Transcript Slide 1

Pendulum Mock Lab
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 Controlled -
t = time for one swing (s)
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
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 it’s 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.
Part 2: Data Collection and Processing
Data Table:
Identifying, specific title
Table 1. Length vs. Period For a Pendulum
Multiple tables must be numbered
Mass of the pendulum bob is 103  1 g
Constants are listed above the table
The angle of swing is 5  1 degrees
Length of
Pendulum
L / cm
L =  1cm
Trial 1
Time For 5 Cycles
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
Appropriate data set
Large spread of data
Units given per measured
quantity
Uncertainty recorded
to 1 SF
Processing Raw Data:
(Example calculations put under the data table)
Determining the period:
Average for 5 cycles = time1 + time2 + time3
3
= 14.3 s + 14.4 s + 14.4 s
=
14.4 s
3
Period = Av. Time for 5 swings
=
14.4 s
5
= 2.9 ± 0.1 s
5
Calculated uncertainty in the period:
T = t / number of swings = 0.6 s / 5 = 0.1 s
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
 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”.
 Language of physics
- Graphs (recognizing functions)
So you’ve 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
Independent
y=k
y does not depend on x
b
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
 Language of physics
- Graphs (recognizing functions)
Inverse Proportional
Square
Square root
y  1/x
y  x2
y  √x
y = k/x
y = kx2
y=k√x
y = k x-1
y is proportional to the
square of x
Y = k x1/2
y is inversely proportional to x
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
 Language of physics
- Graphs (recognizing functions)
Exponential Growth
Exponential Decay
Periodic
y = anbx
y = an−bx
y = A sin (Bx + C)
y increases exponentially with x
Y decreases exponentially with x
Y varies periodically with x
 Language of physics
- Graphs (recognizing functions)
Let’s 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 can’t 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 (y=kx1/2)
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
 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
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
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.
Determining uncertainty in the slope:
Determining max slope:
mmax = (Tmax + T) – (Tmin - T)
= (2.9s+0.1s) – (1.3s-0.1s)
Lmax1/2 – Lmin1/2
mmax = 1.8 s / 0.88 m1/2
(1.47 m1/2 – 0.59 m1/2)
= 2.0 sm-1/2
Determining min slope:
mmin = (Tmax - T) – (Tmin + T)
Lmax1/2 – Lmin1/2
mmin = 1.4 s / 0.88 m1/2
= (2.9s-0.1s) – (1.3s+0.1s)
(1.47 m1/2 – 0.59 m1/2)
= 1.6 sm-1/2
So slope of the graph is 1.8 ± 0.2 sm-1/2
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
= 12 m/s2
= (2 / 1.8 sm-1/2)2
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.
Part 3: Conclusion and Evaluation
Concluding:
I’ll 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 doesn’t 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
so…. gav = L (2 / T)2 = (1.15m) (2 / (2.2s))2
= 9.4 m/s2
Using uncertainty rules for multiplication and powers….
g / g = L / L + 2 (T / T)
g = (0.0996) x g
= (1 / 115) + 2 (0.1 / 2.2)
= (0.0996) x 9.4 m/s2
= 0.0996
= 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.
Part 3: Conclusion and Evaluation
Concluding:
You can calculate a percentage difference between the
accepted value of g and your calculated average value using..
Percent Difference
Your value - Actual value
Percent Difference 
Actual value
 100%
9.4 - 9.8
 100%  4.1%
9.8
This shows a good correlation but it doesn’t reflect the larger range in
possible values of g that 12 ± 3 m/s shows.
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.
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.
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.