Notes for the experimental study of the air-damped motion of
a mass on a spring using a ULI system
Basic Theory
The problem of the motion of a mass on a spring is well worked in the elementary math
and physics literature1. For a mass m on a perfect massless spring with spring constant k,
equilibrium position x=0, and in the absence of drag and friction, the differential equation
is
x
k
x  0 and the solution is x(t )  xmax cos 0 t where 0 
m
k
. (1.1)
m
Damping corrections and estimations
When velocity-dependent damping is added, the problem can become much more
complex. One simple behavior that is also treated in elementary textbooks1 is when the
damping force is linear in velocity FD   x . (We will review below under what
circumstances this is a good assumption.) The differential equation becomes:
x

m
x
k
x0
m
(1.2)
whose solution has the form, x(t )  xmax (0)e  t / 2 m cos  ' t (1.3)
k   

 . So for simple linear damping we expect the amplitude to
m  2m 
decay exponentially in time. The rate of decay depends on 1/m. There is also a (usually
small) shift in the natural frequency.
2
where  ' 
There is a large literature on drag and frictional forces acting on a body moving through a
viscous liquid. There are two important limits of behavior, low velocity and high
velocity, corresponding to whether turbulent flow is present. The complexities of the
system are hidden in a fudge factor known as the coefficient of drag CD . The drag force
is thus given by
F  CD
1 2
v A
2
(1.4)
where A is the cross-sectional area of the object,  is the mass density of the fluid, and
v is the velocity of the object in the fluid.
An empirical measure of low and high flow is the Reynolds number, defined as
Re 
 vL
,

(1.5)
where L represents a characteristic length of the object and  is the viscosity of the fluid.
For very low Reynolds numbers (Re<1) and for a spherical object of radius R0, fluid
mechanical theory yields Stokes’ Law,
F  6 R0v
(1.6),
corresponding to a drag coefficient:
CD 
12
.
 vR0
(1.7)
In words, the drag force is linear in velocity for low Reynolds numbers.
Under these conditions, the differential equation governing the motion is,
x

m
x
k
x  0 where   6 R0 .(1.8)
m
whose solution is given above (1.3). We thus expect the maximum amplitude of the
motion to decay exponentially with time. The rate of decay depends inversely on the
mass.
For Reynolds numbers between 10 and 106 the coefficient of drag is nearly constant and
has a value between 0.1 and 2, depending on shape (~0.5 for a sphere and 1 for a disk).
(There is no simple closed form solution to find CD in this range; it must be done by finite
element computation.) The drag coefficient for a sphere is plotted as a function of
Reynolds number in the attached figure.
Figure 1
Coefficient of drag computed for a spherical object moving in a viscous irrotational fluid.
(from ref. 2)
Some numbers:
(air )  1.8x105 (Pa-s)
 (air )  1.3 kg/m3
For a typical frequency of f=1 Hz and amplitude y of 10 cm, the maximum velocity is
2 fymax  ~0.3 m/s. Using a disk of 0.1 m diameter, we find the Reynolds number:
 vL 1.3x0.3x0.1
Re 

 2167 .

1.8 x105
Thus eq. (1.4) probably applies and the basic differential equation that describes the
motion of a mass on a spring is thus:
mx  x 2  kx  0 where  
CD  A
. This is a nonlinear equation, which cannot be
2
easily solved in closed form.
We don’t want to solve the full equation unless we have to, but we can analyze the
problem by looking at some of its parts. Since we want to find out how the amplitude
decays with time, we could analyze the lost kinetic energy per cycle and compare it to the
1 2
maximum potential energy kxmax
.
2
xmax
xmax
EK
1
2  Fdx  4  CD  v 2 Adx
T
2
 xmax
0
and assuming that CD is independent of position,
xmax
EK
2
2
 2CD  A 2 xmax
 cos  tdx
T
0
and using, x(t )  xmax sin  t and dx   xmax cos  tdt , we get
T /4
EK
3
3
 2CD  A 3 xmax
 cos  tdt
T
0
 / 2
3
3
 2CD  A 3 xmax
 cos  tdt
0
 / 2
 2CD  A x
3
3
max
EK 4
3
 CD  A 2 xmax
T
3
 sin  t sin 3  t 
   3 

0
1 2
( kxmax
)
dx
EK
4
3
2

kxmax max  CD  A 2 xmax
T
T
dt
3
dxmax
4
(1.9)
  CD  A 2 dt
2
3
xmax
 1
1  8
2
so 

