Download Accuracy of Measurement and Propagation of Uncertainty

Accuracy of Measurement and Propagation of Uncertainty
Objectives:


Experimental objective
– Students will identify materials by calculating their densities and the associated uncertainty.
Learning objectives (students should learn to…)
– Use various measuring tools to accurately measure physical properties of an object
– Calculate uncertainty and keep track of it propagating through calculations
– Keep track of significant figures in calculations
Equipment list:
Vernier caliper, ruler, micrometer, 400g balance, block set, rod set
Apparatus:
1: Vernier caliper
2: Micrometer caliper
3b: Rod set
3a: Block set
1. Vernier caliper
a. Measure inside diameter
b. Measure outside diameter (use this one)
2. Micrometer caliper
a. Precision (0.01mm) and range (0-25mm)
b. Reading scale (see theory for explanation)
c. Small knob (use this one only!)
3. Samples
a. Block set
b. Rod set
Theory: Density – The concept of density is useful in comparing the relative masses of materials or calculating
the mass of an object from a known density and volume. The density of an object is expressed as the mass
per unit volume (g/mL). A milliliter (mL) is equivalent to a cubic centimeter (cm3), so density can also be
expressed in the units g(cm-3). Density is calculated from the equation
𝐷=
𝑀
𝑉
Where M is the mass and V is the volume.
Caliper use – The Vernier caliper typically has two sets of jaws. One set is used for measuring inside
dimensions, and has the sharp edge on the outside (see 1a in apparatus section). The other set of jaws,
used for measuring outside dimensions, has the sharp edges that pinch together (see1b in apparatus
section). In this experiment you will only need to use the outside dimension jaws. Before measuring,
gently slide the calipers all the way closed and press the “zero” button. To take a measurement, gently
close the jaw on the object being measured so that only the sharp edges are touching the object.
A micrometer caliper has both a linear and a rotating scale; the “course” linear scale along the barrel of
the micrometer, reads 0-25mm with a 1mm precision, while the rotating scale reads 0-1mm with
0.01mm precision. Used in conjunction the scales
allow you to measure 0-25mm with 0.01mm
precision. The linear scale (count the visible ticks)
gives the digits before the decimal point, while the
rotating scale (the tick that lines up with the
center line inscribed on the barrel) determine the
two digits right of the decimal point. Consecutive
ticks on the linear scale are separated by 2mm;
above the line is even (i.e. the first tick is 2mm, the
second is 4mm, etc.) and below the line is odd
(1mm, 3mm, and so on). In the image to the right,
the scale is indicating 3.13mm. Before taking a
measurement with the micrometer caliper, you
must check that it zeros properly. To do this, gently close the calipers by turning the small knob until it
clicks. Both scales should indicate zero. If they do not, adjust your measurements to account for the offset.
When taking measurements or zeroing the calipers, only use the small knob to tighten! This prevents
over tightening.
Significant digits – Almost every experiment involves recording and calculating numerical data. In
working with these numbers, it is important to retain only the digits that are “significant”. So, what is a
significant digit? Essentially, it is a digit that is known with some certainty. The terms significant digits
and significant figures are synonymous and interchangeable. The measuring device will ordinarily dictate
how many significant digits should be used. For example, with an ordinary meter stick, one can measure
to within 1 mm. A result of measurement should be recorded, for example, as 0.327m not 0.3270m or
0.33m. This result contains three significant digits. In any kind of measurement, judge all the factors
affecting the accuracy of that measurement and record the data using the appropriate number of
significant digits. The following rules explain which digits are considered significant digits:
1.
2.
3.
4.
5.
6.
