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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
LabPro interface
Palm handheld
Data Pro program
Vernier Motion Detector
spring, 10-20 N/m
slotted mass set with hanger, 50 g to
300 g in 50 g steps
wire basket
ring stand
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. Plug the Motion Detector into the DIG/SONIC 1
port of the LabPro interface. Connect the
handheld to the LabPro using the interface cable.
Firmly press in the cable ends. Firmly press in the
cable ends.
4. Set up the handheld and interface for the Motion
Detector. Turn on the handheld and start Data
Pro, by tapping the Data Pro icon on the
Applications screen. Choose New from the Data
Pro menu or tap
to reset the program.
a. On the Main screen, tap
.
b. While still on the Setup screen, tap
.
c. Enter “0.02” as the time between samples in
seconds using the onscreen keyboard (tap
“123”) or using the Graffiti writing area.
d. Enter “100” as the number of samples.
e. Tap
twice to return to the Main screen.
Figure 1
5. Make a preliminary run to make sure things are set up correctly. Lift the mass upward about
five centimeters and release. The mass should oscillate along a vertical line only. Tap
to begin data collection.
6. After 2 seconds, data collection will stop and a graph of distance vs. time will be displayed.
The distance 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 again,
and then tap
to repeat 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. When you have finished examining the graphs, tap
on the Graph screen, to return to
the Main screen.
b. On the Main screen, tap
.
c. On the Setup screen, tap
, then choose Events with Entry.
d. Enter the Entry Label (Weight) and Unit (N). You can enter this information using the
onscreen keyboard (tap “abc”), or by using the Graffiti writing area.
e. Tap
twice to return to the Main screen.
f. Tap
to get ready to collect data.
In this mode, the position of the mass will only be measured when you choose. You will then
enter the weight in newtons on the handheld.
9. Take your force and distance data.
a. Hang a 50 g mass from the spring and allow the mass to hang motionless.
b. Tap
and enter “0.49” using the onscreen keyboard as the weight of the mass in
newtons (N). Tap
to store the data pair.
c. 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.
d. After entering the last weight, tap
, then tap
to view a graph of distance vs.
force.
10. The graph of distance 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. Tap
, then tap
.
b. From the Fit Equation menu, choose Linear. 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. Tap
to display the distance vs. force plot with the fitted equation.
e. Tap
twice to return to the Main screen.
11. Remove the 300 g mass and replace it with a 200 g mass for the remaining steps.
12. As you did before, set up the handheld for a time graph.
a. On the Main screen, tap
.
b. While still on the Setup screen, tap
.
c. Enter “0.02” as the time between samples in seconds using the onscreen keyboard (tap
“123”) or using the Graffiti writing area.
d. Enter “100” as the number of samples.
e. Tap
to return to the Main screen.
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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. Tap
.
b. With the mass hanging at rest from the spring, tap
. When the
button changes
back to
, the process is complete, and readings for the sensor should be close to zero.
c. Tap
twice to return to the Main screen.
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. Take distance 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. Tap
to begin data
collection.
15. After 2 seconds, data collection will stop and a graph of distance vs. time will be displayed. If
it has flat regions or spikes, reposition the Motion Detector and try again. To collect again,
get the mass moving again, and then tap
to repeat data collection.
DATA TABLE
Slope of Distance vs. Force
m/N
Spring constant
N/m
ANALYSIS
1. From the position and velocity of the mass data you can calculate the spring potential energy
and the kinetic energy. First, you will calculate the kinetic energy KE using ½mv2, storing the
data in a new column.
a.
b.
c.
d.
e.
On the Graph screen, tap
, then tap
.
Enter the column name (KE) and unit (J).
Tap
and choose the formula, A * B * X^C.
Choose X = DIG/SON 1:VEL(m/s) as the Column for X.
For the A = value, enter “0.5” using the onscreen keyboard (tap “123”) or using the
Graffiti writing area.
f. For the B = value, enter the mass of your bob in kg.
g. For the C = value, enter “2”.
h. Tap
to display the graph of KE vs. time.
In a similar way you can calculate the spring’s elastic potential energy (PE) using ½kx2. Here,
the stretch or compression of the spring x comes from your distance data and the spring
constant k from the linear fit done in step 11. The PE result is stored at each time in a new
column.
i. On the Graph screen, tap
. Then tap
and choose New Column.
j. Enter the column name (PE) and unit (J).
k. Tap
and choose the formula, A * B * X^C.
l. Choose X = DIG/S 1:DIST(m) as the Column for X.
m. For the A = value, enter “0.5”.
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Energy in Simple Harmonic Motion
n. For the B = value, enter the value you obtained for the spring constant, N/m.
o. For the C = value, enter “2”.
p. Tap
to display the graph of PE vs. time.
2. Examine the KE and PE graphs.
a. Inspect your kinetic energy vs. time graph for the motion of the spring-mass system by
tapping on the y-axis label of the displayed graph and choosing KE(J). 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.
b. Tap on the y-axis label of the displayed graph and choose PE(J). 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.
c. Record the three energy graphs by printing or sketching.
3. 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.
4. Finally, calculate the total energy KE + PE, and store the result in a third new column.
a.
b.
c.
d.
On the Graph screen, tap
. Then tap
and choose New Column.
Enter the column name (TE) and unit (J).
Tap
and choose the formula, X1 + X2.
Choose X1 = KE(J) as the Column for X1. Use the Scroll buttons on the handheld to scroll
through the options for the Column for X1.
e. Choose X2 = PE(J) as the Column for X2. Use the Scroll buttons on the handheld to scroll
through the options for the Column for X2.
f. Tap
to display the graph of total energy vs. time.
5. Compare your total energy graph prediction (from the Preliminary Questions) 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.
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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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