Lab 5: Pendulum IV
Conceptual Focus: Uncertainty of Indirect Measurements
In the previous lab, you learned how to measure (indirectly) quantities that cannot be measured
directly. This generally involves linearizing your collected data based on a model equation and
calculating a quantity of interest from the slope of this linearized plot. Last week, you used the
theoretical model for the period of a pendulum to indirectly measure gravitational acceleration "g."
You may have observed that your value was not the same as the established value for gravitational
acceleration in Denver (9.798 m/s^2)...or was it? How do you know?
In the previous labs, you saw that uncertainty plays a big role in the validity of experimental results and
the conclusions we can draw from them. Until now, you have used uncertainty to compare two
measured values to each other, or a measured value to a known/expected value. These comparisons
allow you to determine whether these values agree (are consistent) or disagree (are not consistent).
That is what is needed here, but how do we determine the uncertainty of "g"? Unlike in previous labs,
we measured "g" indirectly using calculation from the slope of a linearized plot. Our old methods of
calculating uncertainty are no longer sufficient in this new territory.
Background
Uncertainty of an Indirect Measurement
We know that all measured values have uncertainty, which means a value calculated from measured
values must also have an uncertainty associated with it. There are a few different ways to determine the
uncertainty of an indirectly measured value. We will cover two primary methods in this lab guide.
1. Error Propagation - This method is the formal approach to calculating a value for uncertainty in
an indirectly measured quantity, based on the uncertainty of the directly measured values. In
the simplest of cases (two quantities that are multiplied or divided) it involves adding the
relative (fractional) uncertainties of the measured quantities. When the relationship between
the measured and calculated quantities gets more complex, error propagation gets messy
quickly, and generally requires calculus./n2. Min-Max Method - This method is used to estimate the uncertainty by a visual estimate of the
maximum and minimum possible slope of the trendline on a linearized plot. The min-max
method is generally a conservative over-estimate of the uncertainty, but with the tradeoff that
it is quicker, easier, and does not require calculus.
Relative Uncertainty
Relative (also called "fractional") uncertainty is a way of quantifying the precision of a measurement.
The uncertainty of a measurement is commonly represented by the symbol delta (8). The fractional
uncertainty is a ratio of the uncertainty of a measurement to the measurement (8x/x). The percent
uncertainty is the relative uncertainty reported as a percentage. It is defined as:
δχ
x
percent uncertainty = relative uncertainty 100% =
Equation 1: Percent and Relative Uncertainties
The relative uncertainty helps us understand the significance of our measured uncertainty. An
uncertainty of ± 1 m is very large if you're measuring the length of a lab table - the length is about 2
- 100%
Page 117
Lab 5: Pendulum IV
meters long, resulting in a percent uncertainty of 50%! On the other hand, an uncertainty of ±1 m is
very small if you're measuring the distance from Denver to Boulder - this distance is about 48 km, giving
a percent uncertainty of .002%./nError Propagation
When we add or subtract measured values, we can simply add the uncertainties. However, we can only
add or subtract similar quantities (e.g. the perimeter is the sum of four length measurements). When we
have different kinds of quantities that are related by an equation, they are generally multiplied or
divided by each other; for example, velocity is defined as distance divided by time (v = Ax/At). When
we multiply or divide different quantities together, we can't simply add the uncertainties, instead we
use the fractional uncertainties.
For example, say that you have a toy car moving at a constant speed. You have measured the time that
it takes the car to move a given distance and reported your measured values as t = 3.2 s + 0.1s
(percent uncertainty of 3%) and Ax = 2.20 m ± 0.01 m (percent uncertainty of "0.5%). Based on your
measurements and Equation 1, the calculated velocity is:
Ax
2.20 m
At
Sv
However, you can't divide the uncertainties in the same way you did the calculation, which results in an
uncertainty of ± 0.001 m/s (and percent uncertainty of 0.02%), because your measured value for time
has a percent uncertainty of 3% ! A calculation can never lead you to be more precise than your actual
measurements. You also can't determine the uncertainty in the velocity by adding the uncertainties for
time and distance (they have different units!).
However, you can add the fractional uncertainties (which are unitless):
V
3.2 s
St 8x
+-
x
t
= 0.6875 [m/s]
δυ
v = v [S² + 0 = (0.6875 [m/s]) |
[0.1 s
Sa_8x 8(t²)
+
[3.2 s
+
→&a= a
This makes your reported value v=0.69[m/s] ± 0.03 [m/s], giving a percent uncertainty of 4%, which
seems reasonable based on the precision in your initial measurements. This example is the simplest
possible use of error propagation. If there were more terms, or if any of the quantities were squared,
the calculations would get messy quickly.
0.01 m]
2.20 m
Let's consider a slightly more complicated example. When an object accelerates from rest, the distance
it travels (Ax) is related to the time it's moving for (t): Ax=at², where a is the (constant) acceleration.
Say that you measure the distance and time (with uncertainty) and want to know the acceleration (with
uncertainty). Solving this expression for the acceleration gives a = 2Ax/t². To get the uncertainty in this
acceleration we would again need to add the fractional uncertainties:
x = a[0(2²³)+2
= 0.03 [m/s]
Note that the "2" in the equation for acceleration is a pure number with no uncertainty, so we not
need to include it in our fractional uncertainty calculation. We do however need to figure out how to
write the uncertainty in time-squared 8(t²)./nLab 5: Pendulum IV
As a first guess, it seems reasonable that the uncertainty in t² would just be the square of the
uncertainty in t (8t)². But remember that uncertainties are generally (ideally) small, and if you square a
number less than one it actually gets smaller-e.g. 0.5² = 0.25. Nothing you do mathematically to your
data can ever make the uncertainty smaller, so this can't be the appropriate uncertainty for time-
squared. To figure this out, we need to do error propagation, but fortunately we already know how to
do this.
When you square a measurement (here, time t), it is like multiplying the measurement by itself; as in
our examples above, when you multiply or divide two values you can't directly add their uncertainties,
but you can add their relative uncertainties. Therefore, the uncertainty in t² can be calculated as
follows:
8(2²)=(²+8=20 8(t²)=[2t² = 2-t-St
Equation 2: Uncertainty of time squared
Now that we have the uncertainty in time-squared, we can determine the uncertainty in the
acceleration. But note that even this "simple" equation required doing multiple steps of error
propagation. While this is the formal (and more accurate) way to determine the uncertainty of indirect
measurements, it can get messy and time-consuming. In this lab, it will often be sufficient to estimate
the uncertainty, which allows us to skip over some of the more complicated aspects error propagation.
This is particularly useful when the theoretical models become more complicated.
Estimating Uncertainty of Linearized Data
As you learned last week, there are times where an unknown quantity of interest can be indirectly
measured by directly measuring related variables and finding the slope of a linearized plot. For example,
let's imagine a toy car is moving at an increasing speed, i.e., accelerating. The equation for the change in
position of an object moving with a constant acceleration is Ax=at². You cannot measure
acceleration directly, but you can measure distances and times and then calculate the acceleration of
the car from this equation.
Based on the linearization procedure we used last lab, this equation tells us that a plot of distance
versus time-squared should be linear, and that the slope of this line should be related to the
acceleration-specifically, the slope would be m = a/2. Excel's "add trendline" function can generate a
linear trendline and provide a value for the slope. Once you have an experimental value for the slope,
you can use it to calculate a value for the acceleration since a = 2. m.
This value of a is an indirect measurement, calculated from experimentally measured values of distance
and time, which have uncertainties. Therefore a also has uncertainty. The formal way to determine this
uncertainty would be to complete the calculation for da outlined in the prior section. However, if we
are using a linearized plot to calculate a from the slope, we can estimate the uncertainty in a by
estimating the uncertainty in the slope.
