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ERRORS IN SIMPLE MEASUREMENTS
Objective:
To learn how to estimate the uncertainties in direct and indirect measurements.
In this lab you will determine the volume, mass and density of homogenous and regular
shaped solids as well as the uncertainties associated with those measurements.
Introduction:
Whenever taking a measurement one should be aware that there are two main types of errors
associated with the result. They do not refer to any mistakes you may make while taking the
measurements. Rather they refer to the errors inherent to the instrument and to your own
ability to obtain a precise measurement.
One of the errors is the so-called READING ERROR caused by the limitations associated
with the reading of the scale and the need to interpolate between scale markings. For
example, consider the millimeter (mm) markings on a ruler scale. For a person with a normal
vision it is reasonable to say that the length could be read to the nearest millimeter at best.
Therefore reasonable estimate of the uncertainty in this case would be Δl = ± 0.5 mm which is
half of the smallest division.
A rule of thumb for evaluating the READING ERROR is to use half of the smallest division
(e.g. in case of a meter stick with millimeter divisions, the reading error is 0.5 mm),
The other type of error is the OBSERVATION ERROR.
If you and/or your partner take several measurements of the same quantity, (e.g. the length of
a block), most likely you’ll obtain different values. The “correct” measured value lies
somewhere between the lowest value and the biggest value and we assume that the best
estimate of the measured value is the average value. The OBSERVATION ERROR is
defined as the absolute value of the largest difference between the measured values and the
average value.
For example, let’s say that you measure 4 times the length L of a rectangle, with a ruler, and
you get the following values: 3.4 cm. 3.3 cm, 3.6 cm and 3.5 cm. The average value is 3.45
cm. The observation error is then the largest difference which in this case is 0.15 cm= 1.5 mm
What is then the uncertainty estimated for a direct measurement?
When both errors, reading error and observation error, are available, we have to
compare them and take the larger one as the uncertainty associated with most probable
value of the measurement.
In the example given above, if the smallest division of the ruler is 1 mm, then the reading error
is 0.5 mm and the observation error is larger.
The result of the measurement of a quantity x should be presented as
, where x is the
most probable value (average value) and Δx the uncertainty of the measurement. Thus, the
length of the rectangle should be presented as
In the majority of experiments the quantity of interest is not measured directly, but must be
calculated from other quantities. Such measurements are called INDIRECT. For example, the
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sphere’s material calculated from the ratio of the mass and volume of the sphere.
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As seen above, the quantities measured directly are not exact but have errors associated with
them. What is now the uncertainty associated with an indirect measurement?
In order to calculate the uncertainty of an indirect measured quantity we need first to
calculate the uncertainties of the direct measured quantities involved, and then apply the
appropriate formula of propagation of errors.
These formulas depend on how the indirect measurement quantity depends on the direct
measured quantities. The two rules you need for this lab are:
Rule 1: If two or more direct measured quantities are being added or subtracted to yield
an indirect measured quantity y:
y = x1 + x2 +… or y = x1 – x2-… then
Δy = (Δx1)2 + (Δx2 )2 + ...
where Δx1, Δx2, etc. are the uncertainties of the direct measured quantities x1, x2 etc and Δy is the
uncertainty of the indirect measured quantity y.
€
Rule 2: If two direct measured quantities are being multiplied or divided:
y = x1x2 or y = x1/x2 then
 Δx1  2  Δx 2  2
Δy
≈ 
 +

y
 x1   x 2 
where Δx1, Δx2, etc. are the uncertainties of the direct measured quantities x1, x2 etc and Δy is the
uncertainty of the indirect measured quantity y.
€
For more detailed information you may want to consult the two documents “Introduction to
errors and error analysis” and “Significant Figures”.
Procedure:
There are three parts in this experiment. In the first part you will investigate what is parallax
and how it can affect your measurements. In the second part you will make direct
measurements of the mass and dimensions of a block using a scale, meter stick and a caliper.
From these direct measurements you will determine the volume and density of the block
(indirect measurements). In the last part you will use the measurement of the volumetric
displacement of water, when a solid is totally immersed in it, to find the volume of the solid
and then its density. Note: the displacement of water by an object is a very important aspect
of Archimedes’ Principle.
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A. Parallax
Parallax is a significant source of a reading error which refers to when the readings vary
depending on how your eye is lined up with the object and the reading device, e.g. a meter
stick. In this first part you are going to learn what parallax is so that you can avoid the parallax
error in this and future lab experiments.
1. Take a meter stick and one of the larger rectangular solids.
2. Stand the meter stick at one end on the lab bench and then place the solid a few centimeters
in front of it.
3. Try to measure the height of the block 3 or 4 times, each time with your eye at a different
height above the tabletop.
4. Record these values and let your partner repeat these steps.
5. Now place the object right up against the ruler and repeat the steps above. You should
notice that the first set of readings varied depending on how your eye lined up with the object
and the meter stick. This is called parallax. In order to eliminate parallax, your eye and the
object being measured should form a line that is perpendicular (i.e. at 90o) to the ruler.
Obviously, this is easier to do if the object being measured is closer to the meter stick.
B. Measurements of Volume and Density
1. Take one of the smaller rectangular solids. Make note of its physical properties.
2. Measure and record the length of the block with a meter stick and its width and height with
a caliper. DO three measurements. Determine and record the average value and the uncertainty
associated with each measurement.
