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Evaluation of Data:
length (cm)
0.00 20.0 40.0 60.0 80.0 100
0.00
0.200
0.400
0.600
0.800
1.00
1.20
1.40
1.60
1.80
2.00
2.20period vs. length
mass (g)
0.00 40.0 80.0 120 160
0.00
0.200
0.400
0.600
0.800
1.00
1.20
1.40
1.60
1.80
2.00
2.20 period vs. mass
angle (degrees)
0.00 10.0 20.0 30.0 40.0 50.0
0.00
0.200
0.400
0.600
0.800
1.00
1.20
1.40
1.60
1.80
2.00
2.20
2.40 period vs. angle
This graph needed to be linearized.
It appears to be of the general form
y = k xn where n is less than zero.
This graph did not need to be
linearized. It appears to be of the
general form y = k xn where n is
zero.
This graph did not need to be
linearized. It appears to be of
general form y = k xn where
zero.
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In order to linearize the period vs. length data, the length
data had to be raised to the 0.5
power. As shown in the graph above, the graph became linear.
From the regression line
and statistics calculated by the Graphical Analysis software an
algebraic relationship
could be generated:
T = (.204 s/cm.5) l.5 - .0302 s
By using the 5% rule, the y intercept can be ignored since it is
only 1.5% of the largest
period value. Therefore, the equation reduces to:
T = (.204 s/cm.5) l.5
The period of a simple pendulum is directly proportional to the
square root of the length
of the pendulum. The slope has the units of s/cm.5 and its
significance will be discussed
in the Conclusion section.
=length^.5
0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.0
0.00
0.200
0.400
0.600
0.800
1.00
1.20
1.40
1.60
1.80
2.00
2.20
2.40
0.2040.00317 -0.03020.0251 1.00
Slope Y Intercept C.O.R.Statistics:
period vs.
period vs. length^.
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Conclusion:
Several new terms were introduced through this experiment. The
period of a
simple pendulum is the amount of time required for the pendulum
bob to swing back and
forth one time. This term can actually be used to describe any
event that repeats in a
regular way such as the period of the second hand of a clock
being 60 s and likewise theperiod of revolution of the Earth around
the sun is 365 days.
Another important term introduced through this experiment was
linearization.
Linearization is the process by which the exact algebraic
relationship between two
variables is determined by performing calculations on one
variable until when graphed, a
linear relationship appears. This was evidenced in the case in
this experiment when
initially period vs. length yielded a curved graph (not linear)
but period vs. length to the
0.5 power yielded a straight graph.
Much was learned about the behavior of a simple pendulum. Length
appears to
affect the period of a simple pendulum but mass and angle of
release do not. In the post-
lab discussion, it was revealed that the pendulums behavior is
different for small anglesin comparison to large angles. No precise
relationship was discussed. Perhaps this would
be an appropriate topic for additional exploration.
A relationship between period and length was determined:
T = (.204 s/cm.5) l.5
In the post-lab discussion, it was revealed that the slope does
indeed have significance. It
can be re-written:
T = 2 (l/g.5)
Where g is the acceleration of gravity for Earth = 980 cm/s/s.
The equation can be
rearranged:
T = (2 /g.5) l.5
When evaluated, the term in parenthesis is found to equal 0.2.
Therefore, the slope that
was found must equal the term in parenthesis, 2 /g.5. Since the
results obtained in this
experiment are in agreement with a well-established
relationship, they can be considered
to be good results.
