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Simple
Harmonic Motion
Experiment
Profile:
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1.
Background
Simple
Harmonic Motion is a form of periodic motion in which
a point of body oscillates along a line about a central
point in such a way that it ranges an equal distance on
either side of the central point and is always proportional
to its distance from it. One way of visualising SHM is
to imagine a point rotating around a circle of radius
r at a constant angular velocity w. If the distance from
the centre of the circle to the projection of this point
on a vertical diametre is y at time t, this projection
of the point will move about the centre of the circle
with Simple Harmonic Motion. A graph of y against t will
be a sine wave, whose equation is y = rsinwt.
We will use the mass-spring system to illustrate the concept
of Simple Harmonic Motion in this experiment. During SHM,
the mass moves upwards and downwards, changing the length
of the spring. Three forms of energy are involved in this
motion - gravitational potential, translation kinetic
and elastic potential. In this lab, you will examine the
relationships between these three quantities throughout
a single cycle of motion and test the conservation of
mechanical energy.
2.
Objective
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To
study the characteristics of Simple Harmonic Motion. |
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To investigate the motion of a mass-spring system. |
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To
test the conservation of mechanical energy in Simple
Harmonic Motion. |
3.
Equipment List
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Datalogger
interface connected to PC |
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Motion
sensor |
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Masses
and mass hanger |
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Base
and support rod with clamp |
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Spring |
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| 1.
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Connect the datalogger interface to a PC with the software
installed.
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Connect the motion sensor to the appropriate channel
of the interface. See Figure 1.
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| 3. |
Record the total mass.
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| 4. |
Position the mass so it hangs 60-70 cm above the motion
sensor when it is in equilibrium.
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Launch the software needed for measuring position and
velocity during your experiment.
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| 6. |
Pull
the mass down approximately 10 cm (if the spring will
allow this much extension) and release it, setting it
in to SHM.
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| 7. |
When the motion is smooth and straight up and down,
begin data collection.
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| 8. |
Stop
the data collection after two or three cycles have
elapsed.
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| 9. |
Save the data file collected for further analysis. Repeat
the procedure using a new hanging mass or a new spring.
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Analysis:
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As
the mass moves up and down, what energies are involved?
How did the mass get the original amount of each kind of
energy?
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| 2.
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What relationship gives the amount of elastic energy in
the spring?
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| 3. |
How can you tell that this is a Simple Harmonic Motion?
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| 4. |
Compare the amounts of each kind of energy (in a qualitative
way) at the three extremes of motion - highest, middle
and lowest point.
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| 5. |
Construct a spreadsheet that contains the values of the
following quantities:
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Constants: Mass, Spring Constant |
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* Variables: Position (height), Velocity, Time
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* Calculated Values: |
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Gravitational
Potential Energy (Ug) |
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Kinetic
Energy (K) |
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Elastic
Potential Energy (Ue) |
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Total
Mechanical Energy (Et) |
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| 6. |
Record the position, velocity and time for at least 15 different
positions during a single cycle of the motion.
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| 7. |
Calculate
Ug based on height above the lowest point of the motion.
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| 8. |
Calculate
Ue based on distance below the highest point of the motion.
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| 9. |
Graph all four calculated values as functions of time.
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| 10. |
What is your conclusion regarding the total mechanical energy
during a cycle of motion?
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Extensive:
If one considers that energy must be conserved, and therefore
the total energy at each position must be the same, the lab can
be re-configured to dynamically determine the spring constant
k. What value of k would keep the total energy constant, and how
does this agree/disagree with the value of k determined in a separate
measurement?
Video
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