LabQuest
17
Energy in Simple Harmonic Motion
We can describe an oscillating mass in terms of its position, velocity, and acceleration as a
function of time. We can also describe the system from an energy perspective. In this experiment,
you will measure the position and velocity as a function of time for an oscillating mass and spring
system, and from those data, plot the kinetic and potential energies of the system.
Energy is present in three forms for the mass and spring system. The mass m, with velocity v, can
have kinetic energy KE
KE  21 mv 2
The spring can hold elastic potential energy, or PEelastic. We calculate PEelastic by using
PE elastic  21 ky 2
where k is the spring constant and y is the extension or compression of the spring measured from
the equilibrium position.
The mass and spring system also has gravitational potential energy (PEgravitational = mgy), but we
do not have to include the gravitational potential energy term if we measure the spring length
from the hanging equilibrium position. We can then concentrate on the exchange of energy
between kinetic energy and elastic potential energy.
If there are no other forces experienced by the system, then the principle of conservation of
energy tells us that the sum KE + PEelastic = 0, which we can test experimentally.
OBJECTIVES
 Examine the energies involved in simple harmonic motion.
 Test the principle of conservation of energy.
MATERIALS
LabQuest
LabQuest App
Motion Detector
spring, 10–20 N/m
ring stand
slotted mass set with hanger, 50 g to
300 g in 50 g steps
wire basket
twist ties
PRELIMINARY QUESTIONS
1. Sketch a graph of the height vs. time for the mass on the spring as it oscillates up and down
through one cycle. Mark on the graph the times where the mass moves the fastest and
therefore has the greatest kinetic energy. Also mark the times when it moves most slowly and
has the least kinetic energy.
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2. On your sketch, label the times when the spring has its greatest elastic potential energy. Then
mark the times when it has the least elastic potential energy.
3. From your graph of height vs. time, sketch velocity vs. time.
4. Sketch graphs of kinetic energy vs. time and elastic potential energy vs. time.
PROCEDURE
1. Attach the spring to a horizontal rod connected to
the ring stand and hang the mass from the spring
as shown in Figure 1. Securely fasten the 200 g
mass to the spring and the spring to the rod, using
twist ties so the mass cannot fall.
2. Place the Motion Detector at least 75 cm below
the mass. Make sure there are no objects near the
path between the detector and mass, such as a
table edge. Place the wire basket over the Motion
Detector to protect it.
3. If your Motion Detector has a
switch, set it to Normal. Connect
the Motion Detector to DIG 1 on
LabQuest and choose New from the File menu. If
you have an older sensor that does not auto-ID,
manually set up the sensor.
4. On the Meter screen, tap Rate. Change the datacollection rate to 50 samples/second and the datacollection length to 2 seconds. Select OK.
5. Make a preliminary run to make sure things are
set up correctly. Lift the mass upward about
5 centimeters and release. The mass should
oscillate along a vertical line only. Start data
collection.
Figure 1
6. After 2 seconds, data collection will stop and a
graph of position vs. time and velocity vs. time will be displayed. The position graph should
show a clean sinusoidal curve. If it has flat regions or spikes, reposition the Motion Detector
and try again. To collect again, get the mass moving, and then start data collection. Once you
have a clean run, do not move the Motion Detector or the ring stand for the rest of the
experiment.
7. Sketch or print your graphs and compare to your predictions. Comment on any differences.
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Energy in Simple Harmonic Motion
8. To calculate the spring potential energy, it is necessary to measure the spring constant, k.
Hooke’s law states that the spring force is proportional to its extension from equilibrium, or
F = –kx. You can apply a known force to the spring, to be balanced in magnitude by the
spring force, by hanging a range of weights from the spring. The Motion Detector can then be
used to measure the equilibrium position. You will plot the distance vs. weight to find the
spring constant k.
a. Display a single graph by choosing Show Graph from the Graph menu and selecting Graph
1.
b. On the Meter screen, tap Mode. Change the data-collection mode to Events with Entry.
c. Enter the Name (Weight) and Units (N). Select OK.
In this mode, the position of the mass will only be measured when you choose. You will then
enter the weight in newtons.
9. Collect force and distance data.
a.
b.
c.
d.
Hang a 50 g mass from the spring and allow the mass to hang motionless.
Start data collection.
When the reading is stable, tap Keep. Enter 0.49 and select OK to store the data pair.
In the same manner, hang 100, 150, 200, 250, and 300 g from the spring, recording the
position and entering the weights in N. It is important that the length of the mass not
change as the value of the mass is changed.
e. After entering the last weight, stop data collection.
10. The graph of position vs. force should be linear. The magnitude of the slope is k–1, since force
is on the x-axis. Find the slope of the data.
a. Choose Curve Fit from the Analyze menu.
b. Select Linear as the Fit Equation. The linear-regression statistics for these two data
columns are displayed on the screen.
c. The inverse of the slope magnitude is the spring constant k in N/m. Record the slope and
its inverse in the data table.
d. Select OK.
