CBL
Lab 10

Oscillating Masses on Springs: Characterizing the Motion with an Equation 

6/02/98

Goals:  To investigate simple harmonic motion (i.e., sinusoidal oscillation) with a spring-mass system.

Required reading: sections of your text concerning simple harmonic motion and masses on springs.

Required calculator: TI-83.  You will use several lists in this lab; please save any data that is presently stored there elsewhere in your calculator. A large TI program group called PHYSICS (the group consists of the main PHYSICS program and all sub programs which begin with PHZ) will also be stored on your calculator, so make sure that your calculator memory is not full.

1. Attach the cross bar to the ring stand so that the bar hangs over the edge of a table.
2. Mount the force probe on the cross bar. Hang the spring from the hook on the force probe.
3. Hang the 50g mass hanger from the spring and gently add an additional 300g. Make sure the hanger can’t slip off the spring and the masses can’t slip off the hanger. Make sure the hanger is at least 60cm but no more than 1.00m above the floor.
4. Place the motion detector face-up on the floor directly below the hanging mass. Tape an index card to the bottom of the mass hanger so the motion detector can "lock on".
5. Using the adapters, plug the motion detector into the sonic port, the force probe into channel 1, and the accelerometer into channel 2 of the CBL.
6. Obtain the TI program group PHYSICS from someone else via your TI link.
7. Attach your TI-83 to the CBL using your TI link; make sure that this connection is secure.
8. Plug the CBL ac adapter into the wall outlet and turn on the calculator and CBL.

• Hold the accelerometer still with the arrow pointing straight up. When the reading on the CBL screen is stable (at least the first two digits), press the TRIGGER button. Enter 0 as the reference on the calculator.
• Hold the accelerometer still with the arrow pointing straight down. When the reading on the CBL screen is stable, press the TRIGGER button. Enter -19.6 as the reference on the calculator.

• Check the mass (including accelerometer) is hanging still on the spring. When the reading on the CBL screen is stable, press the TRIGGER button. Enter 0 as the reference on the calculator.
• Remove 200g from the hanger. Check the mass is hanging still. When the reading on the CBL screen is stable, press the TRIGGER button. Enter -1.96 as the reference on the calculator.

Start the total mass (hanger, 300g masses, accelerometer) oscillating on the spring. (To set the mass into oscillating motion, carefully lift the suspended mass a few centimeters straight up and then gently release it. Make sure there is no side to side motion.) Hit ENTER on the calculator to collect the force, accelerometer, and ranger data for the oscillating mass.

In a few seconds, the list locations of the data will be given. In the case of this experiment, L1 is time in sec, L2 is net force in N, L3 is acceleration in N/kg, L4 is position, L5 is velocity, L6 is acceleration again. The acceleration data in L6 is not as dependable as that in L2 because the L6 values come from taking the second derivative of the position data within the CBL unit. There is often a bogus phase shift in the L6 data. Thus you will not be using the L6 data for any analysis.

A menu will appear that allows you to view your graphs. The ANALOG choice will show you the force reading vs. time data and then the accelerometer reading vs. time data. The SONIC choice leads you to the distance, velocity, and (bogus) acceleration vs. time graphs. All of your graphs should be sinusoidal. If you are unsatisfied with this set of data, follow the program menus until you get the option to recollect data in the REPEAT? menu.

Work your way through the menus to exit the program. The data should remain in the lists. Share the data in L1-L5 (via TI link) with your partner. Delete L6 to avoid any confusion. You should save the lists as a program to call up later (you should already know how to do this).

1. Use the top FOURTH (include that lineless part, too) of a fresh journal page to neatly sketch your p-t wave form; your sketch should have the correct number of periods and should start (i.e., at t = 0) at the same place as the data on your TI screen. You will be filling the rest of this page with other graphs later. Leave the rest of this page blank for now, and continue on the next clean page.
2. At present, your goal in the following parts is to come up with an equation that matches the data. You will then superimpose this equation onto the data to see how well it fits.

