SPH3U1
Lesson 01
Introduction
SIGNIFICANT DIGITS AND SCIENTIFIC NOTATION AND UNITS
LEARNING GOALS
Students will:




1
Identify the number of significant digits in a value.
Be able to use the metric system (SI).
Be able to convert units.
Learn how to express values in scientific notation.
SCIENTIFIC NOTATION & SI
1.1
Scientific Notation
In science we frequently encounter numbers which are difficult to write in the traditional way - velocity of light, mass of
an electron, distance to the nearest star. Scientific notation, or standard notation, is a technique, using powers of ten,
for concisely writing unusually large or small numbers.

In scientific notation, the number is expressed by writing the correct number of significant digits with one nonzero digit to the left of the decimal point, and then multiplying the number by the appropriate power (+ or -) of
ten (10). Therefore, if you had a number like 2394 and wanted to use scientific notation, the answer would be
3
2.394 × 10 .
Scientific notation also enables us to show the correct number of significant digits. As such, it may be necessary
to use scientific notation in order to follow the rules for certainty (discussed later).

IN-CLASS PRACTICE
1.
2.
1.2
Express each of the following in scientific notation.
(a) 6 807
(b) 0.000 053
(c) 39 879 280 000
(e) 0.070 40
(f) 400 000 000 000
(g) 0.80
(d) 0.000 000 813
(h) 68
Express each of the following in common notation.
(a) 7 × 10 1
(b) 5.2 × 10 3
(c) 8.3 × 10 9
(e) 6.386 8 × 10 3
(f) 4.086 × 10 -3
(g) 6.3 × 10 2
(d) 10.1 × 10 -2
(h) 35.0 × 10 -3
SI
Metric
Abbreviation Power
Over hundreds of years, physicists (and other scientists) have developed
Prefix
of 10
I
12
tera
T
10
the measurements, we can put no faith in reports of scientific research.
9
giga
G
10
 T “
I
U
“I
6
mega
M
10
descendant of the metric system of measurement with a few
3
kilo
k
10
exceptions. For example, atmospheric pressure in SI is measured in
2
hecto
h
10
kilopascals (k Pa) whereas in the metric system it is measured in
1
deca
da
10
millimetres of mercury (mm H g).
-1
deci
d
10
 In the SI system all physical quantities can be expressed as some
-2
centi
c
10
combination of fundamental units, called base units (refer to Appendix
-3
milli
m
10
C Some S I Derived Units on page 572 of Nelson Physics 12).
-6
micro
10

 The SI system is used for scientific work throughout the world.
-9
nano
n
10
Everyone accepts and uses the same rules, and understands that there
- 12
pico
p
10
are limitations to the rules.
 The SI convention includes both quantity and unit symbols. Note that
these are symbols (e.g., 60 km/h) and are not abbreviations (e.g., 40 mi/hr.).
 When converting units the method most commonly used is multiplying by conversion factors (equalities), which
are memorized or referenced (e.g., 1 m= 100 cm, 1 h = 60 min = 3600 s).
1
SPH3U1
Lesson 01
Introduction
 It is also important to pay c los e attention to the units, which are also converted by multiplying by a conversion
factor (e.g., 1 m/s = 3.6 k m/h).
IN-CLASS PRACTICE
3.
Use the Factor/Prefix chart given to the right to convert each of the following measurements to their base unit.
(a) 5.7 GW
(f) 670 MJ
(b) 72 cm
(g) 62 C
(c) 150 k m
(h) 0.50 cm
(d) 6.8 mm
(i) 548 nm
(e) 0.75 km
(j) 1200 mm
1.3 Unit Conversions
There are a variety of ways of converting units. One possible way that always works is shown below.
Create a multiplication factor that will cancel the unit you have and leave behind the unit you want. This factor must be
equal to 1 so as not to change the value of the quantity.