All nonzero digits are significant:
– 1.234 g has 4 significant figures
– 1.2 g has 2 significant figures
Zeroes between nonzero digits are significant:
– 1002 kg has 4 significant figures
– 3.07 mL has 3 significant figures
Leading zeros to the left of the first nonzero digits are not significant; such zeroes merely
indicate the position of the decimal point:
– 0.0002 °C has only 1 significant figure
– 0.012 g has 2 significant figures
Trailing zeroes that are also to the right of a decimal point in a number are significant:
– 0.0230 mL has 3 significant figures
– 0.20 g has 2 significant figures
When a number ends in zeroes that are not to the right of a decimal point, the zeroes are not
necessarily significant:
– 190 miles may be 2 or 3 significant figures
– 50,600 calories may be 3, 4, or 5 significant figures
The potential ambiguity in the last rule can be avoided by placing a bar over the last significant
digit (i.e. 190̅ has 3 significant figures while 19̅0 has 2), but the more common and often
preferred method the use of standard exponential, or "scientific," notation. For example,
depending on whether the number of significant figures is 3, 4, or 5, we would write 50,600 N as:
– 5.06 × 104 N (3 significant figures)
– 5.060 × 104 N (4 significant figures)
– 5.0600 × 104 N (5 significant figures)
By writing a number in scientific notation, the number of significant figures is clearly indicated by the
number of numerical figures in the 'digit' term as shown by these examples. This approach is a
reasonable convention to follow.
Additionally, calculation often introduces error as an artifact by changing the number of significant digits.
For example, you might take two measurements, each with a different number of significant digits
(suppose you determine an object moved 1.000m in 1.5s). One of the measurements has four significant
digits, and the other only has two, but now, suppose you wanted to calculate the velocity of the object…
your calculator might show something like 0.6666667m/s, but in reality, 1.000 divided by 1.5 is
irrational; it equals 0.66666666666666… forever. So, your result has infinite significant figures, but this
is a problem. First, you can spend your whole life trying to write out all of these digits, and you wouldn’t
even get close. But also, how many of those digits can you really say you are certain about? The following
rules explain how to make sure the result of a calculation has the correct number of significant digits:
1.
When adding or subtracting, the number of significant digits may change… the focus is on
precision instead of significant digits. The result should have the same number of digits to the
right of the decimal as the input with the fewest digits after the decimal.
– For example, 0.94 has 2 significant digits, but 2 digits right of the decimal, while 0.083
also has 2 significant digits, but 3 digits after the decimal. To add these, first round 0.083
to 0.08 so that it also has 2 digits after the decimal:
0.94s
+ 0.08s
1.02s
2.
The result has more significant digits than the inputs, but it has the same number of
digits (2) after the decimal.
When multiplying or dividing, the precision may change, but the focus is on significant digits. The
result must have same number of significant digits as the input with the fewest significant digits.
Some examples:
34.2𝑐𝑚 × 0.57𝑐𝑚 = 19𝑐𝑚2
57.0𝑐𝑚
= 119𝑐𝑚/𝑠
0.4820𝑠
Looking back at the example of the object that moved 1.000m in 1.5s, it should be obvious now
that its velocity was 0.67m/s.
Propagation of uncertainty – Associated with the measurement of every physical quantity, there is an
inherent uncertainty. This uncertainty may arise from any of several causes: non-uniformity in the
quantity being measured (e.g. the surface not smooth or ends not square), lack of precision of the
measuring instrument, etc. It is important that you learn to recognize uncertainties and treat them
properly. Below is a brief discussion of some important techniques in estimation and working with
errors or uncertainties.