1. Use your data (and the TRACE function) to determine the amplitude of the motion. Use ALL of your graphed position data! Describe what you did by referring to the (well-labeled) diagram of your spring’s motion.
2. Use your data to determine the equilibrium position of the hanger bottom. Describe how you did this.
3. Using your calculated spring constant, predict the period for 375g oscillating on your spring. Also determine the actual period of the motion using the graph. Again use (most) ALL of your data. Again, describe how you obtained the value for the period by referring to your diagram. Find the % difference between the predicted and actual period values. As a check, your values should be within 5% of each other.
4. Determine the phase of the motion. We suggest that you use the TRACE function to find the position of the weight hanger at a specific time. Then use these data to determine the phase. Show all work clearly.
5. Once you have obtained the 4 quantities above (amplitude, equilibrium position, period, and phase) to obtain an equation for y (the position of the weight hanger bottom) as a function of x(time). Start with a general equation (all letters/symbols) that includes the 4 quantities above, and then substitute your numerical values and units for the 4 quantities.
6. Put this equation into your TI-83 (Y =) and then plot this function ALONG WITH your original data (obtained from the CBL). The two graphs [your y(x) function and the original data] should nicely overlap. Show your instructor the superimposed plot.
7. Go back to your sketch of position as a function of time in your lab book. Label your axes with numbers/units showing the maximum position, the minimum position, the equilibrium position, and the total time that your plot covers.

1. Again go back to the page in your lab book where you sketched a plot of your spring position as a function of time. In the second fourth of the lab book page, draw axes for a velocity-vs.-time graph, making sure you keep the time axes lined up for the two graphs. Use your position-vs.-time plot to determine

1. all places where the velocity is zero; use an "0" to mark all those times on your p-t plot where the velocity is zero. Explain how you knew.
2. all places where the velocity is the most positive; use a + to mark all those times on your p-t plot where the velocity is most positive. Explain how you knew.
3. all places where the velocity is the most negative; use a - to mark all those times on your p-t plot where the velocity is most negative. Explain how you knew.

2. The next goal is to calculate the velocity of the hanger bottom as a function of time. If you are presently enrolled in Calculus (or successfully completed Calculus last year), then follow the CALCULUS instructions below. If you have never participated in a one-year course in Calculus, follow the NON-CALCULUS instructions. If neither of the above apply, check with the instructor as to which instructions you should follow. In either case, list your present math course(s) and your math course(s) from first semester.

1. Use your equation for y(t) to determine the velocity as a function of time (using calculus !); make sure to include units on all numbers (even those inside the argument of the sine or cosine!).
2. Put the equation for velocity as a function of time into your calculator (Y=). Graph that function by itself.. Does this plot match the time of each zero, most positive velocity, and most negative velocity that you predicted in number 4 above? Sketch your plot in the bottom half of the lab book page where you sketched the position-vs.-time graph.
3. Determine the amplitude and period of the velocity graph. Label these quantities on your sketch in your lab book.
4. What is the phase of the velocity-vs.-time graph relative to the position-vs.-time graph? Answer both in units of pi and in units of fraction of a period.
5. We now use the TI to calculate the velocity-vs.-time function for us. Create another function (Y=): Y2=nDeriv(Y1,X,X) (assuming that your position-vs.-time graph was Y1). Plot both functions (position- and velocity-vs.-time) on your TI. You will probably have to adjust your window to be able to see both plots at once.
6. Use TRACE to find the maximum speed. Find the % difference between this value and that calculated using your function from part (a) above. Your values should agree to within 5%.
7. Superimpose your velocity function over the velocity data collected via the motion probe. How well does it match? Show your instructor the superimposed plot.

