A1261158307uxiYE

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Pendulum Mock Lab
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Part 1 Experimental Design
Focused Research Question
I will investigate how the period of a pendulum
depends on its length
Variables
Independent - L length of the pendulum
string (m) Dependent - t time for one
swing (s) Controlled - angle of swing
(degrees), mass of the bob (g),
Equipment
Ceiling hook, string, hanging mass, stopwatch,
electronic scale, protractor, meter stick marked
in mm, scissors, paper
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Part 1 Experimental Design
Method
The hook is attached to the ceiling and string is
cut to make ten lengths differing by 25cm from
over two meters in length to under half a meter
in length. The pendulum bob is massed and the
string is tied to the ceiling hook at one end and
to the bob at the other. The distance from the
pivot position at the hook to the center of the
bob is taken as the pendulum length.
The angle of swing is set at five degrees. To do
this repeatedly, a sheet of paper is marked with
a vertical plumb line and five degrees is marked
with reference to that line. The paper is taped
to the ceiling behind the string when in its
relaxed vertical position.
The bob is pulled back from its relaxed position
to five degrees and the timer is started on
release. The time for five complete back and
forth motions of the pendulum bob is recorded.
This is repeated three times. The pendulum is
then adjusted for a new length and the process is
repeated.
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Part 2 Data Collection and Processing
Data Table
Identifying, specific title Multiple tables must
be numbered
Table 1. Length vs. Period For a Pendulum Mass
of the pendulum bob is 103 ? 1 g The angle of
swing is 5 ? 1 degrees
Constants are listed above the table
Length of Pendulum L / cm ?L ? 1cm Time For 5 Cycles Trial 1 Trial 2 Trial 3 t / s ?t ? 0.6s Time For 5 Cycles Trial 1 Trial 2 Trial 3 t / s ?t ? 0.6s Time For 5 Cycles Trial 1 Trial 2 Trial 3 t / s ?t ? 0.6s Average Time For 5 Cycles t / s ?t ? 0.6s Period t / s ?t ? 0.1 s
215 14.3 14.4 14.4 14.4 2.9
195 13.8 13.6 14.0 13.8 2.8
175 13.1 13.1 13.2 13.1 2.6
155 12.2 12.4 12.5 12.4 2.5
135 11.5 11.6 11.7 11.6 2.3
115 10.7 10.8 10.8 10.8 2.2
95 9.9 9.8 9.9 9.9 2.0
75 8.8 8.8 8.9 8.8 1.8
55 7.6 8.0 7.7 7.8 1.6
35 6.3 6.4 6.3 6.3 1.3
Units given per measured quantity
Uncertainty recorded to 1 SF
Appropriate data set Large spread of data
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Processing Raw Data
(Example calculations put under the data table)
Determining the period
Average for 5 cycles time1 time2 time3

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14.3 s 14.4 s 14.4 s 14.4 s
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Period Av. Time for 5 swings
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14.4 s 2.9 0.1 s 5
Calculated uncertainty in the period
?T ?t / number of swings 0.6 s / 5
0.1 s
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Full descriptive title
Graph
Curve-fit your data No dot-to-dot
Axes labels with units
Data, column options, options to select error
bars Type in value
Using (0,0) shows data does not fit a straight
line
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• Language of physics
• - Uncertainties in Graphs

Because all data contains uncertainties at the
very least data points should be marked as small
circles or crosses. If the absolute uncertainty
in each measurement is known then uncertainty
bars should be used to turn the data point into a
data area. Never connect data points dot to dot.
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• Language of physics
• - Graphs (recognizing functions)

So youve found a question that needs to be
answered, identified variables, restricted them,
produced a mathematical model and devised an
experiment that will collect data. How do you
know that your data supports or refutes your
model?
The answer lies in graphs It is important to be
able to recognize the shape of a graph and be
able to relate that shape to a mathematical
function. You can then compare this function to
your model. Some functions found in physics are
shown on the next two slides!
Direct
y? x y  kx k
slope of the line y is directly
proportional to x Note If the line does not go
through (o,o) it is linear y kx
b b y intercept
Independent y  k y does not
depend on x
b
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• Language of physics
• - Graphs (recognizing functions)

Inverse Proportional y ? 1/x
y  k/x y k x-1 y
is inversely proportional to x
Square y  ?  x2 y  kx2 y is proportional to the
square of x
Square root y  ?  vx y  k v x Y k x1/2 Y is
proportional to the square root of x
Note all these functions are all power
functions as they fit the general expression, y
A xB where A and B are constants
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• Language of physics
• - Graphs (recognizing functions)

Exponential Growth y  anbx y increases
exponentially with x
Exponential Decay y  an-bx Y decreases
exponentially with x
Periodic y  A sin (Bx  C) Y varies
periodically with x
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• Language of physics
• - Graphs (recognizing functions)

Lets say that you are investigating how the
period (T) of a pendulum (time for one swing)
depends on the length (l) of the pendulum and you
have come up with a mathematical model that says,
T 2? ? ( L/ g). You then test this model
experimentally and plot a graph of T vs. L (shown
below). Which function best describes your data?
The curve through the data cant be a straight
line because as the length decreases, the period
decreases so at zero length we would expect the
pendulum to take no time to swing back and forth.
The model says that T ? ? L or T ?
L1/2 This suggests that we should look at a power
function and in particular a square root function
(ykx1/2)
You can see that the computer generated power
function fit is a good fit to our model as the
data fits T (2.06) L0.44 The power 0.44 is
close to 0.5 (1/2) We can also use our data to
verify the constant g because Comparing our model
with the fit equation we find 2? / ?g
2.06 so g (2? / 2.06)2 9.3
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• Language of physics
• - Graphs (turning a curve into a
  straight line)