Example: Convert 2.3 h to s
Note that the numerator and denominator of the multiplication factor are equal quantities and that the units cancel out
the h and leave the s.
Example: Convert 45 km/h to m/s
Note that two factors are needed to convert both the units km to m and h to s.
2
Example: Convert 120 m to cm
2
Note that the conversion factor has to be used twice (or squared) in order to convert a unit that is squared.
IN-CLASS PRACTICE
4.
Convert the following measurements to the units indicated.
(a) 1.00 a
(s)
(c) 1.0 m/s
(k m/h)
8
(b) 1.0 × 10 s (a)
(d) 180 km/h (m/s)
5.
6.
An athlete completed a 5 km race in 19.5 min. Convert this time into hours.
A train is travelling at 95 km/h . Convert 95 km/h into metres per second (m/s).
2
SPH3U1
2.
2.1
Lesson 01
Introduction
ACCURACY
The Accuracy of Measured Quantities
Every measurement has a degree of certainty and uncertainty. As such, there is an international agreement about the
R
T
-plusT
cy of a measurement is
indicated by the number of significant digits.
Exact N umbers
 All counted quantities are exact and contain an infinite number of significant digits. For example, if we count
the students in a class, and get 32, we know that 32.2 or 31.9 are not possible. Only a whole -number answer is
possible.
 Numbers obtained from definitions are considered to be exact and contain an infinite number of significant
digits. As such, they do not influence the accuracy of any calculation. For example, 1 m= 100 cm, 1 kWh =
3600 kJ, and  = 3.141592654 are all definitions of equalities. Pi has an infinite number of decimal places in
equations such as C = 2r. Note that when you are using constants such as  in a calculation, it is important to
choose enough digits for the constant that it does not restrict your answer. For example, if you have r = 2.34 cm
and you want to calculate the circumference, then choose  to be at least one or two digits more; ie, =
3.1416.
When Digits Are Significant
 All non-zero digits are significant; e.g., 259.69 has five significant digits.
 Any zeros between two non-zero digits are significant; e.g., 606 has three significant digits.
 Any zeros to the right of both the decimal point and a non-zero digit are significant; e.g., 7.100 has four
significant digits.
 All digits (zero or non-zero) used in scientific notation are significant.
When Digits are not Significant
 Any zeros to the right of the decimal point but preceding a non-zero digit (i.e., leading zeros) are not significant;
they are placeholders. For example, 0.00019 and 0.22 both have two significant digits.
 Ambiguous case: Any zeros to the rig h t of a non-zero digit (i.e., trailing zeros) are not significant; they are
placeholders. For example, 98 000 000 and 2500 both have two significant digits. If the zeros are intended to
3
be significant, then scientific notation must be used. For example, 2.5 × 10 has two significant digits and 2.500
3
× 10 has four significant digits. (An exception to this statement is the following: unless the number of
F
significant digits since odometers measure to the nearest km.)
IN-CLASS PRACTICE
1.
How many significant digits are there in each of the following measured quantities?
(a) 353 g
(b) 865.7 cm
(c) 926.663 L
(d) 35 000 s
(e) 76 600 000 g
(f) 7.05 kg
(g) 30.405 ml
(h) 40.070 nm
(i) 0.3 MW
(j) 0.006 ns
(k ) 0.003 04 GW
(l) 0.50 km
(m) 10.00 m
(n) 6 050.00 mm
(o) 47.2 m
(p) 401.6 kg
(q) 0.000 067 s
(r) 6.00 cm
(s) 46.03 m
(t) 0.000 000 000 68 m
(u) 0.07 m
(v) 908 s
(w) 7.60 L
(x) 0.005 0 mm
2.
Express each of the numbers above in scientific notation with the correct number of significant digits.
3
SPH3U1
ANSWERS
1 Scientific Notation & SI
3
1.
(a) 6.807 × 10
-2
(e) 7.040 × 10
2.
(a) 70
(e) 6 386.8
9
3.
