Union College
Winter 2014
Physics 120
Lab #4: The Effect of Spring’s Mass on the Oscillation Period
Introduction:
In this experiment we investigate the oscillating motion of a mass moving under the restoring force of a spring
with non-zero mass. In class, we discussed that the period of the oscillation of a mass on a massless spring is
given by
T  2
m
,
ks
(1)
where m is the mass on the spring and ks is the spring stiffness. However, this equation assumes that the spring
itself has no mass, when in fact this is clearly not the case. The mass of the spring must have some effect on the
motion since the spring also oscillates and therefore contributes to the total momentum of the system which
determines the system’s response to the spring force. We can’t just add in the mass of the spring, though,
because different parts of the spring oscillate by different amounts--the coils of the spring at the end near the
attached mass move almost as much as the attached mass, while the coils at the other end hardly move at all. It
is not obvious, therefore, how to account for the mass of the spring correctly. One way to deal with a concept
like this is to add a term in the equation which includes the spring’s mass multiplied by a correctional factor and
use experimentation to measure the value of the factor. Adjusting the period equation, then, we have
T  2
m  fmS
,
ks
(2)
where mS is the mass of the spring and f can be viewed as the fraction of the mass of the spring to include in the
equation. For this lab, it will be useful to rewrite this equation as
T2 
4 2 fmS
4 2
m
.
ks
ks
(3)
Experimental Procedure:
1. Open DataStudio (the icon labeled “English” on the bottom bar), and click on “Open Activity”, and browse
to the desktop and open the file called “SpringOscillation.”
2. Hang a 100-g mass from the spring, pull down on it to stretch the spring, and then release the mass to make
it oscillate. Do not pull the mass so that you have a very large amplitude when released and try not to let
the mass wildly oscillate. In the Data Studio window, click “Start” and note the data appearing on the graph
of force versus time.
Be sure you understand why the magnitude of the force at the top of the spring also oscillates. Should the force
oscillation have the same period as that of the motion of the mass?
3. Let the motion continue for about 20 periods. Using DataStudio’s curve-fit, fit a sine curve to the plot and
obtain from it the period of the oscillation, and the uncertainty in the period.
4. Open Excel and record the mass and period in a table, and their uncertainties.
5. Repeat this process for at least four other masses, ranging from 10 g to 200 g, recording your results in the
Excel table.
Data Analysis:
6. Open Excel, make a nice labeled, organized table (type in appropriate column headers in the top row, and
use the second row for the units) and input your data with columns for mass and period.
7. In Excel, plot T2 vs. m. Is the relationship linear? According to the equation above it should be--why? Fit a
straight line to the plot and from the slope determine the spring stiffness.
ks = _____________
8. Obtain the uncertainties in the straight-line fit values: Click on the “data” tab, in the upper right corner
select “analysis analysis”, and then choose “regression.” Select all the x-column data, then click in the ycolumn box and then select all the y-column data. A new Excel sheet of numbers will pop up. You are only
interested in the four numbers to the lower left. The bottom two numbers in the first column are the same
numbers as in your linear fit. The bottom numbers in the second column are their uncertainties.
9. From the uncertainty in the slope, calculate the uncertainty in ks. See the “Propagation of Uncertainties”
instructions at the end of this handout.
10. In the table, in any cell not in a data column, type in the value of your inferred spring stiffness. Use a
neighboring to cell to label this as the spring stiffness.
11. Use another column to create the theoretical values of T2, based on Equation (1), ks, and the masses used.
Label the column appropriately, and input the equations into Excel (ask your instructor if you need help.)
12. Add the theoretical periods to the plot (ask for help if you need it). How do the theoretical periods compare
with the measured values? Is there a systematic difference? Could this be due to the mass of the spring?
13. You should, indeed, see a systematic difference, and that all measured values of T2 are off by the same
amount and that this difference extrapolates to the y-intercept. Examine Equation (3) and note that a nonzero y-intercept relates to the mass of the spring and the fractional factor, f.
14. Use the scale to measure the mass of the spring (with uncertainty). Be sure to add this measure to your data
tabel. Determine the fraction, f, of the spring’s mass that contributes to the oscillating mass in the harmonic
motion equation.
15. Using the “Propagation of Uncertainties” instructions and the uncertainties in the y-intercept (from the
regression analysis), mS, and k, infer your uncertainty in f.
Computer Simulations:
How do we know that the reason for the systematic differences between the theoretical and experimental values
of T2 is due to the mass of the spring, and not to some other factor, like friction or air resistance? To confirm
our conclusions, we can also simulate the motion of a mass on a spring, both with and without including the
mass of the spring.
A. Massless spring
Open your VPython code “yourname_motion_ona_spring.py.” Change the value of the spring stiffness to
match your inferred value from the slope of your plotted line. Input one of your mass values, and run the
program for a time long enough (by changing the condition in the while statement) to obtain a plot of about
2
10 periods. On the plot, read the amount of time that passes for some number of periods, divide the two to
obtain a measure of the period in the simulation. Input this number into another column on your Excel data
sheet. Use your code to obtain measures of the period for each of your measured values. Input all these data
into the Excel sheet and add these points to your plot. How do the simulated measures of T2 compare with
