General Physics I Lab
M9 Simple Harmonic Motion
General Physics I Lab
M9 Simple Harmonic Motion
Purpose
In this experiment you will investigate the oscillatory motion of a mass on a spring.
Equipment and components
Science Workshop 750 interface, spring, masses and mass hanger, force sensor, motion
sensor, meter ruler, electronic balance, rod, stand and clamp (2).
Background
Imagine a spring hanging vertically from a support. When no mass hangs at its end, the
spring has a length L (called the rest length). When a mass m is added to the spring, its length
is increased by ΔL. The equilibrium position of the mass is now a distance L + ΔL from the
spring’s support. What happens when the mass is pulled down a small distance from the
equilibrium position? The spring exerts a restoring force, F = -kx, where x is the
displacement of the mass, k is the spring constant. The negative sign indicates that the force
points opposite to the direction of the displacement of the mass. The restoring force causes
the mass to oscillate up and down. The period T of oscillation for simple harmonic motion is
given by
T = 2π
m
k
As the mass oscillates, the energy of the system continually interchanges between the kinetic
energy and some form of the potential energy. If friction is ignored, the total energy of the
system remains constant. When the mass is at its highest point, the gravitational potential
energy is at its maximum. When the mass is at its lowest point, the elastic potential energy is
at its maximum.
Procedure
In Part I of the experiment, a force sensor is used to measure the force that stretches the
spring as a weight is added to one end of the spring. You will measure the amount of distance
that the spring stretches using a meter ruler. The computer program displays the force and the
distance. The slope of the best fitted line to the measured force versus distance curve gives
the value of the spring constant k.
In Part II of the experiment, a motion sensor is used to measure the oscillatory motion of the
mass on the end of the spring. The computer program displays the position of the mass and its
velocity as a function of time. The program is used to determine the oscillation period T,
which is compared to the theoretical value given above.
Part I: Determination of the spring constant
Computer setup
1. Connect the Science Workshop 750 interface to the computer. Turn on the interface and
the computer.
2. Connect the DIN-5 plug of the force sensor into Analog Channel A of the interface.
3. Open the “M9-I” program in the course folder.
• The program will open with a graph display of the measured force (N) versus
Stretch (m) and a digital display of the measured force (N).
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General Physics I Lab
M9 Simple Harmonic Motion
•
•
Set the “Sampling Rate” of the force sensor to 5Hz by clicking the “Setup” button
and then click “Force Sensor” icon. A Sensor properties dialog (where the “Sampling
Rate” can be set) will appear.
The Stretch (in meter) data are set as manual sampling keyboard data (that is, you
need to measure them manually and enter the values). You can check it by clicking
the “Experiment” menu and selecting “Set Sampling Options”.
Equipment calibration
1. Click the “Experiment” Menu and select “Monitor Data” to begin the calibration.
2. Suspend the spring from the force sensor’s hook. Adjust the zero adjust (offset) screw
on the side of the force sensor with a screwdriver until the reading of the force sensor
becomes zero.
3. Suspend a 500 gram mass under the spring. Adjust the sensitivity screw on the side of
the force sensor with a screwdriver until the reading of the force sensor becomes -4.9N
(0.5kg x 9.8ms-2, negative sign means a pulling force).
4. Repeat steps 2 and 3 several times to obtain the precise adjustments.
5. Click the “Stop” button to end the calibration.
Data recording
1. Suspend the spring from the force sensor’s hook, with no hanger and mass added. Use a
meter ruler to measure the position of the bottom end of the spring. Record this
measurement as the spring’s equilibrium position in the data sheet in the lab report.
2. Click the “Start” button to begin data recording. The “Start” button will turn to be a
“Keep” button, and “Run #1” will appear in the Data list.
3. Click the “Keep” button to open a “Please enter a value” window. It prompts for a
keyboard input of the data of stretch. Enter “0” (since the spring is not stretched yet) and
click “OK” to record the value.
Figure 1 Setup for measurement of the spring constant
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General Physics I Lab
M9 Simple Harmonic Motion
4. Add the 100 gram hanger to the end of the spring, as shown in Fig. 1. Measure the new
position of the end of the spring. Record the difference between the new position and the
equilibrium position as Δx or “Stretch” (in meters) in the data sheet.
5. Click the “Keep” button again. Enter the value of Δx (in meters). Click “OK” to record
the value.
6. Add one 100 gram mass to the hanger and repeat the measurement of the new position of
the end of the spring. Click the “Keep” button, enter the new value of Δx, and click
“OK”.
