Jason Fong
702847140
date of experiment: 11-7-00
partner: Rosanne Hong
Physics 4AL Lab 5
Experiment 5
Abstract and Introduction
The purpose of this experiment was to investigate undamped and damped harmonic
motion of a mass suspended by a spring, and to measure the frequency of the oscillations. The
undamped case was setup by allowing a mass to oscillate freely while suspended by a spring.
The damped case was similarly setup, but with the mass traveling through an aluminum tube so
that a eddy current would damp the motion of the mass. The results demonstrated what was
expected for the undamped and damped harmonic motion. For the undamped motion, the ratio
between successive amplitudes of each cycle of motion remained close to 1. For the damped
motion, the ratio between successive amplitudes remained around 0.934. For the undamped
case, the measured angular frequency was 4.738 ± 0.007 rad s , and the predicted value was 4.409
± 0.051 rad s .
Spring Constant
Before beginning the experiment, the value for the spring constant k of the spring needed
to be found. This was accomplished by hanging weights of various masses from the spring and
measuring the amount that the spring stretched. Using the equation F=ma, the masses were
converted to forces by multiplying the masses by 9.85 m/s2. Using that data, a plot of force
versus the distance the spring stretched was made. A line was fitted to the plots, and the slope of
the line gave the spring constant. This is because F=kx, so k=F/x; where F is the force from the
spring, k is the spring constant, and x is the distance the spring stretched. F/x is the slope of the
line, thus the slope gives the value of k. The value of k was found to be 3.46 ± 0.07 N/m. The
uncertainty was obtained from a regression analysis performed on the data.
Given two springs made of the same material, one having 50 turns and the other having
100 turns, the spring with 50 turns would have a larger k. This is because the spring with more
turns will have to be stretched farther in order to exert the same force. Given that the springs are
exerting the same force, if one spring is stretched farther to achieve that force, then it will have a
smaller k. The spring with 50 turns is stretched less to achieve a certain force, and so it has the
larger k.
Formulas
In order to calculate the values for this experiment, a few formulas needed to be derived.
The first formula was the formula for the angular frequency of the free, undamped oscillations,
ωfree. Starting with Newton's Laws:


F
  ma  mx
Using the spring force :

F
  mx  kx
For simple harmonic motion, we have:
x(t )  A cos( t )
Taking the second derivative of x(t) gives:
x(t )   2 A cos( t )
Substituting into a previous equation:
mx   kx
 m 2 A cos( t )  kAcos( t )
m 2  k
k
2 
m
 free 
k
m
The uncertainty for this is:
2
2

k
 1
 
 k
  m

 
k   
m   
k    
m 
 
  

 2 km   2m m

2
 free
2
The corresponding frequency in Hertz is:
f free 
 free 1

2
2
k
m
The uncertainty for this is:

 k   m

k
 1
 
f free   k   
m   
k    
m 
 4 km   4 m m
 f
  f


2
2
2
2
Next, the formula for the angular frequency of the damped oscillation, ω0, needed to be
derived. For the damped cased, there should be two forces at work, the damping force Fd and the
force from the spring Fs:
 F  mx  F
d
 Fs
The damping force is assumed to be proportional to velocity, with b as the constant relating the
velocity to the damping force:
Fd  bv  bx
Substituting that back into the previous equation gives:
mx  bx  Fs  bx  kx
mx  bx  kx  0
One solution for this equation is given by:
x(t )  Aei t
Taking derivatives of x(t) gives:
x (t )  i Aei t
x(t )   2 Aei t
Substituting into the previous equation:
 m 2 Aei t  bi Aei t  kAei t  0
 m 2  bi  k  0
Solving for the quadratic equation:

 bi   b 2  4mk
bi


 2m
2m
4mk  b 2
4m 2
k
b2

m 4m 2
bi


2m
The meaning of the imaginary part is that x(t) is an exponentially damped sinusoidal oscillation:
 bt
x(t )  Ae
2m
cos(0t   )
where
k
b2
k
b2
0 


1
m 4m 2
m
4mk
1
4Q 2
 0   free 1 
Q is the quality factor:
Q
km
b
Q measures how near the oscillation is to true undamped simple harmonic motion. Higher
values of Q indicate that the oscillation is nearer to undamped motion. Very high values of Q
will make the square root portion of the expression be nearly equal to 1, and so ω0 will be nearly
equal to ωfree. In other words, the actual oscillations will be nearly undamped oscillations.
Lower values of Q will make the graph decrease in amplitude faster, while higher values of Q
will make the graph maintain its amplitude longer.
For the above equations, the value of b is needed. b relates the velocity of the moving
mass to the damping drag force it encounters through the magnetic field in the tube. The value
of b can be found by comparing amplitudes of two peaks at time t1 and t2. At the peaks, the
cosine term is 1 for the equation for x(t):
x(t )  Ae
 bt
x(t1 )  Ae