the experimental data? How do they compare with the theoretical equation?
B. Spring with Mass
Converting your VPython program to include a the mass of the spring can be tricky. Since there is limited time
for the lab, your instructor has provided a previously written code, called
“Oscillations_witha_MassiveSpring.py”. Open this code. You may examine the code to see how the mass
of the spring is included, if interested. (In short, the spring is divided into 6 pieces, with the mass evenly
distributed, and each piece applies a force to its neighboring pieces).
Note the lines near the top of the code where the spring constant, the attached mass, and the mass of the spring
are defined. Changes these parameters to match your data (pick one of your masses, to start) and run the
code. Note in the shell window the printout of the calculated period. What is the calculated period? How
does it compare with your results? Run the program for each value of your hanging masses and read the
calculated periods. How do the simulation results compare with the experiment now? Did including the
mass of the spring in a simulation succeed, to some degree, in confirming the apparent effect of the mass of
the spring?
Write this lab as a formal report, due on Thursday, 13-February.
Questions to contemplate while writing your report: (NOTE: Do not just list these questions and follow them
with short answers. You should consider your answers and find a way to include them in an appropriate
place in your report. The discussion section, often, is a good place for this.)
1. What assumptions were made in theoretical derivation of the period for the mass on a spring? Why were
such assumptions made? Which particular assumption did this experiment address? Are the theoretical
expectations outside the ranges of the experimental results? Consider the magnitude of the difference
between the oscillation period, according to the theoretical equation (Eq. 1), and the true period you
measured, and assess in which situations this assumption is reasonable (How large is the error in the
theoretical period equation in comparison to how accurately the period needs to be known for systems
involving oscillations with a spring?)
2. As demonstrated by this experiment, how can one modify the oscillation period equation to include the mass
of the spring? When should this modified equation be used instead of the theoretical equation?
3. Did the simulated periods agree with the measured periods, to within the uncertainties? Did the inferred ‘f’
from the simulations agree with the experimental value, to within the uncertainties? Discuss reasons for
limitations in the accuracy of the simulation and how it could have been improved. Considering the limited
accuracy of the simulations, do you feel that the simulation essentially confirmed your experimental result?
4. Did the experimental part, by itself, prove that the mass of the spring must be the reason for the
discrepancies in the measured periods from the theoretical periods? Did the simulation make the case any
better? What would be the weakness of we had only done the VPython simulation? Discuss whether the
combination of the experiment and the simulation is more convincing than either part by itself.
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Propagation of Uncertainties:
A. If you’re interested in knowing a parameter z, which you get with a calculation involving just one
measurable value, x, then, in general the uncertainty in z, which we’ll write as z, depends on the uncertainty
in x (written as x). The uncertainty in z, though, cannot simply equal the uncertainty in x. For example,
imagine if you want to calculate the force of gravity acting on a body whose mass you measure. To get the
force you simply use F = mg. But, now consider the effect of an uncertainty in m. If the value of the mass
can vary by 1 kg, for example, how will that affect the calculation of the force? The force wouldn’t change
by 1 kg...the units aren’t even right. The extra kg, up or down, will cause the calculation of the force to
change by g*1kg, or 9.8 N. In short, for every kg that the mass can change, the force will change by 9.8 N.
In other words, the uncertainty in F is the amount that F would change if m changed by an amount equal to
its uncertainty. Speaking in more general terms, the uncertainty in z is equal to the uncertainty in x times the
rate that z changes with x...which is also known as the derivative of z with respect to x. Therefore, the
uncertainty in z is given by:
z = |dz/dx| x.
Here are two specific examples.
1. If x and z are related by a direct proportionality, i.e
z = cx, where c is a constant,
then an error in x, of magnitude x, will cause an error in z, of magnitude z, as follows.
z ± z = c(x ± x) = cx ± cx.
And so, the uncertainty in z due to an uncertainty in x, in this case, is
z = cx.
Now, let’s apply the equation above and see if we get the same thing:
dz/dx = d/dx (cx) = c. So, z = |dz/dx| x = cx. Yes, this is the same expression.
2. If z depends on x to a power p, i.e. z = cxp,
then z = [d/dx (cx p)] x = pcxp-1 x.
For example, if z = cx, then z = 2cx x
Note, the value you put in for x is the measured value of x.
B. If your parameter of interest, z, depends on two measurables, x and y, and these variables are completely
independent of each other, then, in general, the uncertainty in z is given by:
z2 = [(dz/dx) x]2 + [(dz/dy) y]2
In this lab:
A. You need to calculate the uncertainty in k, where k is related to the slope, s, of the linear fit by
s = 42/ks,
i.e.
ks = 42/s.
So ks = |dks/ds| s, or ks = 42/s2s
Put in your values of s (the slope of the line), and the s that you obtained from the regression analysis tool.
B. You also need to calculate the uncertainty in f. The equation for f is
bk s
b
f 

2
4 mS smS
where b is the y-intercept. There are three uncertainties (b, s (or ks), and mS) contributing to the
uncertainty in f. So, the uncertainty in f is
f 2 = [
1
b
b
b]2 + [ 2
s]2 +[
mS]2
2
sm S
s mS
sm S
or
 b
f  
 smS
2



2
2
2
2
 b  2  s  2  m  2 

s   mS  

2  b 
S
   f       
 
      
 b   s   mS  
 b   s   mS  
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