(a) 5.7 × 10 W
2
(e) 7.5 × 10 m
-7
(i) 5.48 × 10 m
7
4.
(a) 3.15 × 10 s
5.
0.325 hr
6.
26 m/s
2 Accuracy
Page 3
1.
(a) 3
(e) 3
(i) 1
(m) 4
(q) 2
(u) 1
2
2.
(a) 3.53 x 10 g
7
(e) 7.66 x 10 g
-1
(i) 3 x 10 MW
1
(m) 1.000 x 10 m
-5
(q) 6.7 x 10 s
-2
(u) 7 x 10 m
Lesson 01
(b)
(f)
(b)
(f)
(b)
(f)
(j)
(b)
(b)
(f)
(j)
(n)
(r)
(v)
(b)
(f)
(j)
(n)
(r)
(v)
-5
5.3 × 10
11
4 × 10
5 200
0.004 086
-1
7.2 × 10 m
8
6.7 × 10 J
0
1.2 × 10 m
3.2 years
4
3
1
6
3
3
2
8.657 x 10 cm
7.05 kg
-3
6 x 10 ns
3
6.05000 x 10 mm
6.00 cm
2
9.08 x 10 s
(c)
(g)
(c)
(g)
(c)
(g)
3.987 928 × 10
-1
8.0 × 10
8 300 000 000
630
5
1.5 × 10 m
-5
6.2 × 10 C
Introduction
10
(d)
(h)
(d)
(h)
(d)
(h)
-7
8.13 × 10
1
6.8 × 10
0.101
0.0350
-3
6.8 × 10 m
-3
5.0 × 10 m
(c) 3.6 km/h
(d) 50 m/s
(c) 6
(g) 5
(k ) 3
(o) 3
(s) 4
(w) 3
2
(c) 9.26663 x 10 L
1
(g) 3.0405 x 10 ml
-3
(k ) 3.04 x 10 GW
1
(o) 4.72 x 10 m
1
(s) 4.603 x 10 m
(w) 7.60 L
(d) 2
(h) 5
(l) 2
(p) 4
(t) 2
(x) 2
4
(d) 3.5 x 10 s
1
(h) 4.0070 x 10 nm
-1
(l) 5.0 x 10 km
2
(p) 4.016 x 10 kg
-10
(t) 6.8 x 10 m
-3
(x) 5.0 x 10 mm
4
SPH3U1
Introduction
PRACTICE QUESTIONS (ON A SEPARATE SHEET)
1.
Using conversion factors complete the following metric conversions:
a)
323
mm =
m
2
c)
43.2 cm =
e)
3300 dm3 =
g)
0.037 m3 =
i)
2.5 min =
k)
2.5 h =
2
m
mm3
mL
s
s
b)
22
km =
cm
2
2
d)
3.2 x 10 dam =
f)
5.8 x 10-2 m3 =
h)
249.7 cm3 =
j)
540 s =
l)
8500 s =
mm2
cm3
L
min
h
2. Complete the following conversions:
a)
60. km/h =
m/s
b)
9 m/s =
c)
120 km/h =
m/s
d)
2.5 m/s =
e)
9.8 g/cm3 =
kg/m3
f)
5600 kg/m3 =
g)
2 m/s2 =
h)
20 km/h/s =
km/h/s
km/h
km/h
g/cm3
m/s2
3. Use dimensional analysis to solve each of the following. Show your work.
a) Change 2.36 x 10-4 h to seconds.
b) Change 0.058 m/h to metres per second.
c) Change 3.49 km/h to millimetres per second.
d) Change 7.3 mL/s to litres per hour.
e) Change 6500 N/m2 to newtons per square centimetre.
f)
Change 32 m/s to kilometres per hour.
g) Change 2.6 m/s2 to kilometres per hour per second
4. How many millilitres are in one decilitre?
a. 100, or 102
b. 10, or 101
c. 1000, or 103
d. 1000000, or 106
5. Which of
a.
b.
c.
d.
the following is the longest distance?
