Download Energy in SHM - Ryerson Department of Physics

Simple Harmonic Motion
PART I: POSITION AND VELOCITY IN SHM
Lots of things vibrate or oscillate. A vibrating tuning fork, a moving child’s playground swing, and
the loudspeaker in a radio are all examples of physical vibrations. There are also electrical and
acoustical vibrations, such as radio signals and the sound you get when blowing across the top of an
open bottle.
One simple system that vibrates is a mass hanging from a spring. The force applied to the hanging
mass by an ideal spring is proportional to how much the spring is stretched or compressed. Given
this force behavior, the up and down motion of the mass is called simple harmonic and its position
can be modeled with
y = Asin(2πft + φ )
In this equation, y is the vertical displacement from the equilibrium position, A is the amplitude of
the motion, f is the frequency of€the oscillation, t is the time, and φ is a phase constant. This
experiment will clarify each of these terms.
Figure 1
OBJECTIVES
Measure the position and velocity as a function of time for an oscillating mass and spring
system.
• Compare the observed motion of a mass and spring system to a mathematical model of simple
harmonic motion.
• Determine the amplitude, period, and phase constant of the observed simple harmonic motion.
•
MATERIALS
computer
Vernier computer interface
Logger Pro
Vernier Motion Detector
200 g and 300 g masses
ring stand, rod, and clamp
spring, with a spring constant of
approximately 10 N/m
twist ties
PRELIMINARY QUESTIONS
1. Attach the 200 g mass to the spring and hold the free end of the spring in your hand, so the mass
and spring hang down with the mass at rest. Lift the mass about 10 cm and release. Observe the
motion. Sketch a graph of position vs. time for the mass.
2. Just below the graph of position vs. time, and using the same length time scale, sketch a graph of
velocity vs. time for the mass.
PROCEDURE
1. Attach the spring to a horizontal rod connected to the vertical 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. Connect the Motion Detector to the DIG/SONIC 1 channel of the interface.
If the Motion Detector has a switch, set it to Normal.
3. 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.
4. Open the file “15 Simple Harmonic Motion” from the Physics with Vernier folder.
5. Make a preliminary run to make sure things are set up correctly. Lift the mass upward a few
centimeters and release. The mass should oscillate along a vertical line only. Click
to
begin data collection.
6. After 10 s, data collection will stop. The position graph should show a clean sinusoidal curve. If
it has flat regions or spikes, reposition the Motion Detector and try again.
7. Compare the position graph to your sketched prediction in the Preliminary Questions. How are
the graphs similar? How are they different? Also, compare the velocity graph to your prediction.
8. Measure the equilibrium position of the 200 g mass. To do this by allow the mass to hang free
and at rest. Click
to begin data collection. After collection stops, click the Statistics
button, , to determine the average distance from the detector. Record this distance as (y0) in
your data table below.
9. Now lift the mass upward about 5 cm and release it. The mass should oscillate along a vertical
line only. Click
to collect data. Examine the graphs. The pattern you are observing is
characteristic of simple harmonic motion.
10. Using the position graph, measure the time interval between maximum positions. If you drag the
mouse from one peak to another you can read along the x-axis the time interval. This is the
period, T, of the motion. The frequency, f, is the reciprocal of the period, f = 1/T. Based on your
period measurement, calculate the frequency. Record the period, frequency and initial phase
constant of this motion in your data table (to find the initial phase constant remember that a
phase of π/2 would correspond to a quarter of a period).
11. The amplitude, A, of simple harmonic motion is the maximum distance from the equilibrium
position. Estimate values for the amplitude from your position graph. Enter the values in your
data table.
12. Repeat Steps 8–11 with the same 200 g mass, moving with larger amplitude than in the first run.
13. Change the mass to 300 g and repeat Steps 8–11. Use an amplitude of about 5 cm. Store a good
run made with this 300 g mass on the screen. You will use it for several of the Analysis
questions.
DATA TABLE
Run
Mass
(g)
y0
(cm)
A
(cm)
T
(s)
f
(Hz)
φ
(rad)
1
2
3
ANALYSIS
1. View the graphs of the last run on the screen. Compare the position vs. time and the velocity vs.
time graphs. How are they the same? How are they different?
2. Turn on the Examine mode by clicking the Examine button, . Move the mouse cursor back
and forth across the graph to view the data values for the last run on the screen. Where is the
mass when the velocity is zero? Where is the mass when the velocity is maximum?
3. Does the frequency, f, appear to depend on the amplitude of the motion? Do you have enough
data to draw a firm conclusion?
4. Does the frequency, f, appear to depend on the mass used? Did it change much in your tests?
5. You can compare your experimental data to the sinusoidal function model using the Manual
Curve Fit feature of Logger Pro. Try it with your 300 g data. The model equation in the
introduction, which is similar to the one in many textbooks, gives the displacement from
equilibrium. However, your Motion Detector reports the distance from the detector. To compare
the model to your data, add the equilibrium distance to the model; that is, use
y = y 0 + Asin(2πft + φ )
where y0 represents the equilibrium distance.