If only a single measurement is taken, one can only assume the uncertainty is the smallest division on the
measuring device, but this rarely is actually the correct value for the uncertainty. In order to better
determine the uncertainty in a measurement, find the mean value of several determinations of the
quantity. The formula for the mean value of N numbers x1, x2, … ,xN is:
𝑥̅ =
∑𝑁
𝑥1 + 𝑥2 + 𝑥3 + ⋯ + 𝑥𝑁
𝑖=1 𝑥𝑖
, which is the same as 𝑥̅ =
𝑁
𝑁
To find the numerical uncertainty, which will be denoted as “σ”, first find all the squared differences of
the measurements from the mean (i.e. find {x̅ - x1}2, {x̅ - x2}2, etc). Finally use these in the formula:
𝜎=√
2
∑𝑁
(𝑥̅ − 𝑥1 )2 + (𝑥̅ − 𝑥2 )2 + ⋯ + (𝑥̅ − 𝑥𝑁 )2
𝑖=1(𝑥̅ − 𝑥𝑖 )
, or 𝜎 = √
(𝑁 − 1)
(𝑁 − 1)
As an example, suppose four measurements of a length are 2.1cm, 2.2cm, 2.0cm, and 2.5cm, then the
mean is:
𝑥̅ =
2.1𝑐𝑚 + 2.2𝑐𝑚 + 2.0𝑐𝑚 + 2.5𝑐𝑚
= 2.2𝑐𝑚
4
And the numerical uncertainty is:
(2.2𝑐𝑚 − 2.1𝑐𝑚)2 + (2.2𝑐𝑚 − 2.2𝑐𝑚)2 + (2.2𝑐𝑚 − 2.0𝑐𝑚)2 + (2.2𝑐𝑚 − 2.5𝑐𝑚)2
𝜎=√
= 0.2𝑐𝑚
3
(Notice that if you perform this calculation, σ=0.187cm, but only one significant digit is kept, giving
σ=0.2cm). This value is also called the standard deviation. Now, any of the measurements can be
accurately expressed as x±σ… for example, (2.1±0.2)cm. If your calculated value of s is smaller than the
smallest division on your measuring device, that is, if σ=0, then you should replace it with the smallest
division.
Now suppose you need to add two measurements, but those values also have uncertainties… take
(2.2±0.3)cm and (4.5±0.4)cm. Adding the values isn’t tricky, but what should you do with the
uncertainty values? The theory of errors gives the steps shown in the following calculation for this
example:
(2.2 ± 0.3)𝑐𝑚 + (4.5 ± 0.4)𝑐𝑚 =?
First, get the easy part out of the way; find the sum of the values without the uncertainty.
2.2𝑐𝑚 + 4.5𝑐𝑚 = 6.7𝑐𝑚
Now, calculate the new uncertainty by finding the square root of the sum of the uncertainties squared…
you should recognize this as the distance formula (this is not a coincidence).
𝜎 = √(0.3𝑐𝑚)2 + (0.4𝑐𝑚)2 = 0.5𝑐𝑚
Thus, the final result of the problem is…
(2.2 ± 0.3)𝑐𝑚 + (4.5 ± 0.4)𝑐𝑚 = (6.7 ± 0.5)𝑐𝑚
The numerical uncertainty for a subtraction is found exactly the same way. If more than two numbers are
being added or subtracted (or a mixture of adding and subtracting), the numerical uncertainty of the
result is the square root of the sum of the squares of all of the individual numerical uncertainties.
In the case of multiplying or dividing two measurements, a fractional uncertainty, s, is needed. Given a
value in the form x±σ,
𝑠=
𝜎
𝑥
For a product or quotient, for example, the following:
𝑣 = (𝑥 ± 𝜎𝑥 ) × (𝑦 ± 𝜎𝑦 ),
or
The fractional uncertainty of F (sv) is found in a familiar way:
𝑠𝑣 = √𝑠𝑥 2 + 𝑠𝑦 2
The numerical uncertainty of F (σv) is found from sv.