Sometimes its nicer to see a relationship from a
straight line graph rather than a curved graph,
especially when we use uncertainty bars (next
slide). To turn a curve into a straight line you
look at the proportional statement. If you wanted
to turn the pendulum curved graph into a straight
line graph what would you plot on each axis?
Now T ? ? L so our data should fit a straight
line if we plot T vs. ? L instead of T vs. l
Length of Pendulum L/m 0.01 2.15 1.95 1.75 1.55 1.35 1.15 0.95 0.75 0.55 0.35
Length L1/2 / m1/2 1.47 1.40 1.32 1.24 1.16 1.07 0.97 0.87 0.74 0.59
Period t/s 0.1 2.9 2.8 2.6 2.5 2.3 2.2 2.0 1.8 1.6 1.3
Processing Raw Data
?Length ?(2.15m) 1.47 m1/2
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Straight Line Graph
You can see that the data now fits a nice
straight line
The slope (1.8) can be related back to the
proportionality constant which is 2? / ?g
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Analyzing Straight Line Graph
The graph opposite shows possible slopes within
an uncertainty range
We can easily sketch the best-fit line (black)
and the two worst (min slope (blue) and the max
slope (green)) acceptable lines by using the
extremes of the uncertainty bars on the first and
last points.
The graph thus performs the function of averaging
the data.
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Determining uncertainty in the slope
Determining max slope
mmax (Tmax ?T) (Tmin - ?T)
Lmax1/2 Lmin1/2
(2.9s0.1s) (1.3s-0.1s) (1.47 m1/2
0.59 m1/2)
mmax 1.8 s / 0.88 m1/2
2.0 sm-1/2
Determining min slope
mmin (Tmax - ?T) (Tmin ?T)
Lmax1/2 Lmin1/2
(2.9s-0.1s) (1.3s0.1s) (1.47 m1/2
0.59 m1/2)
mmin 1.4 s / 0.88 m1/2
1.6 sm-1/2
So slope of the graph is 1.8 0.2 sm-1/2
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Part 3 Conclusion and Evaluation
Concluding
The slope of the graph represents 2? / ?g
gmax (2? / mmin)2
(2? / 1.6 sm-1/2)2
15 m/s2
gmin (2? / mmax)2
(2? / 2.0 sm-1/2)2
9.9 m/s2
gfit (2? / m)2
(2? / 1.8 sm-1/2)2
12 m/s2
gactual 9.81 m/s2
g can thus be quoted as 12 3 m/s2. The actual
value of g (9.81 m/s2) is within this range. It
can be seen that small differences in the slope
can lead to large differences in the value of g
because of the squaring operation.
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Part 3 Conclusion and Evaluation
Concluding
Ill show you another way another way of
determining ?g without using graphical
interpretation. This method is limited in that it
is based on one point and so doesnt reflect the
range of your data.
From your table choose a mid point, L 115
1cm, T 2.2 0.1s.
T 2? ?(L/g)
so. g L (2? / T)2
9.4 m/s2
so. gav L (2? / T)2
(1.15m) (2? / (2.2s))2
Using uncertainty rules for multiplication and
powers.
?g / g ?L / L 2 (?T / T)
(1 / 115) 2 (0.1 / 2.2)
0.0996
?g (0.0996) x g
(0.0996) x 9.4 m/s2
0.93 m/s2
gav can thus be quoted as 9.4 0.9 m/s2. The
actual value of g (9.81 m/s2) is within this
range.
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Part 3 Conclusion and Evaluation
Concluding
You can calculate a percentage difference between
the accepted value of g and your calculated
average value using..
This shows a good correlation but it doesnt
reflect the larger range in possible values of g
that 12 3 m/s shows.
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Part 3 Conclusion and Evaluation
Concluding The direction of systematic errors
Looking at the Period vs. Square Root of the
Length Graph, it can be seen that the best-fit
line does not go through the origin, (0,0). It
gives a positive y intercept. If the slope
(gradient) was steeper, gfit would have been
smaller than 12 m/s2.
If the string length was measured before being
hung from the ceiling support with the mass
attached, the mass could have stretched the
string giving rise to a systematic uncertainty in
the length of the string. This would be more
significant for shorter lengths.
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Part 3 Conclusion and Evaluation
Evaluating Procedure(s)
Although the mass of the pendulum bob should not
be a factor in this experiment, air drag on the
string and bob might increase the period of swing
thus giving a non-zero y intercept on the
straight-line graph.
Timing the swing time of shorter pendulum lengths
gives a larger uncertainty in determining g than
for longer lengths. String lengths are limited by
the height of the room.
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Part 3 Conclusion and Evaluation
Improving the Investigation
Use of as thin a string as possible, such as
fishing line and a lead weight as the bob with
large mass and small volume should offer the
least amount of aerodynamic drag.
To reduce reaction-time random uncertainties, a
photogate timer may be used at the bottom of the
pendulum swing. Again, a series of swings should
be timed and the average time for one swing
should be determined
The string length should be measured after the
bob is mounted before and after taking a series
of measurements. A higher ceiling room giving
rise to longer pendulum lengths would reduce the
effects of uncertainty in L and t but swinging
through more air would cause drag to factor more.