Lab Water Springs
This lab was designed in 1995 for "A Day in the Life of a Student in the 21st Century" - a teleconference with the US House committees on science, economics, and educational opportunities.

Purpose

The purpose of this lab is to produce an oscillation that has a varying amplitude yet constant period. As the mass is dragged through the water, the amplitude of the spring's oscillation decreases as the energy stored in the spring is transferred to the water. This is an example of a damped oscillation in which the amplitude is experiencing an exponential decay.

Lambda, λ, represents the decay function of this damped oscillation. During our analysis, we will first graph the spring's amplitude vs time and then examine a second graph of the natural logarithm of its amplitude vs time. The slope of LN(y) vs t will be negative lambda, -λ. Using this slope we will determine the half-life of the spring's decay and verify its value with the data collected from the LabPro distance probe today.

A typical graph of this type of oscillation would look like the following sample.

Equipment

• LabPro motion detector
• plastic ½ gallon milk jug with water
• spring
• 700 grams of slotted masses
• one mass hanger (50 grams)
• white index card
• tape

Procedure (set-up):

Under the Start Menu go to Programs, Math, Logger Pro 3.1 to launch the program. Logger Pro should automatically set up the graphs according to the connected sensor. With the Motion Detector properly connected, the program should display graphs of position vs time and velocity vs time. Press Collect to test your connections. Make sure that you are collecting data to at least 3 decimal places.

A second member should verify that the probe can see the white book card attached to the bottom of the spring - just above the top of the milk jug. Note that only slight adjustments should be necessary! When the spring is gently oscillated, the card should not rub against the container or twist violently. Note that the masses (750 grams total) MUST remain completely submerged in the water during each trial and NO water should be splashed out of the container. Verify that the probe is seeing the card by moving the masses carefully up and down -- Do NOT use large amplitudes! 5-6 cm is MORE THAN ENOUGH! When everything is working, you are ready to start collecting data.

Procedure (data collection):

Since the spring needs a few seconds to stabilize, the person releasing the spring should tell the Logger Pro operator when to start the probe. Try to release your spring with a small steady amplitude and a minimal amount of rotation (twisting). Watch your graphs. The oscillations should minimally have either a constant set of smooth crests OR smooth troughs -- it is not absolutely necessary to have perfect oscillations in both places. When you have a good trial, highlight a "good section" of your position vs time graph. The data selected will be highlighted in the accompanying data table. Copy and paste your data into an EXCEL spreadsheet. Rename the sheet Data I.

Finally, carefully rerun the experiment and obtain a second trial. Save this one on a sheet called Data II. Run your trials as accurately AND quickly as possible, remember that at least one other group needs to use your lab station before the period is over.

When you are finished with both trials, leave the spring, card & masses suspended in the water jug. Please wipe up any water that may have been dripped onto the table. Exit from Logger Pro so that the next group can begin.

Analysis (regression theory):

Input your best data into the following EXCEL spreadsheet, WaterSpringAnalysis.xls. From there, we will determine the half-life of your damped oscillation.

Conclusions

 What is the slope of your line of best fit?

 Using this slope, what is the half life, T½, of your spring's oscillation?

In EXCEL, open your spreadsheet and fill out the following data table by scrolling down your data table in column A until you find the time for the fifth maximum in column F. Note the corresponding value of its amplitude in column C and record both in your chart. Then scroll down to the time closest to the passage of one half-life. Record that time from column A and its corresponding value for the amplitude in column C. Continue this procedure until a total of five values have been listed. Then use the definition of half-life to complete the column entitled ideal amplitude value based on the value for the amplitude of your initial time.

 actual ideal description time amplitude amplitude (col A) (col C) (half-life)
 initial time
 initial + 1 T½
 initial + 2 T½
 initial + 3 T½
 initial + 4 T½

Did the decay in the spring's amplitude, as reflected by the changes in its values in the previous table, behave according to the definition of half-life?
 Support your answer by using you data analytically to calculate a percent error for your closest example.