Simple Harmonic Motion
I. Introduction: Simple Harmonic Motion (SHM) is a common and very important type of
motion in which the position of an object repeats regularly with time. Examples of SHM include
the vibration of a guitar string, brain waves, and approximately the swing of a playground swing.
The motion can be described as a sine function. We will study the oscillation of a mass on a
spring; the expected period is T =2  m/ k where m is mass and k is the spring constant.
Study I: Period You will measure T as a function of m use your results to determine k.
Study II: Hooke's Law You will determine k directly by measuring the static extension of the
spring as a function of the applied force.
Study III: Simple Harmonic Motion You will study the periodic nature of SHM by creating
graphs of position, velocity, and acceleration and measuring the Amplitude, Period, and Phase.
Study IV: Kinetic and Potential Energy You will calculate the change in kinetic and potential
energy of the system as height and velocity change.
II. Required Equipment: Photogate, Motion Sensor. Pasport interface, Vertical Post, Crossbar,
Clamp, Hook. Spring. 3 Pendulum bobs. Meter stick.
Preliminary Sketch: Before beginning formal data taking, attach the mass to the spring and
start it oscillating. Draw the mass at a few points in its motion and write down the velocity (in
terms of v = 0, max, and min) and acceleration (a = 0, max, min) at those points:
A) Highest point
B) Lowest point
C) Midpoint (equilibrium)
III. Procedure:
Study I: Period as a function of mass.
1. Hang a mass on the end of the spring so that the bottom of the mass
is positioned just above the photogate. You will probably have to
reposition the photogate.
Check your alignment: start the mass oscillating by gently lifting it
and releasing. If it hits the photogate, keep adjusting.
2. Open DataStudio and enter the Setup screen. If the blue Pasport
box is not displayed, click Interface and select the Pasport
interface. Choose File->New Activity. Choose “Photogate
Timing” from the list.
3. Drag the Graph icon from the lower left box to the “Time between
bands (s)” icon in the upper left box.
Spring
M
Motion
Photogate
Figure 1: Spring/Mass system
4. Determine the period T: Start the mass oscillating so that the bottom edge of the mass just
passes through the photogate, cutting the light beam (the red light should blink). Only a
small amplitude is required. Avoid striking the photogate.
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5. Press Start. Hit the Statistics button  and use the drop down menu to add Count and
Standard Deviation. Record enough oscillations until the standard deviation is sufficiently
small. Record the mass of the bob and T in Table 1.
6. Repeat the process until you have made measurements for all three pendulum bobs of
different mass.
Study I Analysis: Graphical Determination of the Spring Constant (k).
1. Open Graphical Analysis. Enter your three mass values (in kg) and the corresponding
period (T in sec). Create a new calculated column for the square of the period (T2 = T^2).
2. Plot T2 versus M. The errors in the T2 values and M values are sufficiently small that you
can ignore them. Use Linear Fit and double-click to add the error in fit.
Since
2
2
T =4  / k∗M for a mass on a spring, the slope should be:
slope=
4
k
2
3. Calculate k from the slope. The error  slope will allow you to calculate the error in k:
 slope /slope = k /k
4. Print ONE copy of this graph for your group, including fit and errors.
Study II: Hooke's Law: Static Extension as a Function of Mass.
Hooke's Law states that the force exerted by a spring is proportional to the amount it is stretched:
F=−k  x We can measure the k from static extension of the spring and compare to Study I.
For each mass, measure the change in length  x of the spring as pictured in Figure 2. Record
your values in Table 2.
Study II Analysis: Graphical Determination of the Spring
Constant (k).
Using Graphical Analysis, plot  x on the x-axis and the
weight of each mass F = mg on the y-axis. A straight line
relationship verifies Hooke's Law for the spring. Add a linear
fit (with errors). The slope of the line should be equal to k, the
spring constant, and the error in the slope will give the error in
k. Print out a plot of your data. Record k and  k .
Figure 2: x is the static extension
due to F=mg
Study III: Motion of the Spring Oscillator
1. Move the photogate completely out of the way. Place the motion sensor below the spring
oscillator. Choose one of your masses to hang on the spring. Create a new activity in
DataStudio. Don't add any additional sensors (click Okay or Cancel in the pop-up window).
2. DataStudio should automatically find the motion sensor and make a Position vs. Time
graph. If not, double-click on the digital plug and add it. Click Setup and place checks next
to Position, Velocity, and Acceleration. Increase the sampling rate from 10 Hz to 25 Hz.
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3. Make a graph that records position, velocity, and acceleration: Drag Graph 1 (in the lower
left-hand box) up to Velocity (m/s) and Acceleration (m/s2). All three graphs should be in
the same window. Drag the Table icon to Position (m).
4. Check the the mass hangs at least 15 cm from the motion sensor. Get the spring oscillating
and hit Start. You should see a smooth sine curve (Hit Autoscale). If the curve isn't smooth,
check whether the bob is coming too close to the sensor. Check your alignment. If it really
isn't working, try a different hanging mass.
5. Once you have a smooth curve, record around 10 periods and click Stop.
6. Click the Position vs Time Plot. Use the drop down Curve-Fit menu to select Sine. Double
click on the box to see the form of the equation (including helpful labels for A,B,C, and D):
2  x−C
y  x =A sin[
]D
B
7. Record these in your data table. Also record the DataStudio labels which define the values.
Compare your period with the period measured in Study I.
8. As the mass continues to oscillate, hit START and STOP several more times. You'll see very
similar plots, but they don't overlap. You are modifying one of the parameters: which one?
9. Sketch x(t), v(t), and a(t): Highlight a short portion of your graph (about 1 second). Sketch
a position, velocity, and acceleration graphs for one period (each period has one maxima and
one minima) all with the same time axis. Pay particular attention to the alignment of the
graphs (i.e. when the position graph is zero, the velocity graph is... zero? maximum?
Minimum?). Include helpful labels.
Figure 3: Using the Smart Tool will help you match up
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minima and maxima.
10. Energy Loss: Tape an index card to the base of the pendulum and repeat the measurement of
position vs. time. Qualitatively sketch the new behavior of the Position graph.
You're going to have to work at this a bit. You may need to try a different mass, or a
different amplitude to see the effect – it will be very obvious once you get it.
IV. Discussion:
Study I: Period of the Oscillator
Are your results consistent with T =2  m/ k ? Discuss the evidence for your conclusion.
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.
Study II: Hooke's Law
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. Do the two values agree? If not, can you
think of any possible reasons for the difference?
Study III: Motion of the Oscillator
Is your period consistent with the value from Study I? Which parameter where you modifying
by repeatedly hitting start and stop? Describe this physically. How did attaching a paper card
introduce energy loss into the mass-spring system?
Energy of the Oscillator
When is the kinetic energy maximum? When is the potential energy maximum? What are the
three types of energy considered in the system?
Figure 4: The motion of the spring-mass system
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TABLE 1. Period data for the Spring Oscillator
BOB
M=mass of the
bob (kg)
 T  (s)
T= period
(s)
Count
1
2
3
TABLE 2. DATA ON Hooke’s Law
mass hung (kg)
extension, Y (m)
F=mg (N)
1
2
3
Table 3. Motion of the Spring Oscillator
Label (period, phase)
Parameter Values +/- Error (ROUND THESE)
A=
B=
C=
D=
Before you leave the lab: You should have filled out all data tables, printed graphs for Studies I
and II. Your goal for Study III is to understand the relationships between position, velocity,
acceleration, and energy (both physically and graphically). Be sure that you completed the
Preliminary Sketch and the additional sketches required in Studies III (x,v,a vs. t and energy
loss x vs. t).
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