Physical Measurement
Order of Magnitude Estimations: estimations to the nearest power of ten
Measurement
The ranges of magnitude that occur in the universe:
Sizes: 10-15 m to 10+25 m
Masses: 10-30 kg to 10+50 kg
Times: 10-23 s to 10+18 s
(subnuclear particles to extent of the visible universe).
(electron to mass of the universe).
(passage of light across a nucleus to the age of the universe).
Size of a proton ≈ 10-15 m
Size of an atom ≈ 10-10 m
Fundamental Units
Quantity
Units
Symbol
Length
Mass
Time
Electric current
Temperature
Amount
meter
kilogram
second
ampere
Kelvin
mole
m
kg
s
A
K
mol
Luminous intensity
candela
cd
Significant Figures and Decimal Places
4003 m
160 N
160. N
30.00 kg
0.00610 m
Significant
Figures
4
2
3
4
3
Decimal
Places
0
0
0
2
5
Sig Fig Rules
Addition and Subtraction Rule
The sum or difference of
measurements may have no more
decimal places than the least
number of decimal places in any
measurement.
Multiplication and Division Rule
When multiplying or dividing, the
number of significant figures
retained may not exceed the least
number of significant figures in
any of the factors.
Example:
Add 2.34 m, 35.7 m and 24 m
= 62 m
Example:
0.304 cm x 73.84168 cm.
= 22.4 cm2
Derived units are a combination of fundamental units (that
may have an alternate name). Example: 1 kg·m/s2 = 1 N
Accuracy: An indication of how close a measurement is to the accepted value (a measure
of correctness)
Precision: An indication of the agreement among a number of measurements made in the
same way (a measure of exactness)
Systematic Error:
An error associated with a particular instrument or experimental technique that causes the measured value to be off by the same amount each time.
(Affects the accuracy of results - Can be eliminated by fixing source of error – shows up as non-zero y-intercept on a graph)
Random Uncertainty: An uncertainty produced by unknown and unpredictable variations in the experimental situation.
(Affects the precision of results - Can be reduced by taking repeated trials but not eliminated – shows up as error bars on a graph)
1. Multiple trials can reduce the random uncertainty but not a systematic error. (Repeated measurements can make your answer more precise but not more accurate.)
2. An accurate experiment has low systematic error.
3. A precise experiment has low random uncertainty.
1
Metric Prefixes and Conversions
3.
Convert 0.0340 pm into kilometers.
.0340 pm
1 x 10-12 m
1 km
1 pm
1 x 103 m
= 3.40 x 10-17 km
Prefix to prefix --- must make pit stop at base unit first
4.
Convert 12.8 cm2 into m2.
12.8 cm2
1 x 10-2 m 1 x 10-2 m
1 cm
1 cm
Squared units --- two conversions
Cubed units --- three conversions
Factor-Label Method for Converting Units
1. Convert 45 centimeters into meters.
45 cm · 1 x 10-2 m = 0.45 m or 4.5 x 10-1 m
1 cm
= 1.28 x 10-3 m2
5. Convert 4700 kg/m3 into g/cm3
4700 kg
1 m3
1 x 103 g
1 kg
1 x 10-2 m 1 x 10-2 m 1 x 10-2 m
1 cm
1 cm
1 cm
= 4700 x 10-3 = 4.7 g/cm3
2. Convert 1.9 A into microamps.
1.9 A
1 µA
1 x 10-6 A
6. Convert 55 mph into m/s. (1.0 mile ≈ 1.6 km)
= 1.9 x 106 µA
55 miles 1.61 km 1 x 103 m 1 hr
1 hr
1 mile
1 km
3600 s= 27 m/s
2
Every measured value has uncertainty.
a)
Uncertainties
on Raw Data: b)
c)
Uncertainties
on Processed
Data:
a)
b)
c)
Data Processing
A child swings back and forth on a
swing 10 times in 36.27s ± 0.01 s. How
long did one swing take?
Measurements of time are taken as: 14.23 s, 13.91 s, 14.76 s, 15.31 s. 13.84 s, 14.18 s. What value should be reported?
Voltage ±
uncertainty
Absolute Uncertainty Fractional Uncertainty Percentage Uncertainty
V ± ΔV
11.6 V ± 0.2 V
Calculations with Uncertainties
1. Addition/Subtraction Rule:
When two or more quantities are added or subtracted, the overall uncertainty is equal to the
sum of the absolute uncertainties.
Example: The sides of a rectangle are measured to be (4.4 ± 0.2) cm and (8.5 ± 0.3) cm. Find the perimeter of the rectangle.
ON REFERENCE TABLE:
3
2. Multiplication/Division Rule:
When two or more quantities are multiplied or divided, the overall uncertainty is equal to the
sum of the percentage uncertainties.
Example: The sides of a rectangle are measured to be (4.4 ± 0.2) cm and (8.5 ± 0.3) cm. Find the area of the rectangle.
ON REFERENCE TABLE:
3. Power Rule:
When the calculation involves raising to a power, multiply the percentage uncertainty by the power.
1
(Don’t forget that x = x 2 )
Example: The radius of a circle is measured to be 3.5 cm ± 0.2 cm. What is the area of the circle with its uncertainty?
ON REFERENCE TABLE:
4
Exercises
1. Five people measure the mass of an object. The results are 0.56 g, 0.58 g, 0.58 g, 0.55 g, 0.59 g. How
would you report the measured value for the object’s mass?
2. Juan Deroff measured 8 floor tiles to be 2.67 m ±0.03 m long. What is the length of one floor tile?
3. The first part of a trip took 25 ± 3 s, and the second part of the trip took 17 ± 2s.
a. How long did the whole trip take?
b. How much longer was the first part of the trip than the second part?