2m
bt1
cos( 0t   )
2m
and x(t2 )  Ae
Taking the ratio R of the two amplitudes:

bt 2
2m

bt2
bt1
b
t1 t2 
x(t ) Ae 2 m e 2 m
R 2 
 bt 2  e 2 m
bt1
x(t1 ) Ae  2 m e 2 m
b
t1  t2 
ln R 
2m
2m ln R
b
t1  t 2
Procedure
The experiment was setup by hanging a mass vertically from a spring that was connected
to a force transducer. The voltage readings from the force transducer would give an indirect
measurement of the mass's position since the force exerted on the force transducer by the spring
would be a linear function of the mass's position. For undamped simple harmonic motion, the
mass was raised a few centimeters and released to set it into simple harmonic motion. Readings
were then taken from the force transducer for 20 seconds.
For damped harmonic motion, a similar setup was used, except the mass would travel
within an aluminum tube during its motion. The mass is made of a permanent magnet encased in
a brass jacket. When a conductor is near a moving magnetic, an electric field is induced. An
eddy current flows in the aluminum tube while the mass's magnet is moving within the tube.
This induced electric field interacts with the magnet in the mass to slow down its motion. This is
the source of the damping force in the damped harmonic motion. As in the undamped case, the
mass was raised a few centimeters and dropped, with care taken to see that the mass did not
brush up against the side of the tube since this would slow its motion. Readings were taken from
the force transducer for 20 seconds during the motion.
The mass was only raised a few centimeters when being set in motion so that the coils of
the spring would not touch. If the mass were raised too high when being set into motion, the
mass would bounce back high enough so that the coils of the spring touch. If the coils of the
spring touch, it would no longer be simple harmonic motion because the mass's upward
movement would be slowed or stopped before it reached its full distance traveling up. At one
extreme when the spring is completely compressed, the distance will stop changing with
increasing force since the spring cannot compress any more. At the other extreme when the
spring is completely uncoiled into a wire, the distance will also stop changing with increasing
force because the spring cannot be stretched any more.
Data Analysis and Error Analysis
For both the undamped and damped cases, a graph of the force transducer volts versus
time was made. The voltage values did not need to be converted into force because the voltage
has a linear relationship with the position of the oscillating mass, and only the position of the
mass relative to its previous positions needed to be known. For each of the cases, a table was
made with the position of each peak in the graph, the distance between each peak, the ratio of
successive distances between peaks, and the error in that ratio. Those graphs and tables are
included on attached pages. The position of each peak was found by examining the graph. The
error in their ratios were derived as follows:
First, find the uncertainty in the peak to peak values P:
P  V2  V1
P  (V1 ) 2  (V2 ) 2
where P is the peak to peak value, and V1 and V2 are the values of the voltages. By observing
the values from the force transducer when the mass is not moving, there is some random noise
observed in the force transducer readings. The noise gives approximately a 0.001 volts
uncertainty to the voltage readings. Thus the uncertainty in the peak to peak values are:
P  (V1 ) 2  (V2 ) 2  (0.001) 2  (0.001) 2  0.0014 volts
The error in the ratio R is derived as follows:
R
P2
P1
2
2
2
1
 P

1
 P

R   P2    2 P1    0.0014    2 0.0014 
 P1
  P1

 P1
  P1

2
where R is the ratio of the peak to peak values, and P1 and P2 are the values of the peak to peak
distances.
For the undamped motion, the peak to peak ratio stayed near the expected value of 1.
This means that the amplitude of the oscillation was staying nearly the same. The graph of the
ratios show that all of the ratios agree with the value of 1 within the range of error allowed for
each value. For damped motion, the average peak to peak ratio was 0.934. Almost all of the
ratios agreed with that value within the range of error allowed for each value. The 0.934 ratio
means that each successive peak is 93.4% of the previous peak. Thus, the amplitude is
decreasing at an approximately steady percentage rate.
The period of the oscillation can be determined by examining the times for the peaks of
the graph:
T
tn  t1
n
where T is the period, t1 is the time at the first peak, tn is the time at the nth peak, and n is the
number of peaks. The error in this formula is:
2
T 
1

1

 t1    tn 
n

n

2
The uncertainty for each time measurement is taken to be a random error of 0.01 seconds.
For the 8 top peaks covered in the table for the undamped motion, the period of the oscillation is
calculated by:
T
11.81 s  1.20 s
 1.326 s
8
2
2
1
 1

