The Damped Spring

D-term 2004

**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 t_{1/2},
in this case 5cm from the center mark.

**A bit of theory:**

The damped harmonic oscillator can be expressed as

m d^{2}x/dt^{2} + b dx/dt
+ kx = 0,

where F_{damping}= -bv is the damping force and F_{restoration}
= -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:

d^{2}x/dt^{2} + γ dx/dt + ω_{0}^{2}
x = 0,

where γ = b/m and ω_{0}^{2}
= k/m

What is more important here is to know the form of the
solution which has two parts:

x = Ae** ^{-γt/2}**cos(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

ω^{2} = ω_{0}^{2}
– γ^{2}/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 T_{1/2} 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 T_{1/2} 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 tabl**e. **

**For the Write-up now due anytime between this Tuesday,
****4/13/04**** and
next Tuesday, 4/20 in Fred Hutson's mailbox:**

**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 T_{1/2}^{ } 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 T_{o} value). Also present your T_{1/2}
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 T_{o}).
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 δT_{o}). 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 T_{1/2}
damping times for.

Continuing
from your tabular presentation of your 5 T_{1/2} 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 T_{1/2} 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 T_{1/2}, one for each radius value, plus
the corresponding T_{1/2} uncertainties. The Lab Coordinator will
obtain these values just as you obtained the average T_{1/2} 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 T_{1/2}-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. r^{2}.

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 r^{2}. 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. r^{2}. 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. r^{2} 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. r^{2} (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, T_{o} ± δT_{o}. The
undamped angular frequency, ω_{o}, is simply ω_{o} =
2π/T_{o}. In order to evaluate the uncertainty of the
angular frequency in the simplest possible way, we can simply compute
δω_{o} = |2π/T_{o} – 2π/(T_{o} +
δT_{o})|. 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.