When Excel calculates a value for the slope, it uses a method called "least squares" to determine the
best-fit line for the data. But in prior classes, it's very likely you drew best-fit lines by hand and
calculated the slope using the rise and run of this trendline. Since you drew this line by hand, it's very
possible you could get a slightly different answer from someone else looking at the same data, simply
because you drew slightly different "best-fit" lines. This means there is a range of slopes that could/nLab 5: Pendulum
IV
reasonably represent the data. This range in reasonable slopes means that there is an uncertainty in the
slope, and if we can estimate the uncertainty in the slope we can relate this to the uncertainty in the
indirect measurement we calculate from the slope.
While there is no perfect way to estimate the uncertainty in the slope, one approach is to use the
uncertainty in the individual data points. In prior labs, you've indicated uncertainty in directly measured
values using uncertainty bars on a plot. Before, these uncertainty bars were either related to the
resolution of the measurement device, or the standard error of repeated measurements. However, to
draw uncertainty bars on a linearized plot, we may need to go beyond this.
Figure 1 below shows a linearized plot of distance and time data for an accelerating car. As we said
before, to linearize these data we need to plot distance versus time-squared. This means that while the
uncertainty in distance is likely just the resolution of our device, the horizontal uncertainty bars must be
the uncertainty in time-squared-which we know should be given by Equation 2. Uncertainties are
displayed in Figure 1 as error bars (the error bars for the first few data points are so small that they are
hidden behind the points). These uncertainty bars show the reasonable range for each data point, and I
can use them as a way to estimate the reasonable range for the slope of the best-fit line.
60.00
Distance (m)
50.00
40.00
30.00
20.00
10.00
0.00
y = 0.2548x -0.2223
0
100
150
Time Squared (s^2)
Figure 1: Distance vs. Time-Squared for a toy car accelerating along a flat surface.
50
200
250/nMin-Max Method
The plot in Figure 1 shows a trendline and a model equation for the data. The trendline generated by
Excel is the "best-fit" trend in the data and is based primarily off the values of the individual points.
However, the uncertainty bars on each data point provide a range in which each data point is most likely
to fall, so the trendline could reasonably be drawn a little steeper or a little less steep and still "fit" the
data. The extent to which the trendline could be altered is something that the uncertainty bars
communicate, but that Excel does not account for.
For a plot that has uncertainty bars, you could draw two additional trendlines: one with the maximum
(steepest) slope and one with the minimum (flattest) slope that still reasonably represents your data.
Lab 5: Pendulum IV
An example is shown in Figure 2, where maximum and minimum slope lines are drawn for the toy car
acceleration data. Neither of these lines are the best trendline for the data. The max line is clearly
steeper than the best-fit line from Excel, and it looks like this max line is above most of the data points.
However, it could still reasonably fit the data, since it passes through the acceptable range for each data
point.
It's tempting to think of the max-slope line as being a best-fit line for the left side of the error bars - but
this is not accurate since the best fit line for the left edge of the error bars will often have exactly the
same slope as the best fit line for the data points! In Figure 2, both the max and min lines start from
zero, so the max line is always above the best-fit line. More generally, the max line will cross the best-fit
line and will be below it on the left side of the plot and above it on the right side. Likewise, the min line
is not a best-fit line for the right side of the error bars, it is an estimate that shows the shallowest line
that could reasonably fit the data. Note that the best-fit trendline for the plot falls somewhere between
these two extremes.
Distance (m)
60.00
50.00
40.00
30.00
20.00
10.00
0.00
y = 0.2548x -0.2223
MAX
50
Page 417
MIN
100
150
Time Squared (s^2)
Figure 2: Maximum and minimum slope lines drawn onto the toy car acceleration data.
200
250/nOnce you draw maximum and minimum slope lines, you can calculate their slopes using the rise-over-
run method, picking any convenient points that fall directly on the lines. Once the maximum and
minimum slopes have been determined, the uncertainty of the slope can be reported as the average
difference (similar to a "standard deviation") between the max and min slopes.
