Name________________________________________ Date___________ Partner(s)_______________________
_______________________
Physics 21 Laboratory
Measurements and Uncertainties
Purpose:
1)
2)
3)
4)
5)
6)
To learn how to use the meter stick to measure lengths and how to determine the uncertainty in the
length measurements for the meter stick
To learn how to use the vernier caliper to measure lengths and how to determine the uncertainty in the
length measurements for the vernier caliper.
To learn how to use the digital balance to make measurements of mass and how to determine the
uncertainty in the mass measurements for the digital balance
To calculate the density of a solid metal cylinder from measurements made of its height, diameter, and
mass, first using the meter stick measurements then the vernier caliper measurements
To learn how to calculate the uncertainty in the density calculations of the metal cylinder using the data
collected for its height, diameter, and mass
To compare the experimental values of the density of the metal cylinder with the true value of the
density of the metal showing the comparisons in a table and on a one-dimensional error bar graph
Theory:
1)
Length measurement using the meter stick (ruler).
When measuring the length of an object, the measurement should not be taken using the end of the
meter stick, i.e., from the zero on the meter stick. Why not?
Notice that the smallest division on the meter stick is
one millimeter or one-tenth of a centimeter.
However, you can estimate a reading between the
millimeter markings. For example, the diagram
shows the position of the right end of an object. A
meter stick is placed next to the object, and its length
is measured.
object whose length is
being measuredbook
end of
object
meter stick
25 cm
26 cm
Using the position of the meter stick as shown, the uneven edge of the object, and the thickness of the
millimeter lines on the meter stick, an estimate for the minimum length and maximum length of the object
can be made. (My estimate for the imnimum length is 25.71 cm and for the maximum length is 25.77
cm.) Be sure you write the correct number of significant figures and include the unit of measurement.
By calculating one-half of the difference between the maximum and minimum lengths, an estimate of
the uncertainty in length can be found. (In my case, the uncertainty is (1/2) of (25.77 – 25.71) = 0.03 cm.)
This quantity is called the measurement uncertainty.
An additional uncertainty must be included to account for the measuring instrument being imprecise.
This uncertainty is called the instrumental uncertainty and is generally supplied by the manufacturer of
the instrument.. In this case, the instrumental uncertainty of the meter stick is  0.02 cm. This
uncertainty is added to the measurement uncertainty to get the total uncertainty. (For my measurement
it is  0.03 cm.)
2)
Length measurement using the vernier caliper
The vernier caliper can be used to measure length of an object, the inside diameter of a hole and the
depth of a hole. In this laboratory, the vernier caliper is used to measure length.
To measure length, place the object between the jaws of the caliper and lightly close the jaws on the
object. Notice that there is a fixed scale and a sliding scale. The numbers on the fixed scale are in
centimeters, so the smallest division on that scale is a millimeter (1/10 of a centimeter).
Notice that the sliding scale also has divisions from 0 to 10. And when reading the vernier scale, you
start from the 0 on the sliding scale. An example is shown below.
Note that the 0 mark on the sliding scale is lined up between the 2.1 and 2.2 cm marks on the fixed scale.
The reading so far is 2.1 cm.
Next, the hundredths of a centimeter is found by seeing which line on the bottom sliding scale aligns
with a line on the fixed scale. In this case, the third line on the sliding scale aligns. This gives a 3 for the
hundredths place, and so the vernier caliper reading is 2.13 cm.
Note that there is no ambiguity in the reading of the caliper so the reading uncertainty is zero. However,
there is an instrumental uncertainty of  0.005 cm. So the actual reading in this case is (2.130  0.005)
cm.
3)
Measurement of mass using the digital balance
The digital balance is a very convenient and quick way to find the mass of an object. The digital balance
is turned on making sure that the units in which it is reading is grams. The object is then placed on the
pan and the mass is shown on the display. Because it has a digital readout, there is no reading
uncertainty but the instrumental uncertainty for out digital balance is  0.01 grams according to the
manufacturer.
4)
The density of a homogeneous material is defined as the mass per unit volume, or

mass
M
=
.
volume
V
(1)
The units associated with the density are kg/m3, g/cm3, and slug/ft3.
The equation for the volume of a right circular cylinder is
V  R 2 h 
D2 h
,
4
(2)
2
where R is the radius of the cylinder, D is the diameter, and h is its height. When this equation is
substituted into (1), we get

4M
.
D2 h
(3)
This is the equation that is used to find the density of the material from which the cylinder is made.
5)
Find the uncertainty in the density calculation, and note how uncertainties propagate through
calculations (also referred to as “propagation of errors” and “finding the greatest possible error”)
Two methods are shown that demonstrate how the uncertainties associated with measurements are
propagated through calculations to result in an uncertainty in the final result.
We wish to find the density of a right circular cylinder given by equation (3) above. The values 4 and 
are constants, and the measured quantities are mass M, diameter D, and height h. The uncertainties
associated with the measured quantities are M, D, and h .
The first method requires the substitution of the uncertainties into the expression for . That is,
M  M 
4
     
.
   D  D2 h  h 
(4)
The two terms,    and    represent the maximum and minimum values  can have when the
uncertainties are taken into account. The maximum value,    , is found by choosing either the plus
or minus sign preceding the uncertainties so as to maximize . In this case, the plus sign is chosen for
the term in the numerator and the minus signs in the denominator. The maximum value of  is then
M  M 
4
     
