Find the "phase difference" during oscillation between
force and displacement
force and acceleration
force and velocity
kinetic energy and spring potential energy
Check to see if total energy is constant over
several full cycles
Find the "time constant" of energy loss over
a period of several minutes
Optional:
Investigate the effect of air resistance on energy loss.
Setting up the experiment
Set the slide switch on the force probe to 10 N.
Place the motion sensor on the floor.
Attach the force sensor to CH 1,
and the motion sensor to Dig/Sonic 1.
If that fails, try instead grabbing a file
from a shared folder:
Go to "Student Shares" -> "University Physics" ->
"Team Physics 311" -> "Lab Pro",
and drag the file Conservation_of_Energy
(note the underscores) onto your local desktop.
Double-click on the file on your Desktop
to start LabPro.
Calibrate the force sensor, by
Choosing "Experiment -> Calibrate Sensors"
Choose the force sensor
Hold the force sensor vertically, with nothing hanging
from it, and enter a force of 0 Newtons, then press "Keep"
Hang a 50-g hanger, plus 500-g weight, from the force sensor;
enter a force of 5.39 Newtons, then press "Keep"
Choose one of the tiny springs.
Attach the aluminum mass hanger (50 grams) plus
an extra 50-gram weight, making the total mass 100 grams.
The mass should be about 15 or 20 cm below the bench top
at equilibrium.
Zero both sensors with the block at rest.
Check your set-up by lifting the mass
straight up about 8 cm and releasing it.
Avoid side-to-side motion.
What do you expect to see for position as a function of time?
Collect data; do you see what you expected?
If not, adjust the sensors until you get what you expect.
You should see the position oscillate around 0 meters
(to within about +/- 0.001 m),
and the force oscillate around 0 Newtons
(to within about +/- 0.0003 N).
The amplitude of the motion should be around 7 or 8 cm.
Look carefully; if the centers of oscillation aren't zero,
you may have to adjust the readings manually to force them
to zero.
Adjust to zero manually:
Return the system to equilibrium and zero all sensors.
Collect data with the system at rest.
Use Analyze:Statistics to determine the average
position of the mass.
Click on the oval Labpro icon to bring up the Sensors Window,
and click on the motion sensor. On the pop-up menu,
click Set Offset. A box will pop up with an initial value
of the offset.
Subtract the average position from the offset and enter this
difference as the new offset, then click OK.
(E.g. if the average position is 0.0123 and
the initial offset is -0.612345, the new offset is -0.624645.)
Collect data again and the position should have an average
very close to zero. Close the Sensors window.
Each time you zero the motion detector you will need to repeat this.
Look at the highest and lowest displacement the mass
reaches. If these two extremes are not the same size,
make appropriate adjustments and take new data.
If sometimes helps to wait one minute after starting
the block in motion before you collect data.
Be absolutely certain that you have the equipment properly zeroed or your
results will be bogus!
Collecting a first dataset
Start the mass moving and let it oscillate for about twenty or thirty
seconds;
higher order vibrations will damp out during this time,
leaving very smooth, simple, harmonic motion.
Press Collect to gather data.
You should see nice smooth sine wave shapes for position
and velocity.
Before you go any further, ask an instructor to verify
that your data is good enough.
You can (and should) print out several graphs
to illustrate your report.
If you want to plot more than one measurement on the vertical axis,
click the label on the vertical axis, click More on the pop up menu,
and check the boxes for the additional quantities you wish to plot.
Draw on a sheet of graph paper diagram with three panels,
in which the panels show the mass
as it moves downwards through the equilibrium position
after it has moved half-way downwards from the equilibrium
position to the lowest position
when it is at the lowest position
On your panels, use arrows to show the displacement from equilibrium,
velocity, acceleration, and net force on the mass.
Look at the graph of Position versus Time.
It looks sort of like a sine curve, doesn't it?
Suppose that we represent the mass' height
above equilibrium by the equation
y(t) = A sin (ω t + φ)
What is the constant A for your data?
Write an equation for the vertical
velocity as a function of time, v(t).
Write an equation for the vertical
acceleration as a function of time, a(t).
Write an equation for the vertical
force as a function of time, F(t).
Compare the Force and Position graphs,
and give a verbal description of their relation.
Are force and displacement
"exactly in phase",
"exactly out of phase", or something else?
Explain, using your diagram from question 1,
and your equations from question 2.