30 000 millimetres
0.400 kilometres
500 metres
5 000 centrimetres
6. Convert 500 kilograms to grams
a. 600 kg = 6 000 g
b. 600 kg = 0.600 g
c. 600 kg = 600 000 g
d. 600 kg = 0.006 g
7. Convert 350. mL to Litres
a. 350. mL = 3.50 L
b. 350. mL = 35 000 L
c. 350. mL = 0.350 L
d. 350. mL = 0.00350 L
5
SPH3U1
Lesson 02
Introduction
PRECISION AND SIGNIFICANT DIGITS
LEARNING GOALS
Students will:
•
•
•
•
Identify the number of significant digits in a value.
Be able to determine the number of significant digits produced in a calculation.
Learn how to express values in scientific notation.
Learn how to convert units.
3 PRECISION
3.1
The Precision of Measured Quantities
Measurements are always approximate. They depend on the precision of the measuring instruments used, that is, the
amount of information that the instruments can provide. For example, 2.861 cm is more precise than 581.86 cm because
the three decimal places in 2.861 makes it precise to the nearest one-thousandth of a centimetre, while the two decimal
places in 581.86 makes it precise only to the nearest one-hundredth of a centimetre. Precision is indicated by the
number of decimal places in a measured or calculated value.
•
•
•
•
•
•
All measured quantities are expressed as precisely as possible. All digits shown are significant with any error or
uncertainty in the last digit; e.g., in 87.64 cm the uncertainty is with the digit 4.
The precision of a measuring instrument depends on its degree of fineness and the size of the unit being used.
Using an instrument with a more finely divided scale allows us to take a more precise measurement.
Any measurement that falls between the smallest divisions on the measuring instrument is an estimate. We
should always try to read any instrument by estimating tenths of the smallest division; e.g., for a ruler calibrated
in centimetres, this means estimating to the nearest tenth of a centimetre or to 1 mm.
The estimated digit is always shown when recording the measurement; e.g., in 6.7 cm the 7 would be the
estimated digit.
Should the object fall right on a division mark , the estimated digit would be 0; e.g., if we use a ruler calibrated
in centimetres to measure a length that falls exactly on the 6 cm mark , the correct reading is 6.0 cm, not 6 cm.
IN-CLASS PRACTICE
1. U s e the three centimetre rulers to measure and record the length of the pen graphic.
(a) Child’s ruler
(b) Elementary ruler
(c) Ordinary ruler
1
SPH3U1
Lesson 02
Introduction
2.
An object is being measured with a ruler calibrated in millimetres. One end of the object is at the zero mark of
the ruler. The other end lines up exactly with the 5.2 cm mark . What reading should be recorded for the length
of the object? Why?
3.
4.
Which of the following values of a measured quantity is most precise?
(a) 4.81 mm,
0.81 mm,
48.1 mm,
0.081 mm
(b) 2.54 cm,
12.65 cm,
126 cm,
0.54 cm,
0.234 cm
Refer to the page 3. State the precision of the measured quantities (a) to (x) in practice question #1.
4 ROUNDING OFF
4.1
Rounding Off Numbers
When making measurements, or when doing calculations, the final answer should have the same number of significant
digits as the least accurate number in the calculation. For example, 0.6 + 0.32 = 0.9 not 0.92. The procedure for
dropping off digits is called rounding off.
Rounding Down
When the digits dropped are less than 5, 50, 500, etc., the remaining digit is left unchanged.
Example: 4.123 becomes
4.12
rounding based on the “3"
4.1
rounding based on the “23"
Rounding Up
When the digits dropped are greater than 5, 50, 500, etc., the remaining digit is increased or rounded up.
Example: 4.756 becomes
4.76
rounding based on the “6"
4.8
rounding based on the “56"
Rounding with 5, 50, 500, etc.