8 stope
(max slope - min slope)
2
Equation 3: Uncertainty of slope using min-max method
Now that we know the uncertainty in the slope, we can relate this to the uncertainty in any quantity
that is calculated from the slope. In our example (the accelerating car), the slope of the linearized plot is
a/2, so the acceleration is a = 2-slope. Since the acceleration is related directly to the slope, with no
other variables, the uncertainties are also directly related: 8 = 2-stope. In other cases, it may not be
so simple, and we should formally relate the relative uncertainties-just as we did in the velocity
Page 517
Lab 5: Pendulum IV
example above. In this case, we would write:
Sa stope
=
a
slope
Equation 4: Uncertainty of a value of interest from the uncertainty of the slope
This expression has four variables, & (the uncertainty we want to solve for), 8 stoper slope, and a. The
value for the "slope" comes from the best-fit trendline of our data and the value of slope comes from
the min-max estimation using Equation 3, but what is "a"? This is the calculated value of a that comes
from your data. In lab this week, the quantity of interest is g, so this will be your calculated value of g.
Note that we did use error propagation to determine the uncertainty bars in our linearized plot - and if
the slope had not been directly related to the acceleration, we might have needed to do another step of
error propagation in the end-so this approach does not eliminate error propagation entirely. However,
in our estimate of uncertainty we only had to do the simplest form of error propagation. In more
complicated situations, this can save a lot of effort./nAdditional Notes on Min-Max Method
There are several conventions for how to draw min and max lines. One common convention says that
you connect the right error bar of the smallest data point to the left error bar of the largest data point.
We will not use this convention in this lab. While this convention offers a repeatable way to calculate
the min and max lines, it completely ignores most of the data and is basing your uncertainty on only two
data points. In this lab, you will need to eyeball these min-max lines. This may feel "un-scientific," and
it is very likely that you will draw slightly different lines than your groupmates. While this may not be
repeatable, recall that min-max is supposed to be an estimate. It is not the most accurate way to
calculate uncertainty, it is a quick estimate that will allow us to avoid more complicated calculations.
The subjectivity of the min-max method may cause doubt about this method's accuracy, and that is a
reasonable objection. However, min-max generally over-estimates the uncertainty. As you know,
uncertainty is used to compare values and draw conclusions, so it is better to overestimate uncertainty,
because this protects against hasty or false conclusions./nLab 5: Pendulum IV
In-Lab Activities
Uncertainty of Gravitational Acceleration
In the previous lab, you used a theoretical relationship between the period of a pendulum and the
length of a pendulum to indirectly measure gravitational acceleration. This required linearizing your
direct measurements of period and length and relating the slope of this linearized plot to "g". To draw
any conclusions from a comparison of this value of g with the established value of 9.81, you will need
to determine the uncertainty of g. Note that while you are welcome to collect new data, you can re-use
your data from last week.
T² = 4x²²
Equation 5: Theoretical period of a pendulum, linearized form
> Note: Include any equation rearrangement and calculations you do in the Methods / Data
Analysis sections of your lab notebook.
Based on your data (either new or from last week) report the following key expressions/values:
o The slope of your linearized form of Equation 5 - this is an expression
o The slope of your trendline - this is a value
o Your indirect measurement of g.
➤ Determine the uncertainties for each data point for each axis (i.e. 8 and 8,2). Add error bars to
your plot in Excel - you will need to select your calculated uncertainties as custom errors.
➤ "Hand-draw" minimum and maximum lines on your plot that will allow you to estimate the
uncertainty in the slope. Use the line tool-add shape-in Excel to do this.
> Calculate the uncertainty of the slope.
> Calculate the uncertainty of g.
► Make a dot-and-whisker plot of your value of g with uncertainty versus the established value for
gravitational acceleration in Denver. Hint: Make sure the units of the two values match!
Calculate a T-score to compare your value of g with the established value.
➤ Write a conclusion that summarizes your results, describes the major outcomes of this lab, and
addresses the following questions:
o Do your values agree or disagree with the established value? Why?
•
What forms of error in your experiment and/or analysis may be skewing your
results?
• What are some improvements you could make to your experiment?
• Is the established value correct? Should it (does it) have an uncertainty?
o Why are we using min-max method to estimate uncertainty as opposed to using error
propagation?