𝜎𝑣 = 𝑠𝑣 × 𝑣
Here’s an example of the full calculation:
𝑣=
(4.7 ± 0.3)𝑚
𝑥
=
𝑦 (0.74 ± 0.05)𝑠
𝑣=
(𝑥 ± 𝜎𝑥 )
(𝑦 ± 𝜎𝑦 )
𝑠𝑥 =
𝜎𝑥 0.3𝑚
=
= 0.06,
𝑥
4.7𝑚
𝑎𝑛𝑑
𝑠𝑦 =
0.05𝑠
= 0.07
0.74𝑠
𝑠𝑣 = √0.062 + 0.072 = 0.09
𝑣=
4.7𝑚
= 6.4𝑚/𝑠,
0.74𝑠
𝑎𝑛𝑑
𝜎𝑣 = 𝑠𝑣 × 𝑣 = 0.09 × 6.4𝑚/𝑠 = 0.6𝑚/𝑠
Finally,
𝑣 = (6.4 ± 0.6)𝑚/𝑠
On the other hand, often it is necessary to multiply or divide a measurement (with an associated
uncertainty) by an exact number (without an uncertainty), for example, the radius (r) of an object of
(10±0.2)cm diameter (d) is written as
𝑟=
𝑑 (10 ± 0.2)𝑐𝑚
=
2
2
As you can see, the multiplication rule above will not work for exact numbers. Depending on the
situation, is best to approach the problem in one of two ways. First, if fractional uncertainties are being
used, the fractional uncertainty of the measurement is simply kept the same:
𝑠𝑑 =
0.2𝑐𝑚
= 0.02
10𝑐𝑚
Thus the radius is 5cm with a fractional uncertainty of 0.02, and finding the numerical uncertainty,
𝜎𝑟 = 0.02 × 5𝑐𝑚 = 0.1𝑐𝑚
The equation for the radius of the circle becomes
𝑟=
(10 ± 0.2)𝑐𝑚
= (5 ± 0.1)𝑐𝑚
2
Alternatively, if fractional uncertainty is not used, the equation can be rewritten as the measurement
multiplied by a constant (here the constant is simply ½).
𝑟=
1
1
× 𝑑 = × (10 ± 0.2)𝑐𝑚
2
2
Then the constant can be distributed through the measurement, giving the following:
1
1
𝑟 = [( × 10) ± ( × 0.2)] 𝑐𝑚 = (5 ± 0.1)𝑐𝑚
2
2
Notice both approaches give the same result. Another example is the circumference (c) of a circle with
radius (r) (3.0±0.1) cm is calculated as (remember, the uncertainty only has a single significant digit):
𝑐 = 2𝜋𝑟 = 2 × 𝜋 × (3.0 ± 0.1)𝑐𝑚 = 6.28 × (3.0 ± 0.1)𝑐𝑚 = (19 ± 0.6)𝑐𝑚
Another important operation is exponents.
𝑦 = 𝑥3
In the case of an exponent, the fractional uncertainty of x (sx) is multiplied by the power in the exponent:
𝑠𝑦 = 3𝑠𝑥
The conversion between fractional and numerical uncertainties is the same as described above. In the
case when a calculation involves a mixture of the operations discussed above, the uncertainty is
calculated in the same order of operations that you would use to normally solve the equation. It is often
helpful to break the equation into parts, each with only a single type of operation (i.e.
addition/subtraction, multiplication/division, or exponent), and calculate the value and uncertainty for
each part in succession.
Equations:
As a reminder, the volume of a cylinder is
𝑑 2
𝑉 = 𝐿×𝜋( )
2
Procedure:
Part 1:
1.
2.
3.
4.
5.
6.
Measure the width of the block along each of the
four edges (see image). Repeat for width and
height, so that all twelve edges have been
measured. Record these values in table 1.
Calculate and record the mean value, numerical
uncertainty, and fractional uncertainty for the
length, width, and height of the block.
Calculate the volume of the block from the mean
dimensions, and calculate the fractional and
numerical uncertainty.
Measure and record the mass (including
numerical and fractional uncertainty)
Calculate the density of the block, and calculate
the fractional and numerical uncertainty.
Identify the block material from the density table.
Part 2:
1.
2.
3.
4.
5.
6.