When the digits dropped are exactly equal to 5, 50, 500, etc., the remaining digit is rounded to the closest even number.
This is sometimes called the Even-Odd Rule.
Example:
4.850 becomes 4.8 rounding based on the “50"
4.750 becomes 4.8 rounding based on the “50"
IN-CLASS PRACTICE
1.
2.
3.
Round off the following numbers to two significant digits.
(a) 7.8361
(b) 85.51
(c) 2.85
(d) 9.865
(e) 4.67
(f) 2.75
(g) 6.750
(h) 490.0
(i) 0.006 070
(j) 648 700
(k ) 704
(l) 0.5990
(m) 435
(n) 0.000 16
(o) 0.032 45
(p) 6 807
(q) 0.000 053
(r) 0.070 40
(s) 68
(t) 0.080 5
Round off each of the following to the number of significant digits specified.
-5
0
7
(a) 3.786 × 10 L (2)
(b) 6.805 × 10 m (3)
(c) 6.049 98 × 10 s (2)
(d) 1876 m
(3)
(e) 0.077 25 g
(2)
(f) 67.083 5 L
(2)
(g) 68 725 m
(3)
(h) 0.087 255 L
(3)
(i) 7.326 s
(2)
(j) 70 326 g
(3)
Convert a measured length of 2 897 cm to metres and show the correct number of significant digits in the
answer.
5 CALCULATIONS
5.1
Calculations Involving Measured Quantities
If measurements are approximate, the calculations based on them must also be approximate. Scientists agree that
calculated answers should be rounded so they do not give a misleading idea of how precise the original measurements
were.
2
SPH3U1
Lesson 02
Introduction
• For multiple-step calculations, leave all digits in your calculator until you have finished all your calculations, then
round the final answer. Otherwise you could be introducing error into your calculations. If you have to re-enter
a number, enter 1-2 digits more than you will be rounding off to at the end.
• When multiplying and/or dividing, the answer has the same number of significant digits as the measurement
with the fewest number of significant digits (e.g., 14.25 x 6.43 = 91.6275 which rounds to 91.6 because 6.43 has
three significant digits whereas 14.25 has four significant digits).
• When adding and/or subtracting measured values of known precision, the answer has the same number of
decimal places as the measured value with the fewest decimal places (e.g., 6.6 + 18.74 + 0.766 = 26.106 which
rounds to 26.1 because 6.6 has one decimal place whereas 18.74 has two decimal places and 0.766 has three).
IN-CLASS PRACTICE
1.
Perform the following operations.
(a) 67.8 + 968 + 3.87
(d) 68.7 - 23.95
(g) (2.6)(42.2)
5
-2
(j) (3.275 × 10 )(1.0375 × 10 )
-2
3
(m) (7.8213 × 10 )÷(3.80 × 10 )
(p)
.
(s) 3.5
2.
Express your answer to the correct accuracy or precision warranted.
(b) 463.66 + 29.2 + 0.17
(c) 16.36 + 287.345 + 682.23
(e) 0.09875 - 0.0875
(f) 0.1927 - 0.035
(h) (65)(0.041)(325)
(i) (0.0060)(26)(55.1)
-3
-4
-3
7
(k ) (1.625 × 10 )(7.3 × 10 )
(l) (3.9087 × 10 )(4.920 × 10 )
3
-2
-4
-7
(n) (8.2875×10 )÷(7.9227×10 )
(o) (5.019 × 10 )÷(3.10 × 10 )
(q)
2
.
(r)
.
..
.
(t) √4.9
Simplify each of the following, using scientific notation where appropriate.
2
1
4
2
-2
5
(a) 10 × 10
(b) 10 × 10
(c) 10 × 10
-6
2
2
5
4
7
(e) 10 ÷ 10
(f) 10 ÷ 10
(d) 10 × 10
-6
2
-6
-7
2
1
(g) 10 ÷ 10
(h) 10 ÷ 10
(i) (1.4 × 10 )(3 × 10 )
4
-3
(j) (3.5 × 10 )(2.0 × 10 )
.×
-5
(m)
(n)
(p)
(q)
.×
..× .×
-3
(k ) (5.0 × 10 )(4.00 × 10 )
.× .× .×
.× .
(l)
(o)
(r)
.×
.×
.× .× .×
.× .× .× 3.
Each of the following questions has some measured data. Perform the required calculations with the data. Give
your answer to the accuracy or precision warranted by the data.
(a) Find the perimeter of a rectangular carpet that has a width of 3.56 m and a length of 4.5 m.
(b) Find the area of a rectangle whose sides are 4.5 m and 7.5 m.
(c) A triangle has a base of 5.75 cm and a height of 12.45 cm. Calculate the area of the triangle.
-9
(d) If a gold atom is considered to be a cube with sides 2.5 × 10 m, how many gold atoms could be stacked on
-7
top of one another in a piece of gold foil with a thickness of 1.0 × 10 m?
25
(e) On the average, 1.0 kg of aluminum consists of 2.2 × 10 atoms. How many atoms would there be in a
3
3
3
block of aluminum10 cm by 1.2 cm by 15.6 cm? (DAluminum = 2.7 × 10 kg/m , or 2.7 g/cm .)
4.
Solve each of the following word problems. Watch your accuracy/precision and show all your work!
(a) On the planet Zot, distances are measured in zaps and zings. If 3.9 zings equal 7.5 zaps, how many zings are
equal to 93.5 zaps?
12
8
(b) Neptune is 4.50 × 10 m from the Sun. If light travels at 3.0 × 10 m/s, how long does it take light from the
Sun to reach Neptune?
24
27
(c) The Earth has a mass of 5.98 × 10 k g while Jupiter has a mass of 1.90 × 10 kg. How many times larger is
the mass of Jupiter than the mass of the Earth?
9
(d) The average distance between the Sun and Pluto (the farthest planet) is 5.9 × 10 km, and the speed of light
5
is 3 × 10 km/s. Approximately how many hours does it take sunlight to reach Pluto?
3
SPH3U1
Lesson 02
Introduction
ANSWERS
3 Precision
1.
(a) 4.4 cm
(b) 4.3 cm
(c) 4.35 cm
2.
5.20 cm
3.
(a) 0.081 mm
(b) 0.234 cm
4.
(a) 1 g
(b) 0.1 cm
(c) 0.001 L
(d) 1000 s
(e) 100 000 g
(f) 0.01 kg
(g) 0.001 ml
(h) 0.001 nm
(i) 0.1 MW
(j) 0.001 ns
(k ) 0.00001 GW (l) 0.01 km
(m) 0.01 m
(n) 0.01 mm
(o) 0.1 m
(p) 0.1 kg
(q) 0.000 001 s
(r) 0.01 cm
(s) 0.01 m
(t) 0.000 000 000 01 m
(u) 0.01 m
(v) 1 s
(w) 0.01 L
(x) 0.0001 mm
4 Rounding Off
1.
(a) 7.8
(b) 86
(c) 2.8
(d) 9.9
2
(e) 4.7
(f) 2.8
(g) 6.8
(h) 4.9 x 10
-3
5
2
-1
(i) 6.1 x 10
(j) 6.5 x 10
(k ) 7.0 × 10
(l) 6.0 x 10
2
-4
-2
3
(n) 1.6 x 10
(o) 3.2 x 10
(p) 6.8 x 10
(m) 4.4 x 10
-5
-2
-1
-2
(q) 5.3 x 10
(r) 7.0 x 10
(s) 6.8 x 10
(t) 8.0 x 10
-5
0
7
3
2.
(a) 3.8 × 10 L (b) 6.80 × 10 m (c) 6.0 × 10 s
(d) 1.88 x 10 m
-2
1
4
-2
(e) 7.7 x 10 g
(f) 6.7 x 10 L
(g) 6.87 x 10 m (h) 8.73 x 10 L
(i) 7.3 s
4
(j) 7.03 x 10 g
1
3.
2.897 x 10 m, 4 significant digits
5 Calculations
1.
Perform the following operations.
3
(a) 1.040 x 10
1
(d) 4.48 x 10
2
(g) 1.1 x 10
3
(j) 3.393 x 10
-5
(m) 2.06 x 10
2
(p) 1.6 x 10
1
(s) 1.2 x 10
2.
3.
4.
Express your answer to the correct accuracy or precision warranted.
2
2
(b) 4.930 x 10
(c) 9.8594 x 10
-2
-1
(e) 1.12 x 10
(f) 1.58 x 10
2
(h) 8.7 x 10
(i) 8.6
-6
5
(k ) 1.2 x 10
(l) 1.923 x 10
5
3
(n) 1.0460 x 10
(o) 1.62 x 10
1
2
(q) 3.8 x 10
(r) 3.9 x 10
(t) 2.2
Simplify each of the following, using scientific notation where appropriate.
3
6
3
(a) 10
(b) 10
(c) 10
-4
-3
-3
(d) 10
(e) 10
(f) 10
-8
-1
3
(g) 10
(h) 10
(i) 4 x 10
1
-7
4
(j) 7.0 x 10
(k ) 2.0 x 10
(l) 1.7 x 10
-25
8
-17
(m) 1.4 x 10
(n) 1.8 x 10
(o) 3.33 x 10
22
2
(p) 8.3 x 10
(q) 6.0 x 10
(r) 4
1
1
2
1
2
(a) 1.61 x 10 m
(b) 3.4 x 10 m
(c) 3.52 x 10 cm
1
25
(d) 4.0 x 10
(e) 1.1 x 10 atoms
1
4
2
(a) 4.9 x 10 zaps
(b) 1.5 x 10 s
(c) 3.18 x 10
(d) 5 h
4
SPH3U
Lesson 02
Introduction
SIGNIFICANT DIGITS – EXTRA PRACTICE
1.
2.
3.
4.
5.
6.
How many significant figures are there in each of the following?
a)
146.32 cm
b)
1.8 cm
c)
28.0 cm
d)
28 cm
e)
700
m
f)
700. m
g)
1.00346 kg
h)
0.00346 kg
i)
3.46 x 10-3 kg
j)
3.00 x 108 m/s
k)
2 640 m/s
l)
2.640 m/s
Round off the following to two significant digits:
a)
5.934 m
b)
4.863 km
c)
76.9 s
d)
0.3451 h
e)
0.0181 m/s
f)
0.0346 km/h
Express the following numbers in scientific notation with the proper number of significant
digits:
a)
954 km
b)
321.0 m
c)
6 874 s
d)
6 400 J
e)
6 400. m
f)
0.000 64 min
g)
0.800 KJ
h)
0.000 15 W
i)
300 000 000 V
Express the following numbers in decimal notation:
a)
3.2 x 105 m
b)
3.2 x 10-5 m
c)
3.2 x 100 km
d)
8.791 x 102 s
e)
4.96 x 10-4 m
f)
3 x 108 s
Calculate the following expressing your answers with the proper number of significant
figures:
a)
21.36 g + 2.8 g + 35.348 g + 2.9164 g
c)
54.612 m - 3.0 m
d)
b)
147.845 m - 121.1 m
the area of a rectangle whose length is 35.66 cm and
whose width is 9.1 cm.
