Name________________________ Block_____________ Date Due__________________
Pre-calculus Physics
Lab – Calculations & Significant Figures
Introduction
Most often, the measurements we take in the lab do not give us the answer that we are
looking for until we use them in some sort of calculation. In the next lab we will earn how to
include information about the precision of the data (via the decimal place) in each data record.
We now want to use this information to show the informed reader how confident we are in the
results of a calculation using this data. In all cases, the precision of a calculation is limited by the
last precise datum used in the calculation.
Addition/Subtraction Rule
When the calculation involves simple addition or subtraction, the rule is simple. The result of
the calculation is rounded off to the same decimal place as the datum having the least number
of decimal places.
Examples:
5.074 (to thousandths)
+ .10 (to hundredths)
5.203 = 5.20 (round to hundredths)
17.453 (to thousandths)
-1.2 (to tenths)
16.253 = 16.3 (round to tenths)
When the calculation involves multiplication or division the precision of the result is given by
counting significant figures (SF). The rules for counting the number of significant figures in
a number are:
Rule #1
All non-zero digits and all zeroes surrounded by non-zero digits are significant. Beginning
zeroes are never significant.
Examples:
112.2 has 4 SF
25 has 2 SF
1002.6 has 5 SF
1.0026 has 5 SF
0.0028 has 2 SF
1.0028 has 5 SF
Note that rule #1 covers all numbers except those which end with a zero. You’ll see in the
next lab that all measurements taken with the same device are recorded to the same decimal
place even when the last place is a zero. When we calculate with a datum of 9.80, for example,
we mean that our measurement was accurate to the hundredths place and, therefore, our
confidence in our calculation is influenced by the ending zero, i.e. the zero is significant.
Another example of a measurement ending with zero is (1500). If the smallest division on
the measuring device used for this measurement was 1000 (therefore the 5 was estimated),
then, only the 1 and 5 are significant. However, if the smallest division was 10, then the last zero
was estimated and all four digits are significant (i.e. they increase our confidence in the
accuracy of our calculation). There are two common way to record measurements so that a
reader knows how precisely we measured numbers such as this.
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The first is to write the number in scientific notation keeping only the significant zeroes.
Therefore, writing 1.500 x 103 indicates 4 SF whereas writing 1.5 x 103 is the same as writing
1500 and indicates only 2 SF.
The second method is to follow the number with an explicit decimal point (note: only useful
when all of the ending zeroes are significant). Thus, 1500. indicates 4 SF while 1500 indicates
only two. The for ending zeros can be summarized as follows
Rule #2
Ending zeroes are significant only when they are the last digit to the right of the decimal
point or are followed by an explicit point.
Examples:
1200 = 1.2 x 103 has 2 SF
1200.0 = 1.2000 x 103 has 5 SF
0.0003400 = 3.400 x 10-4 has 4 SF
1,200,000. = 1.200000 x 106 has 7 SF
27.0 has 3 SF
0.2800 has 4 SF
400. has 3 SF
30 has 1 SF
Note that all significant zeros have to be written in scientific notation even though it
sometimes looks awkward, such as in the example 1,200,000.)
Multiplication/Division Rule
When we perform multiplication or division with data, the answer is rounded off to the same
number of significant figures as the datum having the least number of significant figures.
Examples:
13.2 x 2.0 = 26
(26.4 rounded to 2 SF)
(note 13.2 has 3 SF; 2.0 has 2 SF)
500./0.03654 = 1.37 x 104 (13683.6 rounded to 3 SF)
(note: 500. has 3 SF; 0.03654 has 4 SF)
61.32 x 1.8 = 110
(110.376 rounded to 2 SF)
(note: 61.32 has 4 SF; 1.8 has 2 SF)
1.2365 x 30 = 40
(37.095 rounded to 1 SF)
(note: 1.2365 has 5 SF; 30 has 1 SF, only the ‘3’ is significant)
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Summary: rules for counting significant figures

Non-Zero digits are always significant (e.g. 39.24 has 4 SF)




Zeros in the middle of a number are always significant (e.g. 48.02 has 4 SF)
Zeros at the front of a number are never significant (e.g. 0.0482 has 3 SF)
Zeros at the end of a number are sometimes significant
Zeros at the end of a number after the decimal point are always significant (e.g.
36.00 has 4 SF and 3.600 x 101 has 4 SF)
Zeros at the end of a number followed by a decimal point are always significant (e.g.
3600. has 4 SF)
Zeros at the end of a number without a decimal point are never significant (e.g. 3600
has 2 SF)
Zeros:


Summary: rules for using significant figures in basic calculations
Addition and Subtraction
round to the “least decimal place”
Multiplication and Division
round to “least number of significant figures”
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Graphing Data
In Pre-calculus Physics graphs will sometimes be made by hand and sometimes in Excel. In
either case, the following criteria should be met.
Graphing Requirements
1. The title lists the vertical quantity (the dependent variable) as a function of the
horizontal axis quantity (the independent variable).
2. The axes are labeled with the variable name (e.g. velocity) and the proper units (e.g.
m/s).
3. The choice of what numbers are used on the axes must take into account the
following
a. Easy to read: The choice of what to make the smallest division on the graph
paper must make the graph easy to read and plot. For example, if the large
divisions are labeled (0, 8, 16, 24…) then, the small divisions between 16 and
24 are (16, 16.8, 17.6, 18.4, 19.2, 20, 20.8, 21.6, 22.4, 23.2, 24). Trying to
plot a data point of 19.45 is awkward this awkwardness often leads to
mistakes in plotting or future mistakes when reading a point for a slope
calculation. The general rule is to make the smallest division some decimal
multiple of 1, 2, or 5 (e.g. 0.1; 20; 0.005, but NOT 0.4; 3.1; 1.5)
b. Spread out: The range of numbers used on the axis allows all of the data to
be plotted and spread the data out as much as possible. The graph should
comfortably fill the entire page.
c. Data Precision: The decimal place of the numbers on the axes must match
the decimal place of the data in the data table.
4. Data points are fine dots which are highlighted by “point protectors” (circles, squares,
etc.)
5. The data points are not “connected.” The purpose of a graph is to enable us to “see
the flow” of the data, i.e. to average the individual data points’ contributions so that
we can see the general rule which governs the interactions of the two variables. To
accomplish this we draw the best smooth curve or the best straight line which
averages the data. In some plots, few (if any) of the data points will actually be on
the line or curve.
6. If the graph is a straight line, we may need to find its slope. To determine the slope of
the graph we must:
a. Choose two points – never use data points – on the line. Preferably points
that are not too close to each other and where the coordinates are easily
readable.
b. Label the coordinates of the points (x,y).
c. Show a slope calculation using “Uncle Al’s Three Steps.” Make sure that the
decimal place of the coordinates matches that of the axes (which, in turn,
matches the original data) so that the proper number of significant figures can
be used. Always include units in the calculation.
4
Model Graph
Distance as a funtion of time
700
600
Distance (m)
500
(42s, 500m)
400
300
200
100
0
0
(2s, 20m)
10
20
30
Time (s)
40
50
Slope Calculation
Slope = y/x = distance/ time
Slope = (500 m – 20 m)/(42 s – 2s)
Slope = 12 m/s
5
Graph with 10 mistakes – can you find them all?
184
161
138
Velocity
115
92
69
46
23
0
0
2.7
5.4
8.1
10.8
13.5
16.2
18.9
Time
Slope Calculation
Slope = y/x
Slope = (125.6 – 51.6) / (14 – 4.0)
Slope = 7 m/s/s
Mistakes
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
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Graph Corrections – ANSWERS
1) Y-axis: units are missing (Velocity (m/s) or Velocity (m/sec)
2) X-axis: labels are missing (Time (s) or Time (sec))
3) There are awkward small divisions on the y-axis (smallest division = 23 m/s /5 = 4.6 m/s).
Smallest divisions must be 1, 2, or 5.
4) There are awkward small divisions on the x-axis (smallest division = 2.7s /5 = 0.54 s = 4.6
m/s). Smallest divisions must be 1, 2, or 5.
5) The title is missing. The title for this graph is “Velocity as a function of Time.”
6) There are no “point protectors,” i.e. a small circle, square, or triangle, must surround data
points. Note: if your graph is produced in Excel, using a large circle or other shape for each
data point is fine.
7) The coordinates of the points used for finding slope must be shown.
8) The points used to find the slope must never be data points.
9) The slope formula must be written in “physics words or letters” e.g. (slope =  velocity/ 
time or slope = y/ x = v/t)
10) The slope calculation must follow Uncle Al’s Three Steps. In this case, there are two errors:
a) units were left out of the “plug-in” step. (125.6 m/s – 51.6 m/s) / (14 s – 4 s)
b) the number of significant figures in the answer must match the least number of
significant figures in the plug-in step. The correct answer is 7.4 m/s2.
7
Name________________________ Block_____________ Date Due__________________
Lab work to turn in: Graphing Data and Practice with Significant Figures
Graphing Data: Exercise
Graph this data, by hand, using the format discussed in this
lab.
The data represents an experiment where the velocity of an
object was measured at various times. Plot a graph of Velocity as
a function of Time and calculate the slope of the line.
Velocity (m/s)
0.0
Time (s)
0.0
9.7
1.0
19.4
2.0
30.2
3.0
39.2
4.0
49.1
5.0
58.6
6.0
68.5
7.0
78.7
8.0
88.2
9.0
97.5
10.0
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Practice: Calculations and Significant Figures
1) In the second column, write the number of significant figures in each number. In the third
column, write each number in standard scientific notation (i.e. coefficient has only one digit
to the left of the decimal place). Note: all significant figures are written in the coefficient.
Number
# of Significant Figures
Scientific Notation
Remember: all SF’s are
shown in the coefficient
a) 1,707
b) 320.
c) 4200
d) 72.07
e) 0.00060
f)
0.00723
g) 98.60
h) 3.20
i)
500.0
j)
10,000.
2) Add/subtract the following – round to the “least decimal place”
a) 306 - 12
b) 4.20 + 0.004
c) 1306.50 – 15.3
d) 175 + 30
e) 175 + 30.
f)
Multiply/divide the following – round to “least number of SF.” Rewrite all answers in
scientific notation
Answers (to the correct # of
SFs)
Scientific Notation (remember:
all SFs are shown in the
coefficient
3) 601 x 40. =
4) 3000.0 x 1.7235 =
5) 720 / 1375 =
6) 111.0 / 32.3 =
7) 0.007270 / 0.01 =
8) 0.0000032 / 0.17 =
9
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