Compare the Force and Acceleration graphs,
and give a verbal description of their relation.
Are force and acceleration
"exactly in phase",
"exactly out of phase", or something else?
Explain, using your diagram from question 1,
and your equations from question 2.
Compare the Force and Velocity graphs,
and give a verbal description of their relation.
Are force and velocity
"exactly in phase",
"exactly out of phase", or something else?
Explain, using your diagram from question 1,
and your equations from question 2.
Find the spring constant k by making a graph
of one particular quantity versus another.
Print the graph.
On your printed graph, make a linear fit to the graph
and determine the slope and its uncertainty.
(Hint 1: the left-hand side should be "Force", in Newtons).
(Hint 2: the force constant is a positive value)
Find the "total mass" m of the moving system
by making a graph
of one particular quantity versus another.
Again print the graph and make a linear fit.
(Hint the left-hand side should again be "Force", in Newtons).
How does this "total mass" compare to the mass of the
block plus the hanger? If they are not the same,
explain how big the difference is, and explain where
the difference ("extra" mass or "missing" mass) originates.
Modify the formulae within LabPro for kinetic energy (KE) and
potential energy (U).
To modify the column for kinetic energy, go to
Data: Column Options and choose the name
of the column you want to modify: Kinetic Energy.
In the Equation box,
you should see the formula for kinetic energy, which will
look something like this:
0.5 * 0.111 * "velocity"^2
You should know the effective mass from part 5.
Replace the incorrect mass value of 0.111
with your derived mass from part 5.
In a similar fashion, replace the
bogus spring constant in the equation for
the Potential Energy column
with your actual spring constant.
the total mechanical energy, E = KE + U.
Make graphs that allow you to discuss conservation of energy.
Examples of graphs might be any one of the energies
versus time (or position, or velocity, or acceleration),
one energy versus another energy, etc.
You need to try different graphs and decide what
they tell you about energy conservation.
Is energy conserved during your measurements?
There are two ways to answer this question:
what is the "peak-to-peak" change in the total energy,
expressed as a fraction of the average total energy?
You should be able to state something like
"total energy bounces around its mean by around +/- 2 percent."
what is the secular change in the average total energy
during the period you measured? You might compare the
average energy during the first second of data
to the average energy during the final second of measurements.
Analysis over a very long period
You ought to find that total mechanical energy is almost
exactly the same over a few cycles of oscillation.
But what if we monitor the system over many, many
cycles?
Change the Experiment->Data Collection
parameters so that you measure 10 samples per second
over a period of 180 seconds (yes, that's about 3 minutes).
Reset the system to equilibrium and re-zero the sensors
Pull the weight down about 10 cm and release it carefully.
Wait about 30 seconds to let the wiggles dissipate
Collect measurements over the long period
While you are waiting for this to finish, you might perform
calculations for the previous portions of the experiment,
or start writing your report.
Look at (and print) a graph of Total Energy versus Time. Is energy
conserved over this long period?
What is the "half-life" of energy? That is, how long does it
take for the total energy to decay to half its original value?
What is the "half-life" of the displacement's amplitude?
That is, how long does it
take for the maximum displacement during each cycle
to decay to half its original value?
Is this the same as the "half-life" of energy?
Should it be?
How much energy would be left if the spring continued to
oscillate for twenty minutes?
Some of the energy lost by the spring and hanger is due
to work performed by the force of air resistance.
Recall that this force depends on several parameters,
one of which is the cross-section area A of the
moving object.
What is the cross-section area of your moving object?
Express the area in square centimeters.
Create a "sail" by cutting a piece of paper so that it has
a larger area -- twice the original A of your
object. Tape this sail onto the bottom of the hanger
so that the moving object now has double its original
area. Repeat the Analysis over a long period
procedure (steps 10 to 17 above).
Determine the half-life for this modified object.
(If you have time)
Now make the "sail" three times the original area A.
Determine the half-life again.
(If you have time) Do it again for a sail four times the
original area.
In theory, how should the amount of energy
lost to air resistance depend on the area of the object?
In practice, how does the amount of energy
lost to air resistance depend on the area of the object?
Discuss.
For more information
You can save your data file(s) and analyze them outside
of class by
downloading a demo version of Logger Pro
or by grabbing a copy of LabPro from the
SVPHY01 server in Science Bldg Zone.