4. A car traveled 600. m ± 12 m in 32 ± 3 s. What was the speed of the car?
5. The time t it takes an object to fall freely from rest a distance d is given by the formula:
where g is the acceleration due to gravity. A ball fell 12.5 m ± 0.3 m. How long
did this take?
t=
5
2d
g
Analyzing Data Graphically
The masses of different volumes of alcohol were measured and then plotted (using Logger Pro). Note there are three lines drawn on
the graph – the best-fit line, the line of maximum slope, and the line of minimum slope. The slope and y-intercept of the best-fit line
can be used to write the specific equation and the slopes and y-intercepts of the max/min lines can be used to find the uncertainties in
the specific equation. The specific equation is then compared to a mathematical model in order to make conclusions.
General Equation: y= mx + b
Specific Equation: M = (0.66 g/cm3)V + 0.65 g
Uncertainties: slope: 0.66 g/cm3 ± 0.11 g/cm3
y-intercept: 0.65 g ± 3.05 g
Mathematical Model: D = M/V so M = DV
Conclusion Paragraph:
1. The purpose of the investigation was to determine the relationship between volume and mass for a sample of alcohol.
2. Our hypothesis was that the relationship is linear. The graph of our data supports our hypothesis since a best-fit line falls within the
error bars of each data point.
3.
The specific equation of the relationship is M = (0.66 g/cm3)V + 0.65 g.
4.
We believe that enough data points were taken over a wide enough range of values to establish this relationship. This relationship
should hold true for very small volumes, although if it becomes too small for us to measure with our present equipment we won’t
be able to tell, and for very large volumes, unless the mass becomes so large that the liquid will be compressed and change the
density.
5.
Zero falls within uncertainty range for y-intercept (0.65 g ± 3.05 g) so our results agree with math model and no systematic error is
apparent
6.
By comparison to the mathematical model we conclude that the slope of the graph represents the density. Therefore the density of
the sample is 0.66 g/cm3 ± 0.11 g/cm3.
7.
The literature value for the density of this type of alcohol is 0.72 g/cm3. Our results agree with the literature value since the
literature value falls within the experimental uncertainty range of 0.66 g/cm3 ± 0.11 g/cm3 .
6
Graphical Representations of Relationships between Variables
y
y
y
x
x
Constant
y=c
y
y
x
Direct (linear)
y = mx + b
Quadratic
y = cx2
y
x
x
Inverse
y = c/x
Inverse quadratic
(inverse square)
y = c/x2
x
Square root
y = c√x
Independent Variable: the measurable quantity that you vary intentionally
Dependent Variable: the measurable quantity that changes as a result
Control Variables: other important measurable quantities (not pieces of equipment) that could be independent variables and so must be held constant (controlled).
Graph Straightening
Linearizing a graph:
Purpose:
Transforming a non-linear graph into a linear one by an appropriate transformation of the variables and a re-plotting of the data points.
To be able to find the constant of proportionality and write the specific equation so the relationship can be compared to a mathematical model.
y
y
x
y
y
x
x
To linearize this . . .
To linearize this . . .
To linearize this . . .
y
y
y
X2
. . . graph this.
To linearize this . . .
y
1/X2
1/x
. . . graph this.
x
. . . graph this.
√x
. . . graph this.
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Example of Graphically Analyzing Non-Linear Data
Research Question: What is the relationship between kinetic energy and speed for a uniformly accelerating object?
Data Collection
General Equation:
Specific Equation:
Mathematical Model:
Data Processing
Comparison:
The graph can be
straightened by plotting KE
vs. the square of the speed.
Conclusion Paragraph:
Note that no error bars are required on the straightened graph, nor are max/min
lines, but these could be found if desired.
The purpose of the investigation was to determine the relationship
between the kinetic energy and the speed of a uniformly accelerating
object. Our hypothesis was that the relationship is quadratic. The
graph of our data supports our hypothesis. The specific equation of
the relationship is KE = (4.9 J/(m2/s2)) v2. By comparison to the
mathematical model of KE = ½ mv2 we conclude that the slope of the
graph represents one-half the mass of the object. Therefore the mass
of the object is 9.8 kg. The measured mass of the object was 10.0 kg.
This is consistent with our experimental value and represents a percent
difference of 2.0%.
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Linearizing Graphs Using Logarithms
This is a special linearizing (straightening) technique that works with general equations that are power functions.
Power Function:
Method of straightening:
Derivation:
Examples
1.
2.
3.
4.
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Exercises – Linearizing Data with Logarithms
In each example below, straighten each graph by logarithms. Then, write the specific equation for each relationship.
What is the most probable type of relationship in each case?
1.
2.
3.
Time (s)
± 0.2 s
0
Displacement (m)
1.0
3
2.0
13
3.0
30
4.0
41
5.0
72
±2m
General Equation:
Specific equation:
0
Slope:
Y-intercept:
Mass (kg)
± 0.1 kg
Acceleration
(m/s2)
±0.1 m/s2
1.1
12.0
2.1
5.9
3.0
4.1
3.8
3.0
5.0
2.5
6.2
2.0
Distance (m)
± 0.1 m
1.5
Force (N)
± 0.2 N
6.7
2.0
3.8
2.4
2.4
3.1
1.7
3.6
1.2
3.9
0.9
4.6
0.7
5.2
0.6
General Equation:
Type of relationship:
Specific equation:
Slope:
Y-intercept:
General Equation:
Type of relationship:
Specific equation:
Slope:
Type of relationship:
Y-intercept:
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