Use the micrometer calipers to measure the diameter of the rod in six places, distributed along the
length. Turn the rod slightly between measurements. Record these values in table 2. Note: turn the
micrometer screw by the small end-knob (it will click to let you know when it is tight enough. Do not
try to tighten it further). Correct for any zero error when the jaws are closed.
Measure the length of the rod in at least four places, and allow for sloping ends.
Calculate the mean value, numerical uncertainty, and fractional uncertainty for the diameter and
length of the rod.
Calculate the volume of the rod, and calculate the fractional and numerical uncertainty.
Measure and record the mass (including numerical and fractional uncertainty) of the rod
Calculate the density of the rod, and calculate the fractional and numerical uncertainty.
7.
Identify the rod material from the density table..
Prelab:
Note: Be sure to use the correct number of significant figures, and units, where necessary in your
calculated results.
1.
2.
Why are multiple measurements often taken in an experiment?
How many significant figures in the following numbers?
a. 1030.0
b. 02.6
c. 0.0
d. 500̅00
Add the following: 1.23+0.42000+1.00923
Multiply the following: 3004.027x2
Find the total displacement (sum) from the following measurements: (51±2)m, (48±6)m, (-27±3)m
Find the momentum (multiply) for a (20.0±0.6)Kg object traveling at (15±0.6)m/s
Find the kinetic energy for two masses, m1=(64±3)Kg and m2=(136±4)Kg, traveling together at a
velocity v=(5.0±1.5)m/s. Hint: the equation for kinetic energy is
1
𝐸 = 𝑚𝑣 2
2
Where m is the total mass (m1+m2).
3.
4.
5.
6.
7.
Report:
You should complete these tables in your report (don’t forget units). Note: do not write directly in these
tables… all notes and records must be kept in the notebook section of this book. These are for your
reference only.
Table 1
Length (L)
Width (W)
Height (H)
Mean
Numerical
Uncertainty
Fractional
Uncertainty
L=
W=
H=
Mass =
Volume (with numerical uncertainty)=_____________________________
Fractional uncertainty=________________
Density (with numerical uncertainty)=_____________________________
Fractional uncertainty=________________
Sample #:
Material:
Length (L)
Table 2
Mean
Diameter (d)
Numerical
Uncertainty
Fractional
Uncertainty
L=
d=
Mass =
Volume (with numerical uncertainty)=_____________________________
Fractional uncertainty=________________
Density (with numerical uncertainty)=_____________________________
Fractional uncertainty=________________
Sample #:
Material:
Density Table
Material
Pine
Poplar
Cedar
Maple
Oak
Teak
Polyethylene
Latex
Nylon
Acrylic
Polycarbonate
Buna-n (nitrile rubber)
Polyester
Polyvinyl Chloride (PVC)
Delrin
Viton
Density (g/cm3)
0.43±0.08
0.43±0.08
0.48±0.01
0.65±0.05
0.75±0.15
0.81±0.16
0.90
0.94±0.02
1.13
1.175±0.005
1.21±0.01
1.25±0.05
1.40
1.405±0.015
1.41
1.8±0.10
Material
Teflon
Quartz (fused silica glass)
Pyrex (borosilicate glass)
Common glass
Aluminum
Titanium
Tin
Iron
Steel
Admiralty Bronze
Brass
Copper
Silver
Lead
Gold
Platinum
Density (g/cm3)
2.20
2.203
2.21
2.60±0.20
2.7
4.5
7.28
7.9
8.25±0.25
8.25±0.55
8.450±0.045
8.9
10.5
11.3
19.3
21.4
Questions:
1.
2.
3.
4.
What is the smallest part of a centimeter that can be read or estimated on the 30 cm scale? What is
the smallest part of a centimeter that can be read off the Calipers? Which reading is more reliable and
why?
What is the smallest part of a centimeter that can be read or estimated with your micrometer?
In measuring the length and diameter of a cylinder, for which dimension is more important that it be
measured carefully? Why?
Why use a micrometer rather than the calipers in determining the diameter of the wire?