3. Calculate the volume and its uncertainty using the appropriate propagation rule. Reminder:
In this calculation and in all others in all labs you must use the correct number of significant
digits.
4. Determine the mass (take three measurements) of the object using the balance and record its
most probable value together with the uncertainty.
5. Calculate the density (mass per unit volume) of the object and its uncertainty using the
appropriate propagation rule. Display the results properly.
6. From the table given at the end, determine the material the object is made of.
C. Volumetric Displacement of Water
1. Take the same object as in part B, a beaker and a graduated cylinder.
2. Put enough water in the beaker so that the solid will be completely immersed when placed
in the water; fill also the graduated cylinder with about 200 ml (millilitres) of water. 3. Record
the exact volume of water in the cylinder.
4. Mark the height of the water in the beaker, when the solid is submerged, using a piece of
masking tape.
5. Remove the solid from the beaker. Slowly pour water from the graduated cylinder into the
beaker until the water reaches the level indicated by the tape. Record the volume of the water
remaining in the cylinder.
6. Use your initial and final readings of the volume of water (recorded in 3. and 5.
respectively) in the graduated cylinder to determine the volume of the solid and its density. 7.
Determine the uncertainties of the volume (remember it is an indirect measured quantity since
it is the difference between two values (use rule 1 for errors propagation).
8. Determine the uncertainty of the density value calculated by this method.
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Display the results properly.
PRECAUTION: Some of the objects you used are made of lead. Wash your hands
immediately after the lab.
Conclusions:
Compare the volumes measured by using the displacement technique with those calculated
from the measured linear dimensions in part B. Which method of measuring a volume is more
accurate and why? Can you describe the limitations of each method?
APPENDIX I:
Useful conversion factor: 1 ml = 10-6 m3
Various densities:
Brass
8.6x103 kg/m3
Aluminum
2.699x103 kg/m3
Copper
8.96x103 kg/m3
Iron
7.87x103 kg/m3
Lead
11.36x103 kg/m3
Tin
7.2984x103 kg/m3
APPENDIX II: THE VERNIER SCALE
A vernier is a device which makes it possible to read the position of a pointer on a scale with
great precision. Visually, without a vernier, one can subdivide a scale interval to within, at
best, about 1/5 division. Repeatability of 1/10 to 1/30 of a division is easily possible, with a
vernier.
The idea behind a vernier scale is that humans are more accurate at assessing the coincidence
of two lines than the distance between two lines. Hence, the vernier is so designed that the
distance between two tick marks on the main scale is mapped onto the coincidence of a tick
mark on the main scale and one on the vernier scale.
A short "vernier" scale, consisting usually of 10 marks, is attached to the pointer. The zero
of this scale is the true pointer. Divisions on the vernier scale are spaced so that the whole
vernier scale is one small main-scale division longer or shorter than the corresponding
interval on the main scale. Thus, when the pointer (vernier zero) is moved from one main
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scale division to the next, each mark on the vernier will move successively into and out of
coincidence with a corresponding mark on the main scale. Vernier scales of 8, 10, 16, 25
and 30 divisions are commonly found.
Instructions
Read the position of the vernier zero on the main scale, estimating roughly the subdivision of
the smallest scale interval.
Find the line on the vernier which is in coincidence with the main-scale line. This gives the
exact subdivision of the smallest main-scale interval.
Example: Reading = 5.6mm
Pre-lab for Lab #1:
After reading the procedure for this lab, prepare tables for all your measurements. Also,
complete the exercises on significant digits given below.
Significant Figures
Consult the document “Significant Figures” posted on the physics lab web site.
Exercises
)
1. Write the following quantities in standard form, assuming three significant figures in each:
a) 5360 W
b) 20,000 V
c) 9500.5 A
d) 12345.6 m
e) 0.0004177 kg
f) 865.51 cm
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In all the following exercises, only the significant figures have been kept in all the measured
quantities.
2.
Add the following measured lengths:
45.7 cm; 2032 cm; 2.5688×102 m; 12.7×10-2 km; 1.00001×103 mm
3. A certain rod measures 1.0000 m in length at –10.0 oC and 1.0008 m at +25.0 oC. Use a
formula from Example 12 to determine which of the following gives the correct values of α.
a. 2.2857×10-5/Co
b. 2.29×10-5/Co
c. 2.3×10-5/Co d. 2×10-5/Co
4. The time taken by sunlight to reach the Earth is given by:
A calculator gives 8.33917 (verify this). What is the correct answer (including units)?
5. Calculate the weight of 37 cars if each weighs 1.283×104 N.
6. Re-calculate the answer in Example 11, using
a. Δy=36 m b. Δy = 34 m
7. Calculate the surface area of a cylindrical tank whose radius and height were found to be
1.320 m and 0.445 m, respectively. Using the formula
the calculator yields 10.947822 m2 +3.690743 m2 = 14.638565 m2. What is the correct
answer?
8. For a cart rolling down an inclined plane, the formula
gives the displacement Δs as a function of time t. Given the following measurements:
v0 (initial velocity) = 0.12 m/s down the plane, Δs= 1.325 m, t= 1.906 s, find the acceleration
a.
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