T   0.01 s    0.01 s   0.002 s
8
 8

Thus, the period is 1.326 ± 0.002 s.
The angular frequency is:

2
2

 4.738 rad s
T 1.326s
2
2
 2

    2 T   2 T 
0.002  0.007 rad s
T
1.3262
 T

2
Thus, the angular regular frequency obtained from the graph is 4.738 ± 0.007 rad s .
The predicted angular frequency is:
 pred   free 
k
m
The uncertainty in that formula is:
2
2

k
 1
 
 k
  m

 
k   
m   
k    
m 
 
  

 2 km   2m m

2
 pred
2
where k is the spring constant, which was found to be 3.46 ± 0.07 N/m, and m is the mass of the
oscillating object. An ideal spring would obey Hooke's law and would also be massless.
However, the spring does have mass and affects the oscillation frequency. This can be seen in
this experiment because the value of the predicted angular frequency is different if the mass of
the spring is not considered. Since part of the spring moves with the oscillating mass, part of the
spring's mass also needs to be taken into account. A quick approximation for the effect of the
spring's mass is to add half of the spring's mass to the mass of the oscillating object. The
oscillating mass's mass was 173.5 ± 0.1 g, and the spring's mass was 9.1 ± 0.1 g. Adding half of
the spring's mass to the mass oscillating object gives 178 ± 2 g. The uncertainty was increased
due to the crudeness of the approximation. Using this value for the mass gives:
3.46 N m
 4.409 rad s
0.178 kg
 pred 
2
 pred
2

 

3.46 N m
1
 
0.07 N m    
0.002kg   0.051 rad s
 2 3.46 N  0.178 kg
  2  0.178kg 0.178 kg

m

 

Thus, the predicted value for the angular frequency is 4.409 ± 0.051 rad s .
The maximum possible effect the spring could have on the calculation of the frequency is
if the entire mass of the spring is considering to be part of the oscillating object. The oscillating
mass would then be around 183 g. The angular frequency would then be approximately:

3.46 N m
 4.348 rad s
0.183 kg
The angular frequency calculated from the graph and the one calculated from the spring
constant and mass do not agree within their error ranges. However, their values are close enough
to suggest that they would have been the same in the absence of other sources of error. One
source of error is that the motion was not truly undamped. Air resistance would have had a
damping effect, and there was also some energy lost to heat in the spring. Taking the average of
the ratios of the amplitudes of each successive peak, each peak is 99.6% of the amplitude of the
previous peak. In order to find the number of peaks until the amplitude is reduced by a factor of
e, this formula can be used:
1
 e 1
e
n
ln 0.996  ln e 1
n ln 0.996  1
0.996 n 
n
1
 250
ln 0.996
There will be 250 peaks until the amplitude is reduced by a factor of e, so half the number of
peaks is the number of oscillations since there are two peaks (one high, one low) in each
oscillation. So, there will be approximately 125 oscillations until the amplitude is reduced by a
factor of e.
The accuracy of the frequency measurements were fairly good as can be seen by the low
uncertainty in the frequency obtained from the measured values recorded in the graph. The
accurate measurements from the force transducer aided in calculating a fairly accurate value for
the frequency. The reproducibility of the experiment is also good because a very large number
of measurements were taken so that random errors would have less of an effect. Also, the graphs
were created using the averages of five points in order to further minimize the effects of random
errors.
For the damped motion, the value of b is:
b
2m ln R 2  0.178  ln 0.936

 -0.505
t1  t2
0.73  1.39
The quality factor is:
Q
km
3.46  0.178

 1.554
b
 0.505
The angular frequency for the damped motion is:
0   free 1 
1
1
 4.409 1 
 4.175 rad s
2
2
4Q
4   1.554
The angular frequency for the damped motion is 4.175 rad s , which is lower than the angular
frequency for undamped motion. The damping force is slowing the motion of the mass so that it
can not move through each cycle as fast as the undamped movement can.
Conclusion
The results of the experiment demonstrated the behavior expected from undamped and
damped harmonic motion. For the undamped motion, the ratio between successive amplitudes of
each cycle of motion remained close to 1. For the damped motion, the ratio between successive
amplitudes remained around a lower value at 0.934. For the undamped case, the measured
angular frequency was 4.738 ± 0.007 rad s , and the predicted value was 4.409 ± 0.051 rad s . This
discrepancy between the measured and predicted values was most likely caused by factors that
damped the motion. The predicted value assumed there was no damping, but the measured value
would be affected by damping from air resistance and energy loss from the heating of the spring.