Physics 1.2
Measurements and Mathematics
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Unit:
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Standard unit of: time, distance, mass, etc.
SI system
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Scientific measurements
SI prefixes and symbols
Some Units of Measure:
Dimensional Analysis:
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A way to examine the accuracy of a particular
formula.
Also allows one to convert and cancel units.
Example:
Si Prefixes:
Tools for Measurement
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Length
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Mass
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Stop watch
Force
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Electronic balance
Time
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Metric ruler
Spring scale
Angle
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Protractor
trigonometry
Trigonometry
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SOH CAH TOA
Sample Problem

A block is displaced a vertical distance of
0.75 meters as is slides down a 1.25-meter
long plane inclined to the horizontal.
a)
b)
How far is the block displaced horizontally?
What is the measure of the angle of inclination of
the plane to the horizontal?
Uncertainty in measurement
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Precision:
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Multiple measurements are nearly identical.
Accuracy:
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Measurements are very close to accepted value
found in reference table.
Significant Figures

0.040900 has how many sig figs?
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Zeros before nonzero digits are not significant.
Zeros between nonzero digits are significant.
Zeros after nonzero digits are only significant if
they:
a.
b.
Are followed by a decimal point.
Are to the right of a decimal point.
Addition, Subtraction,
Multiplication, and Division:

RULE: When adding or subtracting your answer
can only show as many decimal places as the
measurement having the fewest number of
decimal places.
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RULE: When multiplying or dividing, your
answer may only show as many significant digits
as the multiplied or divided measurement showing
the least number of significant digits.
Sig Fig Problems:
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Example: When we add 3.76 g + 14.83 g + 2.1
g = 20.69 g. We must round our answer, 20.69,
to one decimal place (the tenth place). Our
final answer is 20.7 g.
Example: When multiplying 22.37 cm x 3.10
cm x 85.75 cm = 5946.50525 cm3. Our final
answer becomes 5950 cm3.
Scientific Notation
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
Used to express very large or very small
measurements.
Ex/ The number 123,000,000,000 in scientific
notation is written as :
Rules for Scientific Notation:
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Rule for Addition and Subtraction - when adding
or subtracting in scientific notation, you must express
the numbers as the same power of 10. This will often
involve changing the decimal place of the coefficient.
Rule for Multiplication - When you multiply
numbers with scientific notation, multiply the
coefficients together and add the exponents. The
base will remain 10.
Rule for Division - When dividing with scientific
notation, divide the coefficients and subtract the
exponents. The base will remain 10.
Orders of Magnitude:
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Estimating the answer allows you to check if
your calculated answer is reasonable.
Sometimes, you can steal an answer quickly
and save time.
On the Regents exam, this is rare, so be
careful!
Sample Problem:
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Estimate the magnitude of the gravitational
force between earth and the moon and
compare it with the actual value.
gravitational constant G = 6.67300 × 10-11
m3 kg-1 s-2
mass of Earth = 5.9742 × 1024 kilograms
mass of the Moon = 7.36 × 1022 kilograms
Distance (r)= 3.84E8 meters
Another Problem:
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
As the Voyager spacecraft passed the planet
Uranus, it sent signals back to Earth.
Determine the order of magnitude of the time
in seconds for a signal to reach Earth. The
distance from Earth to Uranus is 2.71E12 m.
The speed of light in a vacuum is 3.00E8 m/s.
From V=d/t
Evaluating Experimental Results:
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Data Analysis
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Range: highest to lowest.
Mean: average
Variance: sum of the squares of differences of
values from mean.
Standard Deviation: square root of the variance
Problem:
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In an experiment, a student determines that the
acceleration due to gravity in the laboratory is
9.98 m/s². Determine the percent error. (The
accepted value for the acceleration due to
gravity is 9.81 m/s².)
Percent Error:

% error = | your result - accepted value | x 100 %
accepted value
Graphing Data:
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Independent variable
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Dependent variable

One that experimenter changes, x or horizontal axis.
One measured for change due to change in independent
variable, y or vertical axis.
Line of best fit
A straight or curved line which approximates the
relationship among a set of data points. May be
extrapolated (extend beyond data).
A common statistical function to execute is something called a linear
regression. Linear regression looks for a correlation between an independent
variable and a dependent variable. Let’s say our independent variable is time,
and our dependent variable is a batter’s home run totals. The following is a time
series plot of home run numbers for Cal Ripken:
More graph stuff:
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The slope of a graphed line is defined as a ratio
of ∆y/ ∆x.
Sample Problem:

The position of a
moving car was
measured at one second
intervals and recorded
in the following table.
Graph the data on graph
paper and draw the line
of best fit. Determine
the slope of the line.
Time (s)
0.0
Position (m)
0
1.0
18
2.0
40
3.0
62
4.0
80
5.0
100
Types of proportions:
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Direct proportion
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Inverse proportion
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Increase in one causes a squared increase in another.
Indirect squared proportion
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Increase in one causes decrease in another
Direct squared proportion
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Increase in one causes increase in another.
Increase in one causes a squared decrease in another.
Constant proportion
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Increase in one causes no change in another.
More Graphs: