Physics 221
EXPERIMENT 6
SIMPLE HARMONIC MOTION
I. Introduction:
In this experiment the behavior of simple oscillating systems will be studied and compared with
the behavior expected for simple harmonic motion (SHM). When a body is subjected to a linear
restoring force (for example, F = -kx, for a mass attached to a spring of force constant k), the
acceleration is proportional to the displacement and is in the opposite direction. Newton’s
Second Law gives: ma = - kx. This may be rewritten as ma + kx = 0, or d2x/dt2 + 2x = 0, were
 = sqrt (k/m). The solution of this equation (the motion of the body) can be written: x(t) = A
cos (t + ) and is called simple harmonic motion. The period of the oscillatory motion is T =
2, and is independent of the amplitude A.
 For a mass m on a spring (spring oscillator), the period is expected to be T = 2 m/k .
You will measure T as a function of m for such a system and use your results to
determine k. You will also determine k directly by measuring the static extension of the
spring as a function of the applied force.
 For a simple pendulum, a small mass suspended by a light string, the restoring force is
approximately linear in its dependence on , the angle the pendulum makes with the
vertical (but not exactly, since it is proportional to sin  rather than ). (The motion is
approximately SHM for small amplitudes where sin  and  are nearly the same.) In this
approximation, the period is independent of the mass and is T = 2 L/g , where L is the
length of the pendulum. You will determine the period for various masses and lengths
and will use your results to determine g, the acceleration due to gravity.
II. Required Equipment:
Lab Bench computer running Science Workshop and Graphical Analysis software.
photogates. Pasco interface.
A. Spring Oscillator. Vertical post, crossbar, clamp and hook. Spring. Set of pendulum
bobs. Meter stick. Ruler.
B. Simple Pendulum. Vertical post, crossbar, and string clamp. String and cutter for string.
Set of pendulum bobs. Protractor.
III. Procedure:
A. Spring Oscillator.
1) Period as a function of mass. Use the set of three pendulum bobs as masses. Hang the spring
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from the stand. Hang a mass m on the end of the spring, so that the bottom of the mass is
positioned just above the photogate. Make sure that the photogate is plugged into input #1 on the
Pasco interface. Start the mass oscillating vertically from its equilibrium position, so that the
bottom edge of the mass just passes through the photogate, cutting the light beam. Only a small
amplitude is required. Avoid striking the photogate.
Determine the period T of the
oscillations, by running the
Science Workshop program and
analyzing
with
Graphical
Analysis.
Spring
4.
5.
6.
7.
1. To measure the period, T, start up the
Data studio program, by double-clicking
on the Data studio icon on the Desktop.
You should see the familiar Control
Window and Menu Bar.
2. Click and drag the digital plug icon to
M
digital channel 1 to logically connect the
photogate to that channel. From the
pop-up timing menu, scroll down and
Motion
select “photogate and picket fence”.
3. Create the data table and graph by
dragging the little table and graph icons
Photogate
to the same channel as you did the plug,
and selecting “period(t)” to display for each. In the table heading, click on the [0.00...] button
and select 4 digits to be displayed.
Start the motion and record at least N = 20 time measurements.
Click on the Stop Button.
Select the data column in the table and click the [] symbol to obtain mean (T) and standard
deviation  (T) values, and N = number of measurements. Record these values.
(note: T = (T)/ N ).
Repeat the process until you have made measurements for all three pendulum bobs of
different mass. For each pendulum bob, determine its mass and estimate the error (m). Also
measure and record the mass of the spring.
Graphical Determination of the Spring Constant (k).
1. Open Graphical Analysis, from Recent Applications or from the desktop. In the blank table
which appears, label successive columns and enter your three mass values (M in kg), the
corresponding period (T in sec), and the square of the period (T2 = T^2).
2. From the Graph Menu, turn on Regression Statistics. Plot T2 versus M. The errors in the T2
values and M values are sufficiently small that you can ignore them.
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3. The slope of the regression (straight) line should be numerically equal to (42)/k. The error
in the slope (slope) can be obtained from the regression statistics.
4. From the slope we can now write: k = (42)/slope. The error in k can be obtained from the
expression: k/k = slope/slope.
5. Print a copy of this graph for each group member, including regression statistics. Record
values of slope, slope, k, and k in your lab report form.
2) Hooke's Law: Static Extension as a Function of Mass. We can also measure the spring
constant (k) from static extension of the spring, and compare the value of k so obtained with that
of the dynamical measurement above.
1. Use the three masses used above. For each, measure the static extension Y of the spring as a
function of the weight W = mg which is hung from it.
2. Estimate the error Y in your measurements of Y.
3. Using Graphical Analysis, plot the static extension Y of the spring as a function of the weight
W = mg. Include error bars on the Y values; you may assume the errors in m and thus in W
are negligible.
4. A straight line relationship would verify Hooke's Law for the spring. Make sure Regression
Statistics is selected under the Graph Menu. The slope of the line should be Y/W = (1/k)
and the error in the slope (slope) will be given from the Regression Statistics.
5. From this result, we may determine the force constant from the expression k = 1/slope and its
error from k/k = slope/slope.
6. Print out a plot of your data with the Regression Statistics for each group member. Record k,
k, slope, and slope in your physics lab report form.
B. Simple Pendulum
1) Period as a function of mass.
Use the three pendulum bobs of identical size but different mass in this part of the experiment.
Keep the amplitude m of the swings smaller than about 20o and try to use the same amplitude
for your repeated measurements. do not let the pendulum strike the photogate
Make a simple pendulum roughly 50 cm long from a piece of string and one of the pendulum
bobs. Measure the length L (from pivot point to center of mass) and estimate the error L in your
length measurement.
You will measure the periods at small amplitude for this pendulum and the other two
pendulum bobs of different mass, being careful to keep the length constant when you change the
bob.
Data for Simple Pendulum
1. Double-click on the Science Workshop icon on the computer desktop (i.e. screen). If Science
Workshop is already open, select New under the File menu. A Control Window with a Menu
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Bar across the top should appear on the screen.
2. Click and drag the digital plug image to the digital channel into which the photogate is
connected to logically connect the photogate to that channel. From the pop-up timing menu,
scroll down and select “photogate and pendulum”.
3. Create the data table and graph by dragging the little table and graph icons to the same
channel as you did the plug, and selecting “period(t)” to display for each. In the table
heading, click on the [0.00...] button and select 4 digits to be displayed.
4. Start the pendulum oscillating, by pulling it sideways several inches and releasing it.
5. Now Click on the [REC] button in the Control Window to begin data recording. The graph
will update dynamically in real time as the data taking progresses. Depending upon
conditions, you may be able to record 30 to 50 oscillations . Then click on the “Stop” button.
6. Select the data column in the table and click the [] symbol to obtain mean (T) and standard
deviation  (T) and number of measurements (N). Record these values .
(note: T = (T)/ N ).
7. Repeat the above procedure for the other bobs and enter these data in your data table.
2) Period as a function of length
Using the bob of intermediate mass, determine T for 3 or more different pendulum lengths L.
Cover as wide a range of L as possible. Adjust the position of the string clamp so that the bob
intercepts the photogate. (Again, keep the amplitude reasonably small and fairly constant.)
Your measurement from part B1 using this bob can be one of the measurements. Measure L and
estimate the error L in each case. (A single length error will suffice for all the measurements
unless you have reason to believe different errors are needed.)
Record about 30 oscillations for each length, using the same procedure as above. Record the
mean (T) and standard deviation (T) for each pendulum length in a data table.
Use the values from this data table for this part. In Graphical Analysis, select New under the File
menu and make 5 more columns. The first 5 columns will be L, T, (T), N and
2
T = (T)/ N . Next 2 columns will be set up as T2 and  T 2 (where T = 2TT) using the same
procedure as in A1.
Plot T2 versus L , Include vertical error bars on the T2 values, with magnitude T2. Turn on
Regression Analysis to get the slope and its error. Make a Print Screen for each partner.
From the equation T = 2 L/g expected for SHM, we see that T2 = (42/g)L and the slope
2
2
T2/L of the T2 versus L line should be 42/g, the error in the slope is (4  / g ) g . Thus
2
g = 4 /slope and g/g =  slope/slope.
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NOTE: You have now completed all the necessary in-class work. Before you leave the
laboratory make sure you have made all the required measurements and estimates of errors
needed to complete the experiment. Get your instructor to initial your work.
IV. Analysis and Discussion:
A. Spring Oscillator.
Are your results consistent with the variation of period with mass which is expected from
theory? Discuss the evidence for your conclusion. Was Hooke's Law obeyed for the
spring you used? Discuss the evidence for this. Compare the two values of the spring
constant k which you found. Compute the difference and the error in the difference. Do
the two values agree? If not, can you think of any possible reasons for the difference?
What was the value of the intercept in your plot of T2 versus m? Do you expect T=0
when m=0? Discuss this briefly.
B. Simple Pendulum
1) Period as a function of mass.
Are your measurements of T for the three pendulum bobs of different masses
consistent with the period being independent of the mass?
2) Period as a function of length.
Calculate a value for g and  g using your fit results. Compare the value of g you
obtain with the standard value. Are they consistent? If not, can you think of any
possible reasons for the difference? Also determine the intercept (the value of L for
which T2 = 0) and the error in the intercept. What would you expect this intercept to
be and why? Are your results consistent with what you would expect?
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TABLE 1. DATA ON Spring Oscillator
BOB ID
M=mass of
the bob (kg)
T= period
(sec)
 T(s)
1
2
3
mspring 
TABLE 2. DATA ON Hooke’s Law
mass hung (kg)
extension, x (m)
F=mg (N)
1
2
3
4
5
x 
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TABLE 3. DATA ON Pendulum
(length kept constant, mass varied)
BOB ID
mass of bob
T= period
(sec)
 T(s)
1
2
3
TABLE 4. DATA ON Pendulum
(mass kept constant ,length varied)
BOB ID
length L (m)
T= period
(sec)
1
2
3
L 
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 T(s)