.
   D  D2 h  h 
(5)
In a similar fashion, the minimum value of  is
M  M 
4
     
.
   D  D2 h  h 
(6)
The value of  is calculated from (3), and  is found either by subtracting (3) from (5) or (6).
The second method of finding how uncertainties are propagated utilizes differentials. Recall that
differentials represent infinitesimal changes, and that small finite changes are approximately equal to
the infinitesimal changes. That is,
dQ  Q
as long as Q is small.
3
To find the uncertainty , the differential d is taken, then all the differentials are replaced by their
appropriate small finite changes. In order to simplify the taking of the differential, the logarithm is
taken first, then the differential. This produces the same result as simply taking the differential. First,
take the natural log of ,
ln  = ln (4/) + ln M – ln  - ln D2 – ln h.
Now take the differential of this expression,
d d4 /  dM
dD dh


2

.
4 /   M

D
h
d 4 /  
is zero. The infinitesimal uncertainties are then replaced with the
4 /  
corresponding small finite uncertainties ,M ,D, and  h with the sign chosen in such a
way that each of the terms is positive,
Note that the term
 M
D h

2

.

M
D
h
Each of terms
 M D
h
,
,
, and
is referred to as a “fractional uncertainty.”
 M Dh
h
Multiply both sides of equation by  and we end up with an equation for the uncertainty in the density
,
D h 
 M
  
2

..
D
h 
 M
(7)
Now that the density,  from (3) and the uncertainty in density,  from (5), (6), or (7) are found, the
range within which the experimental value of density  should fall is from    to    .
6)
The error bar graph is a convenient and very visual way to indicate how good your experimental values
are in comparison to the theoretical values. An example is shown below.
 - 
“true” value

experimental value
 (g/cm3)
5
6
7
The meaning of this range is that the "true" value of  must lie within the range    provided the
uncertainties M, D, and h have been correctly estimated and the measurements and calculations
have been performed correctly. If the "true" or best known value of A lies outside the range, then this
indicates that a measurement or computational error may have been made and/or the uncertainties in
the measured values of M, D, and h were not correctly estimated. Anytime this occurs, the experimenter
should reexamine the calculations, measurements, and the estimations of uncertainties.
4
Equipment and Supplies Needed:




metal cylinder
digital balance,  0.01 grams
meter stick or metric ruler,  0.02 cm
vernier caliper,  0.005 cm
Procedure:
a)
Identify the metal from which your cylinder is made. Record.
b)
Use the meter stick to measure the height and diameter of the cylinder. Record these values together
with their uncertainties in your data table.
c)
Use the vernier caliper to measure the height and diameter of the cylinder. Record these values together
with their uncertainties in your data table.
d)
Use the digital balance to measure the mass of the cylinder. Record this value with its uncertainty.
Data:
Material from which your right circular cylinder is made:
Here are the kinds of data tables you use to record the heights, diameters, and mass of the cylinder.
Measurement of Height:
Height, h (cm)
Instrumental
Uncertainty of
height (cm)
Reading
Uncertainty of
height (cm)
Instrumental
Uncertainty of
diameter (cm)
Reading
Uncertainty of
diameter (cm)
h
 h
(cm)
D
 D
(cm)
Measurements
using the meter
stick
Measurements
using the vernier
caliper
Measurement of Diameter:
Diameter, D (cm)
Measurements
using the meter
stick
Measurements
using the vernier
caliper
5
Measurement of Mass:
Mass, M (g)
Instrumental
Uncertainty of
mass (g)
Reading
Uncertainty of
mass (g)
M
 M
(g)
Measurements
using the digital
scale
Calculations:
a)
Use your meter stick values of height and diameter and the mass value to calculate the density and the
uncertainty in density of your cylinder. Show your calculations.
b)
Use your vernier caliper values of height and diameter and the mass value to calculate the density and
the uncertainty in density of your cylinder. Show your calculations.
6
Results:
Report the two values of density and their uncertainties in a table of results. The construction of the table is
shown below. Also include the book value of density. The error bar is also to be included in this section.
Meter stick value
Vernier caliper value
Book value
Density of Cylinder,
   , (g/cm3)
Write numbers indicating the density scale for the vertical lines on the axis. Also write at the right end of the
axis the quantity that is being graphed (density, ) and the units in which it is expressed (g/cm3).
Slightly above the axis, draw and label an error bar for the meter stick value with its uncertainty. Slightly
above the previous error bar, draw and label a second error bar for the vernier caliper value with its
uncertainty. Above the second error bar graph and label the “true” (or book) value.
Analysis
Now that you have calculated your values of density and have compared them to the book value, you now
need to examine your experiment and your experimental methods to determine what factors might have
contributed to errors in your results. (Even if your value of density is exactly the book value, there still are
error factors that are present but their effects may have cancelled out.)
List the factors that you think contributed significantly to errors in your experimental values for density.
Think about things like, “Was there something about the object (e.g., rounded or rough edges, misshapen,
dents, etc.) that made me unsure of the measurements I was making?” Or, “Was there something about the
instruments I was using or how I was making the measurements that might have produced errors?”
List the factors according to how large an error you think they contribute to your values of density with the
one contributing the most listed first. Also accompany each factor with a brief explanation as to why you
think they contribute significantly to the error. Do not include factors that you can control such as your
unfamiliarity with the instrument (familiarize yourself with the instrument), your misreading the instrument
(check your readings), or errors in calculations (don’t make them).
7
Conclusions
Indicate whether or not you consider the experiment to have successfully determined the density of the
material from which your cylinder was made. Take into account not only how close the experimental values
are to the “true” (book) value, but also whether or not you accounted for the factors that produce the
uncertainties. Finally, suggest some ways to improve the experiment (still using the same instruments) so
that a more precise determination of density can be made.
8