determine a phase shift (an angle) that must be added to the argument of the sine (or cosine) function in order to bring your equation into correspondence with the actual experimental curve. Also describe how you determined the phase shift. Give the equation in symbols first, and then substitute constants with units.
[y vs t equations]
[Method of determing phase shift]
10) Check your equation by substituting the time coordinate for one of the data points to see if you get the correct position. If you are doing the lab with a partner, each of you should select a different time coordinate.
[(t,y) coordinates of selected point =]
[Calculated position =]
[Percentage error =]
11) Physically, what does the phase shift represent about the initial conditions of the experiment?
[Explanation]

12) Return to the main menu and select O for Other Options. Then select R for Sampling Rate. Change this value to 40. (A faster sampling rate provides greater accuracy in velocity and acceleration calculations.) Return to the main menu and then select M. When you get back to the Real Time Graphing-Scaling menu, turn graphs M and B back on. Select the maximum and minimum of the velocity graph to be +1 and -1. Select a time of 0:02 s. Then set the spring (total mass = 0.350 kg) in oscillation with about a 5 cm amplitude and collect data. You should get a smooth curve for the position graph, but the velocity and acceleration graphs may show some scatter in the points. (This has to do with the method used to calculate velocity and acceleration from the position data.) If any graphs are cut off at the top or bottom, select C in the Graph Follow-up Options menu and change the scale maxima and minima as needed. Then collect data as before. When you get good results, sketch all graphs accurately on graph paper, one below the other as on the video display. Label the axes and include scale numberings.
[Graphs]
13) Add the following to your graphs. Draw vertical lines at three values of time, times for which the position is maximum, minimum, and zero. Extend these lines down through the velocity and acceleration graphs. Use them to answer the following.
a) What is the phase difference between position and velocity? position and acceleration? acceleration and velocity?
b) Where is the mass when the velocity is a maximum? What is the acceleration at this point?

A mass, m, is hung on a spring, pulling it downward a distance, y_o, to its equilibrium position. The mass is then pulled downward an additional distance, A, away from the equilibrium position and released.
1) Determine an expression for the maximum speed, v_max, reached by the spring in terms of A, m, and k, the spring constant. Include gravitational as well as elastic potential energy. Identify initial and final states in diagrams. It's recommended that you define all positions (spring with no mass, spring with mass in equilibrium, spring with mass in oscillation) relative to the position of the end of the spring with no mass. (Hint: This is both a conservation of energy problem and a net force problem. What force laws will you use?)
[Diagrams and derivation]
2) Starting with Hooke's law, determine an expression for the maximum acceleration of the spring, a_max, in terms of A, m, and k. Do this as a net force problem.
[Diagram and derivation]