7. Continue to add mass in 100 gram increments each time. Measure the new stretched
position of the end of the spring. Record the new values of Δx until the total mass equals
500 grams.
8. Click the “Stop” button to end data recording.
Data analysis
1. Use the curve fitting tool of the graph display to determine the spring constant k.
2. Print the graph with the data points and the fitted line. Paste the graph and record the
value of k in the data sheet.
Part II: Measurement of the oscillation period
Computer setup
1. Connect the motion sensor to an adapter box which is connected to the interface box with
two stereo phone plug cables. Match the labels ‘1/3’ to Digital Channel 1 and ‘2/4’ to
Digital Channel 2 respectively.
2. Open the “M9-II” program in the course folder. The program will open with a graph
display for plots of Position (m) and Velocity (m/s) versus Time (s).
Data recording
1. Add 200 grams of mass to the hanger (i.e. total mass = 300g), as shown in Fig. 2. Record
their total mass in the data sheet.
2. Place the motion sensor on the floor directly beneath the mass hanger. Adjust the position
of the spring so that the minimum distance from the mass hanger to the motion sensor is
larger than 40 cm (as shown in Fig. 2).
3. Pull the mass down about 2 cm and then release it. The hanging mass begins to oscillate.
4. Click the “Start” button to begin recording data.
5. The plots of the position and velocity of the oscillating mass will appear in the Graph
display.
6. Click the "Stop" button to end data recording after about ten cycles of oscillation are
recorded.
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General Physics I Lab
M9 Simple Harmonic Motion
Figure 2 Setup for simple harmonic motion
Data analysis
1. Click the “Scale to Fit Data” button to rescale the Graph display.
2. Print the position-time and velocity-time graph and paste it in the lab report.
NOTE: Resize the graphs to suitable size before printing them.
3. Click the “Smart Tool” button. The cursor is a cross-hair. The X- and Y-coordinates of
the cursor’s position are shown next to the horizontal and vertical axes.
4. Use the Smart Cursor to find an average value of the oscillation period. Magnify the time
scale to increase accuracy if needed. Move the cursor/cross-hair to the first peak in the
plot of position versus time and record the value of time (shown below the horizontal
axis) in the data sheet.
5. Move the Smart Cursor to each consecutive peak in the plot and record the value of time.
6. Find the period of each oscillation by calculating the difference between the times for
each successive peak. Find the average of the periods. Record your result.
7. Find the theoretical value of T based on the measured values of k and m.
NOTE: Use mass in unit of kilograms for this calculation.
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Name
Date
Lab session
(Day & time)
Lab partner
M9 Simple Harmonic Motion Lab Report
A. Answer the following questions BEFORE the lab session (5 pts each)
1. Write down the equation of motion for a mass on a spring. Find the solution of the
equation and sketch the position versus time plot. On the same graph, sketch the velocity
versus time plot.
2. From the solution of the equation, find the position of the mass relative to its equilibrium
position when the absolute value of the velocity of the mass is greatest. Compare your
result with the plots in (1).
3. When the mass m is removed and replaced by a mass 2m, the period of the motion is
(Circle the correct answer):
(a) 2T ,
(b)
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2T ,
(c ) T ,
(d )
T
,
2
( e)
T
.
2
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General Physics I Lab
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B. Results and data analysis
Part I: Determination of the spring constant (25 pts)
Equilibrium Position = _______
Mass (g)
Stretch Δx (m)
100
200
300
400
500
Spring constant, k = _________
Paste your graph here:
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General Physics I Lab
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Part II: Measurement of the oscillation period (25 pts)
Total mass = _________
Peak #
1
Time (
)
Period (
)
2
3
4
5
6
7
Average period of oscillation = ________
Calculated period of oscillation = ________
Paste your graph here:
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General Physics I Lab
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Answer the following questions after the experiment (5 pts each)
4. How does your calculated oscillation period compare to the measured value? Find the
percent difference between your calculated value and the measured value.
5. What are the possible reasons for the difference between the calculated and measured
values of T?
6. When the position of the mass is farthest from the equilibrium position, what is the
velocity of the mass?
7. What will happen if the mass on a spring is driven by an external force at a frequency
close to its oscillation frequency?
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C. Summary of the experiment (15 pts)
Summarize what you have done and your interpretation of the results, especially in respect to
how they match the goal of the experiment and what you have learned from this experiment
(2-3 paragraphs, less than 450 words).
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