€ graph to select it.
a. Click once on the position
b. Choose Curve Fit from the Analyze menu.
c. Select Manual as the Fit Type.
d. Select the Sine function from the General Equation list.
e. The Sine equation is of the form
y=A*sin(Bt +C) + D.
Compare this to the form of the equation above to match variables; e.g., φ corresponds to C, and
2πf corresponds to B.
f. Adjust the values for A, B and D to reflect your values for A, f and y0. You can either enter
the values directly in the dialog box or you can use the up and down arrows to adjust the
values.
g. The phase parameter φ is called the phase constant and is used to adjust the y value reported
by the model at t = 0 so that it matches your data. Since data collection did not necessarily
begin when the mass was at the equilibrium position, φ is needed to achieve a good match.
h. The optimum value for φ will be between 0 and 2π. Find a value for φ that makes the model
come as close as possible to the data of your 300 g experiment. You may also want to adjust
y0, A, and f to improve the fit. Write down the equation that best matches your data.
6. Does the model fit the data well? How can you tell?
7. Predict what would happen to the plot of the model if you doubled the parameter for A by
sketching both the current model and the new model with doubled A. Now double the parameter
for A in the manual fit dialog box to compare to your prediction.
8. Similarly, predict how the model plot would change if you doubled f, and then check by
modifying the model definition.
9. Click
, and optionally print your graph.
EXTENSIONS
1. Investigate how changing the spring amplitude changes the period of the motion. Take care not
to use too large amplitude so that the mass does not come closer than 40 cm to the detector or
fall from the spring.
2. How will damping change the data? Tape an index card to the bottom of the mass and collect
additional data. You may want to take data for about 2 minutes. Does the model still fit well in
this case?
3. Describe how would you have to extend this experiment to discover the relationship between the
mass attached to the spring and the period of his motion.
PART II: ENERGY IN SHM
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 =
1
2
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 ΔKE + ΔPEelastic = 0, which we can test experimentally.
OBJECTIVES
•
•
Examine the energies involved in simple harmonic motion.
Test the principle of conservation of energy.
MATERIALS
computer
Vernier computer interface
slotted mass set, 50 g to 300 g in 50 g steps
slotted mass hanger
Logger Pro
Vernier Motion Detector
wire basket
spring, 1-10 N/m
vertical stand
PRELIMINARY QUESTIONS
1. On the graphs of position velocity from part I, label the times when the spring has its greatest
elastic potential energy. Then mark the times when it has the least elastic potential energy.
2. Now try to sketch the graphs of kinetic energy vs. time and elastic potential energy vs. time.
PROCEDURE
1. The Motion Detector should still be connected to the DIG/SONIC 1 channel of the interface and its
switch set to Normal.
2. 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. Open the experiment file “17b Energy in SHM.” Logger Pro is now set
up to plot the applied weight vs. position.
3. Click
to begin data collection. Hang a 50 g mass from the spring and allow the mass to
hang motionless. Click
and enter 0.49, the weight of the mass in newtons (N). Press
ENTER to complete the entry. Now hang 100, 150, 200, 250, and 300 g from the spring,
recording the position and entering the weights in N. When you are done, click
to end
data collection.
8. Click on the Linear Fit button, , to fit a straight line to your data. The magnitude of the slope
is the spring constant k in N/m. Record the value in the data table below.
9. Open the experiment file “17c Energy in SHM.” In addition to plotting position and velocity,
three new data columns have been set up in this experiment file (kinetic energy, elastic potential
energy, and the sum of these two individual energies). You may need to modify the calculations
for the energies. Adjust the parameter for mass and spring constant as appropriate.
10. With the mass hanging from the spring and at rest, click
to zero the Motion Detector.
From now on, all distances will be measured relative to this position. When the mass moves
closer to the detector, the position reported will be negative.
12. Start the mass oscillating in a vertical direction only, with an amplitude of about 10 cm. Click
to gather position, velocity, and energy data.
DATA TABLE
Spring constant
N/m
ANALYSIS
1. Click on the y-axis label of the velocity graph to choose another column for plotting. Click on
More to see all of the columns. Uncheck the velocity column and select the kinetic energy and
potential energy columns. Click
to draw the new plot.
2. Compare your two energy plots to the sketches you made earlier. Be sure you compare to a
single cycle beginning at the same point in the motion as your predictions. Comment on any
differences.
3. If mechanical energy is conserved in this system, how should the sum of the kinetic and
potential energies vary with time? Choose Draw Prediction from the Analyze menu and draw
your prediction of this sum as a function of time.
4. Check your prediction. Click on the y-axis label of the energy graph to choose another column
for plotting. Click on More and select the total energy column in addition to the other energy
columns. Click
to draw the new plot.
5. From the shape of the total energy vs. time plot, what can you conclude about the conservation
of mechanical energy in your mass and spring system?
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, position measured upward and labeled y,
whose origin is located at the bottom of the relaxed spring of constant k (no force applied).
2. 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 position 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.