4/22/96

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

Required reading: Giancoli sections 11(2,3)

Required calculator: TI-82; you will use lists L1 and L2 in this lab; please save any data that is presently stored there elsewhere in your calculator. A TI program (SPRING) will also be stored on your calculator, so make sure that your calculator memory is not full.

1) Equipment setup:

a) Place a total mass (including hanger) of 0.300 kg on the spring; use the exact same

spring as you did in lab C1. Also, as in C1, an index card should be taped to the

bottom of the weight hanger; the weight hanger should be displaced far enough

from the table edge that the sonic ranger is "seeing" the hanger and not the table.

b) Use masking tape to hold the weights to the weight hanger securely, to hold the top

of the weight hanger to the spring, and to hold the top of the spring to the support

rod.

c) As in lab C1,

1) attach the sonic ranger to CBL sonic port using the adapter provided.

2) plug the CBL into a 120 v outlet using the AC adapter provided.

3) place the sonic ranger directly underneath the spring/weight-hanger.

d) Obtain the TI program SPRING from someone else who has via your TI link.

e) Attach your TI-82 (an -85 will NOT work) to the CBL using your TI link; make sure

that this connection is secure.

f) Practice setting your system in oscillation: do so by lifting the mass hanger up (NOT

by pulling it down) and then releasing it; this usually results in the spring oscillating

only vertically (rather than also having some horizontal motion). Practice this a few

times to get it right (i.e., with no horizontal motion).

g) Turn on the CBL. Turn on your TI. Access and run the program SPRING. Follow the instructions contained within the program.

If you are successful, your TI will display a graph of the position of the hanger bottom as a function of time. Share the data (via TI link) with your partner.

2) Use the top half of a lab book page to sketch your 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. Leave the bottom half of the page blank for now.

3) 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.

a) Use your data (and the TRACE function) to determine the amplitude of the motion. Use ALL of your data! Describe what you did by referring to the (well-labeled) diagram of your spring's motion.

b) Use your data to determine the equilibrium position of the hanger bottom. Describe how you did this.

c) Determine the period of the motion. Again use (most) ALL of your data. Again, describe how you obtained the value for the period by referring to your diagram. As a check, your value should be within 5% of that on lab C1, part 8. Find the % difference between these two values.

d) 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.

e) 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.

f) Put this equation into your TI-82 (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 instructor the superimposed plot.

g) 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.

4) Now 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 bottom half of the lab book page, draw axes for a velocity-vs-time graph. Use your position-vs-time plot to determine

a) all places where the velocity is zero; use an "o" to mark all those times in your lab book where the velocity is zero. Explain how you knew.

b) all places where the velocity is the most positive; use a + to mark all those times in your lab book where the velocity is most positive. Explain how you knew.

c) all places where the velocity is the most negative; use a - to mark all those times in your lab book where the velocity is most negative. Explain how you knew.

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

a) (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 at that calculated in part (a1) above. Your values should agree to within 5%.

b) (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) 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.

(4) Sketch this graph in the bottom half 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 (b1) above. Your values should agree to within 5%.