The Damped Spring
Objective: This week we will consider the effect of damping of a 2-spring system by air. To our 2-spring system we will attach cardboard disks of sizes ranging from 5 cm. to 10 cm. In particular, we are interested in finding the relation between γ in the damped harmonic oscillator equation and the radius r of the cardboard disks.
Equipment: At each station there are 2 springs, three hangers, 2 labeled cardboard disks, a stopwatch and 400 grams for loading the system. At the scales placed about the room there are also 5, 2 and 1 gram masses so that for each of your experiments the mass of your system should be within 0.5 grams of 400 grams independent of the size of the disk. There is also a clamp and a string with a center mark to full displacement marks at 10 cm and two marks at t1/2, in this case 5cm from the center mark.
A bit of theory:
The damped harmonic oscillator can be expressed as
m d2x/dt2 + b dx/dt + kx = 0,
where Fdamping= -bv is the damping force and Frestoration = -kx is the restoring force. Formal discussions of the solutions to the equation can be found in most first year texts.
In PH-1140 we are using a somewhat different form for the damped harmonic oscillator:
d2x/dt2 + γ dx/dt + ω02 x = 0,
where γ = b/m and ω02 = k/m
What is more important here is to know the form of the solution which has two parts:
x = Ae-γt/2cos(wt), (1)
the first being an exponential representing the damping and the latter cosine function representing the oscillation of the system. For small γ the picture below, borrowed from Tippler, gives a good visual representation.
[ to be inserted later. Sorry!]
Here, the envelope to the diminishing oscillation is a decreasing exponential.
After a bit of algebra one can demonstrate that eq. 1 is a solution of the differential equation if
γ = b/m and
ω2 = ω02 – γ2/4.
Procedure: Now we need some data. Where do we start?
We need to have each student do one measurement of the time for 20 cycles for an un-damped hangar and mass combination of 400gr. Pairs of students should average their values and enter the value for their station on the board to 3-digits.
Next each student will take one of the marked disks and determine the value for T1/2 for a hangar, cardboard disk and sufficient added mass to once again have combined mass of 400gr. There are triple-beam balances placed around the room. Near the balances you will find small weights that will permit you to adjust your mass to 400gr +/- 0.5gr. The determination of T1/2 should be done five times. On the board you will write the average value of your five trials. Note the limited space and recognize that there will be several entries for each disk. Be neat and thoughtful as you make your entry to the table.
For the Write-up now due anytime between this Tuesday,
Instructions for writing-up the Damped Harmonic Oscillation Lab Report.
Report all measurements.
In the action-part of the lab, you and your partner each measured the time duration for 20 cycles of the “undamped” mass/spring oscillator, plus you each measured the T1/2 time for your damping disk 5 times. Present the two 20-cycle time durations, the corresponding average periods, and their average value that was reported to the Lab Coordinator (a To value). Also present your T1/2 times in the form of a table as shown in lab.
Determine the un-damped period of the motion, the undamped angular frequency, and the spring constant for the 2-spring system. Average the several values for the undamped period (sent to you via e-mail) as measured by the several student teams in your section for their respective systems (that gives you a value for To). Calculate the deviation magnitudes (i.e., they’re all positive!) between the average period and each measure period; average those deviations, and call that average of deviations the uncertainty of the period measurement (and write it as δTo). From that average period and average deviation, calculate the average angular frequency and its uncertainty*. Also calculate the average spring constant, together with its uncertainty*.
Determine the damping constant (the γ = gamma-factor) for the one damping disk that you measured the T1/2 damping times for.
Continuing from your tabular presentation of your 5 T1/2 measured values, average these 5 values, and calculate the 5 deviation magnitudes in the manner described in Part 2 above. Obtain also the average of these deviation magnitudes.
Process the T1/2 data (obtained from the whole class and sent to you via e-mail) in order to determine the damping constants and corresponding uncertainties for each disk-size.
Shortly after the completion of the final Lab #2 session, the Lab Coordinator will send you the aggregated results of ALL the damping experiments conducted by PH 1140 students. Those results will take the form of 11 values of T1/2, one for each radius value, plus the corresponding T1/2 uncertainties. The Lab Coordinator will obtain these values just as you obtained the average T1/2 and its uncertainty from your own set of 5 measurements – the only difference is that these aggregated results include the collective experience of the entire class. Construct a table with a row for each of the 11 radius values and columns for the aggregated T1/2-values, their corresponding uncertainties*, the corresponding γ-values, and the corresponding γ-uncertainties*. Be sure to include units for the column values in the header for each column.
Graph your results of gamma vs. r2.
In principle we expect the amount of damping to be proportional to the AREA of the damping disk because the amount of air that affects the motion of the system is directly proportional to the disk area. (But that’s just our simple theory and nature may actually work a bit differently! Let’s find out!!!) The first thing to do is to add 1 column to the table constructed in Part 4 for the values of r2. Once you have completed this table (being sure to include column headings complete with units of the quantity listed in each respective column), you can plot the points representing gamma vs. r2. For graphical precision, be sure to use precision graph paper (that means 1-mm grid graph paper available in the Bookstore), drawing the graph as large as possible while still using a non-awkward scale. Plot gamma on the vertical axis vs. r2 on the horizontal. Through each datum point, draw a vertical line that represents the plus and minus uncertainty amount around the average value of gamma (if somehow that amount is no bigger than the datum point itself, then leave out the error bars and just stick with the datum points). What you will have then is a graph of average measured gammas vs. r2 (which itself is proportional to the disk area). Draw a straight line that best fits these 11 datum point (including uncertainty error bars). If the straight line can be drawn to intersect most of the points-plus-error-bars, then we can say that the damping really seems to be proportional to the area. If the points-plus-error-bars deviate substantially from your best straight line fit, then we can say that he damping is not exactly proportional to area and that for some reason the actual theory is more complicated than we hoped. So which is it? (State your conclusion!)
* OK, let’s now provide some information about this uncertainty business. Let’s say that we have the average period and it’s uncertainty, To ± δTo. The undamped angular frequency, ωo, is simply ωo = 2π/To. In order to evaluate the uncertainty of the angular frequency in the simplest possible way, we can simply compute δωo = |2π/To – 2π/(To + δTo)|. There are other more sophisticated ways to compute uncertainties, but this approach provides a nice simple (and basically correct!) way of looking at it. So whenever you are asked to calculate an uncertainty in some quantity that is related to some other quantity whose average value AND uncertainty are known, this is the pattern for how you are supposed to go about it.