  CD  A t
 xmax (t ) xmax (0)  3
If you plot 1/ xmax against t you should find a straight line for each sample with slope
proportional to frequency squared. You should then be able to deduce the coefficient of
drag.
The Experiment
You will use the ULI system with the ultrasonic motion sensor and LOGGER PRO (tm)
software. Detailed run notes are given in appendix A. You should first calibrate your
electronic measurement systems and determine the limits of the measurement system.
The object of this experiment is not just to get a measurement of frequency or decay rate,
but to qualify the equipment. “How well does the system work?” The checks you should
do should include i) measurement (with a meter stick) of the displacement of the spring
with mass to check deviations from Hooke’s Law, ii) measurement of the quiescent
position using the ultrasonic detector to see what the noise in the computer-based
measurement is, iii) calibration of the length measurement of the ultrasonic/ULI system,
iv) calibration of the time measurement of the ULI system.
Figure 2
Displacement of the spring as a function of applied mass. x- data. From the slope. k=8.03
kg/s2 (Newhall and Krajewski, 2002). (A second group found k=8.24 kg/s2 using the
same system.)
You should now take amplitude vs. time data as discussed in Appendix A for a variety of
measurement conditions aimed at addressing various issues in the theory. For example
one of the most basic questions is whether the frequency of oscillation varies with the
applied mass as predicted by basic theory 0  k / m . Since the spring has mass, we
expect a relationship more like, 0  k /(m  ms ) , so we chose to plot 1/  2 against
applied mass in Fig. 3. The relationship is linear with slope 4.83 kg-1Hz-2s2
corresponding to k=8.165 kg/s2. This differs slightly from the spring constant found by
spring extension measurements and it is important that you include uncertainty estimates
for each of these quantities so that you can determine whether the difference is real.
Notice also that the mass-axis intercept of the plot is not zero, but is actually offset by
about 120 g. This is likely due to the mass of the spring and holder. You might check on
the order of magnitude by weighing the spring.
200 g mass on spring
0.1
0.08
0.06
position (m)
0.04
0.02
0
-0.02
-0.04
-0.06
-0.08
-0.1
0
50
100
150
time (s)
200
250
300
Figure 3
Data taken at 20 samples per second with the 200 g mass added. A CD disk has been
added to the sample in order to act as a damper.
0.04
0.03
0.02
position (m)
0.01
0
-0.01
-0.02
-0.03
-0.04
-0.05
19.5
20
20.5
21
21.5
time(s)
22
22.5
23
Figure 4
Expanded scale for the same data as shown in Fig. 3. The solid line is a fit to a sinusoidal
function.
Figure 5
Data for frequency as a function of applied mass. (Newhall and Krajewski, 2002)
Another question is whether the decay is exponential or algebraic as described above.
This test has not yet been done by a student group. Once the appropriate decay
coefficient is determined you could check to see whether it varies (with mass (frequency)
and amplitude) as predicted by the appropriate theory.
In Fig. x we show the log of the amplitude extremes of the motion plotted as a function of
time. A best-fit straight line is also plotted. Deviations from exponential decay are
clearly visible. One way to quantify the deviation is to compute the root mean square
2
1
deviation from the fit  
  xi  x fit  . (The n-2 is because we have used two
n2
parameters to fit the data.) For the exponential fit the standard deviation is 2.5 mm.
200 g mass on spring
-1.3
-1.4
-1.6
-1.7
-1.8
log
10
position (m)
-1.5
-1.9
-2
-2.1
-2.2
0
50
100
150
time (s)
200
250
300
Figure 6
Logarithm of the amplitude extrema from Fig. 3 plotted against time. The line
corresponds to a least squares fit with slope –0.0025.
200 g mass on spring
140
120
1/amplitude (m -1)
100
80
60
40
20
0
0
50
100
150
time (s)
200
250
300
Figure 7
Inverse amplitude from figure 3 plotted against time for comparison with equation 1.9.
The best-fit slope is 0.35 s-1m-1. The standard deviation is 1.4 mm, almost a factor of two
smaller than the exponential fit.
The fit to equation 1.9 (Fig. 7) is much better than that to 1.3 (Fig. 6), so we will proceed
to extract the coefficient of friction.
3
 1/ x  3
CD  
 0.35
 0.9

2
8  1.3    0.06252  4.92
 t  8 A
This is definitely in the ballpark for an irregularly shaped object.
You can also check to see whether the loss slope in Fig. 7 scales as expected with
frequency.
1 see for example, R. D. Halliday, R. Resnick, and J Walker , Fundamentals of Physics,
5th edition, (Wiley, New York, 1997), Ch 16.
2 W. Arthur and S. K. Fenster, Mechanics, (Holt, Rinehart and Winston, New York,
1969), p243.
P D Persans
March 5, 2002
Appendix A – Measurement notes
I.
Setup
a.
Ensure vertical rod is secured to table
b. Clip horizontal rod approximately 2 meters from floor and extending at least 1 foot
from the table
c.
Hang spring from horizontal rod
d.
Attach mass holder to spring
e. Connect the sonic range finder to the ULI and the ULI to the serial port of your
computer
f.
Ensure that software for the ULI is loaded (logger pro is recommended) and the
serial port is active
g.
Position the sonic range finder directly below the mass holder
II.
Data Collection and Analysis
a.
Check out and calibrate the spring linearity, detector linearity, and interface as
discussed above. Using a variety of different weights, measure the spring
displacement while the mass is at rest (~10 seconds of data collection per mass,
10-40 data points per second is recommended). Determine how linear the
displacement is and deduce the spring constant appropriate for each mass.
b.
Measure the displacement as a function of time for several applied masses. You
should take enough data to determine both the frequency and the rate of decay
(probably a few minutes for each mass.) You may also want to look at the
effect of varying the initial amplitude.
c.
Add the compact disk or other object as a damper and take enough data to
determine the frequency and rate of decay.
d.
Determine the spring constant by plotting 1/  2 against applied mass. Assess
uncertainties and compare this result with the spring constant you deduce from
extension measurements.
e.
Plotting and fit 1/amplitude against time and determine the loss slope for
various experimental conditions. Is the fit within your estimated uncertainties
from preliminary measurements? Test whether the coefficient of friction is
independent of experimental variables such as frequency, mass, and amplitude.
NOTE: Never let the mass come within ~50cm of the range finder.