11. Remove the 300 g mass and replace it with a 200 g mass for the remaining steps.
12. Tap Meter, then tap Mode. Change the data-collection Mode to Time Based. Change the datacollection rate to 50 samples/second and the data-collection length to 2 seconds. Select OK.
13. In order to measure distances from the equilibrium position of the mass, it is necessary to zero
the Motion Detector. Measuring from equilibrium allows easy calculation of the elastic
potential energy, since the distance will correspond directly to spring stretch or compression.
a. Ensure that the mass is hanging at rest from the spring.
b. Choose Zero from the Sensors menu. When the zeroing process is complete the reading
should be close to zero.
Distances will now be measured from the current position of the mass, with displacement
above the current position measured as positive. Displacement below the current position will
be read as negative.
14. Collect position vs. time data as you did before. Lift the mass upward about five centimeters
and release. The mass should oscillate along a vertical line only. Start data collection.
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15. After 2 seconds, data collection will stop and a graph of position vs. time and velocity vs.
time will be displayed. If it has flat regions or spikes, reposition the Motion Detector and try
again. To repeat data collection, get the mass moving and start data collection.
DATA TABLE
Slope of distance vs. force
m/N
Spring constant
N/m
ANALYSIS
1. Display a single graph by choosing Show Graph from the Graph menu and selecting Graph 1.
2. Calculate the bob’s elastic potential energy (PE)
a. Tap Table to display the data table and choose New Calculated Column from the Table
menu.
b. Enter the Name (PE) and Units (J).
c. Select the equation, ABX^C.
d. Select Position as the Column for X.
e. For the A value, enter 0.5.
f. For the B value, enter the value you obtained for the spring constant, N/m.
g. For the C value, enter 2.
h. Select OK to display the graph of potential energy (PE) vs. time.
3. Inspect your elastic potential energy vs. time graph for the motion of the spring-mass system.
Explain its shape. Be sure you compare to a single cycle beginning at the same point in the
motion as your predictions. Comment on any differences.
4. Print or sketch the potential energy vs. time graph.
5. Calculate and graph KE.
a.
b.
c.
d.
e.
f.
g.
h.
i.
Tap Table to display the data table.
Choose New Calculated Column from the Table menu.
Enter the Name (KE) and Units (J).
Select the equation, ABX^C.
Select Velocity as the Column for X.
For the A value, enter 0.5.
For the B value, enter the mass of your bob in kg.
For the C value, enter 2.
Select OK to display the graph of kinetic energy (KE) vs. time.
6. Inspect your kinetic energy vs. time graph for the motion of the spring-mass system. Explain
its shape. Be sure you compare to a single cycle beginning at the same point in the motion as
your predictions. Comment on any differences.
7. Print or sketch the kinetic energy vs. time graph.
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Energy in Simple Harmonic Motion
8. If mechanical energy is conserved in this system, how should the sum of the kinetic and
potential energies vary with time? Sketch your prediction of this sum as a function of time.
9. Calculate the total energy KE + PE, and store the result in a third new column.
a. Tap Table to display the data table and choose New Calculated Column from the Table
menu.
b. Enter the Name (TE) and Units (J).
c. Select the equation, X + Y.
d. Select KE (J) as the Column for X.
e. Select PE (J) as the Column for Y.
f. Select OK to display the graph of total energy (TE) vs. time.
10. Display all three energy plots on the same graph.
a. Choose Graph Options from the Graph menu.
b. Select PE, KE, and TE for the Graph 1 Y-Axis.
c. Select OK to display a graph of PE, KE, and TE.
11. Compare your total energy graph prediction to the experimental data for the spring-mass
system. What do you conclude from the total energy vs. time graph (marked with a closed
circle) about the total energy of the system? Does the total energy remain constant? Should
the total energy remain constant? Why? If it does not, what sources of extra energy are there
or where could the missing energy have gone?
EXTENSIONS
1. In the introduction, we claimed that the gravitational potential energy could be ignored if the
displacement used in the elastic potential energy was measured from the hanging equilibrium
position. First write the total mechanical energy (kinetic, gravitational potential, and elastic
potential energy) in terms of a coordinate system, distance measured upward and labeled y,
whose origin is located at the bottom of the relaxed spring of constant k (no force applied).
Then determine the equilibrium position s when a mass m is suspended from the spring. This
will be the new origin for a coordinate system with distance labeled h. Write a new
expression for total energy in terms of h. Show that when the energy is written in terms of h
rather than y, the gravitational potential energy term cancels out.
2. If a non-conservative force such as air resistance becomes important, the graph of total energy
vs. time will change. Predict how the graph would look, then tape an index card to the bottom
of your hanging mass. Take energy data again and compare to your prediction.
3. The energies involved in a swinging pendulum can be investigated in a similar manner to a
mass on a spring. From the lateral position of the pendulum bob, find the bob’s gravitational
potential energy. Perform the experiment, measuring the horizontal position of the bob with a
Motion Detector.
4. Set up a laboratory cart or a glider on an air track so it oscillates back and forth horizontally
between two springs. Record its position as a function of time with a Motion Detector.
Investigate the conservation of energy in this system. Be sure you consider the elastic
potential energy in both springs.
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