1. Use the principle of conservation of energy to determine the maximum speed. Hint: at what position(s) will the mass have the greatest speed?
2. Use your predictions in part 4 above to determine the period of the velocity function. How does it compare to the period of the position function? Explain.
3. Complete this conceptual statement: To find velocity from a position-time graph, you find _______________ of the graph. We now use the TI to calculate the velocity-vs.-time function for us. Create another function (Y=): Y2=nDeriv(Y1,X,X) (assuming that your position-vs.-time graph was Y1). Plot both functions (position- and velocity-vs.-time) on your TI. (You will probably have to adjust your window to be able to see both plots at once.) Conceptually, what did you just have your calculator do? Explain.
4. Sketch this graph in the next fourth of your lab book page below the position-vs.-time graph. How well did you do at predicting the times when the velocity was zero, most positive, and most negative?
5. Use TRACE to find the maximum speed. Find the % difference between this value at that calculated in part (a) above. Your values should agree to within 5%.
6. Superimpose your velocity function over the velocity data collected via the motion probe. How well does it match? Show your instructor the superimposed plot. Discuss any discrepancies.

1. Just below your v-t graph, use the next fourth of the page to create the corresponding a-t graph.
2. Call your instructor over and explain to him/her how the velocity graph tells you when acceleration is zero, maximum, minimum, negative, and positive. Get the instructor to initial your a-t graph.
3. Use the remaining fourth of the page to create the corresponding Fnet-t graph.
4. Describe (no, we don’t want actual numbers here, except perhaps for 0) the magnitude and direction of the net force on the mass at this/these time(s):

• when the mass is at the top of its motion
• when the mass is at the bottom of its motion
• when the mass is at the equilibrium position

1. Plot the force probe and accelerometer data against one another. Sketch the graph. What is the shape of the graph? Is the shape what you thought it should be--why? Perform the appropriate fit and record the fit coefficients. Show how the math and physics equations match up. Discuss the physical meaning of the fit coefficients and the predicted values. Compare the calculated values of the fit coefficients to your predicted values. Explain any discrepancies.

2. Before taking data, you calibrated the force probe. Review this calibration procedure. Fully explain why you entered 0 and -1.96 for the reference values during the calibration.

Start this work on a clean sheet of paper. Make sure all graphs are properly labeled, including units and maximum and minimum numerical values on vertical axis, and useful times on the horizontal axes. Do not include excess significant figures. Make sure that graphs are not cramped and that you keep your time axes the same for all graphs.

1. Resketch your position vs. time graph. Just below the graph, write out the function that best fits your raw position-time data:

• in symbolic physics form; y = .......
• in numerical physics form with units; y = ......
• in the form used in the Y register of your calculator; Y₁ = ......

2. Create and sketch the PE_grav vs. time graph. Just below the graph, discuss how you had to use/modify the position-time function to find the PE_grav function. Is there a special reference position that you need to discuss? Write out the PE_grav function:

• in symbolic physics form; PE_grav = .......
• in numerical physics form with units; PE_grav = ......
• in the form used in the Y register of your calculator; Y₂ = ......

3. Create and sketch the PE_spr vs. time graph. Just below the graph, discuss how you had to use/modify the position-time function to find the PE_spr function. Write out the PE_spr function:

• in symbolic physics form; PE_spr = .......
• in numerical physics form with units; PE_spr = ......
• in the form used in the Y register of your calculator; Y₃ = ......

4. Create and sketch the KE vs. time graph. Just below the graph, discuss (in some detail) how you had to use/modify the position-time function to find the KE function. Write out the KE function:

• in symbolic physics form; KE = .......
• in numerical physics form with units; KE = ......
• in the form used in the Y register of your calculator; Y₄ = ......

5. Create and sketch the E_tot vs. time graph. Just below the graph, discuss how you had to use/modify the previous functions to find the E_tot function. Write out the E_tot function:

• in symbolic physics form; E_tot = .......
• in numerical physics form with units; E_tot = ......
• in the form used in the Y register of your calculator; Y₅ = ......

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