Calculate the values of the following expressing your answers in scientific notation with the
proper number of significant digits:
a)
4.0 x 105 m/s x
c)
(2.0 m2)3 x 102
e)
2.7 x 103 km
÷
2.0 x 103 s
(3.0 x 10-4 h)
b)
4.0 x 105 cm
d)
(5.0 m/s x 4.0 s)2 x 104 N
f)
8.5 x 103 m - 4.5 x 102 m
÷
(2.0 x 103 s)
1
SPH3U1
Lesson 03
Introduction
SIGNIFICANT DIGITS AND SCIENTIFIC NOTATION
LEARNING GOALS
Students will:
•
•
•
•
Describe the difference between precision and accuracy
Be able to compare values quantitatively
Understand and describe how error arises from measurement
Review the trigonometry ratios
6 ERROR
6.1
Expressing Error in Measurement
In everyday usage, "accuracy" and "precision" are used interchangeably, but in science it is important to make a
distinction between them. Accuracy refers to the closeness of a measurement to the accepted value for a specific
quantity. Precision is the degree of agreement among several measurements that have been made in the same way.
Think of shooting towards a bulls-eye target:
Error is the difference between an observed value (or the average of observed values) and the accepted value. The size
of the error is an indication of the accuracy. Thus, the smaller the error, the greater the accuracy.
Every measurement made on every scale has some unavoidable possibility of error, usually assumed to be one-half of
the smallest division marked on the scale. The accuracy of calculations involving measured quantities is often indicated
by a statement of the possible error. For example, you use a ruler calibrated in centimetres and millimetres to measure
the length of a block of wood to be 12.6 mm (6 is the estimated digit in this measurement). The possible error in the
measurement would be indicated by 12.6 ± 0.5 mm.
Relative error is expressed as a percentage, and is usually called percentage error.
− % =
× 100%
This same formula can also be used to express by what percentage a value has changed.
− × 100%
Sometimes if two values of the same quantity are measured, it is useful to compare the precision of these values by
calculating the percentage difference between them.
%ℎ =
% =
|1 − 2|
× 100%
1 + 2
2
1
SPH3U1
Lesson 03
Introduction
PRACTICE
1.
2.
2
At a certain location the acceleration due to gravity is 9.82 m/s [down]. Calculate the percentage error of the
following experimental values of “g” at that location.
2
(a) 8.94 m/s [down]
2
(b) 9.95 m/s [down]
Calculate the percentage difference between the two experimental values
2
2
(8.94 m/s and 9.95 m/s ) in question #2 above.
7 MEASUREMENTS
7.1
Measuring Reliably
Many people believe that all measurements are reliable (consistent over many trials), precise (to as many decimal places
as possible), and accurate (representing the actual value). But there are many things that can go wrong when
measuring.
There may be limitations that make the instrument or its use unreliable (inconsistent).
The investigator may make a mistake or fail to follow the correct techniques when reading the measurement to the
available precision (number of decimal places).
The instrument may be faulty or inaccurate or in need of calibration; a similar instrument may give different readings.
To be sure you have measured correctly, you should repeat your measurements at least three times. If your
measurements appear to be reliable, calculate the mean (average of the three measurements) and use that value. To be
more certain about the accuracy, repeat the measurements with a different instrument.
7.2
Measurement Errors
Systematic Errors
If you use a measurement instrument that is worn or is not calibrated, you will introduce a systematic error into your
measurement. A systematic error is one that makes your measurement always too small or too large by a certain
amount. For example, a ruler that is worn at the zero end will make all your lengths a bit too large. A voltmeter that has
the needle a bit below zero when not in use (this is what it means to be not calibrated) will make all your measurements
a bit too small.
Systematic errors can be fixed using a variety of techniques:
• If the amount of error is known, add or subtract it from your measurement.
• If you are measuring with a ruler that has a worn end, do not measure from the zero mark on the ruler. Start all
your measurements from the 1.0 cm mark and then subtract 1.0 cm from each measurement. This should be
done even if you think the start of your ruler is not worn
– it might be worn, but not visibly so.
expected result
V
• Use a graph of the data. For example, if you are
result of V too low
measuring voltage and current over a resistor to
or I too high
determine its resistance, and you suspect that the
voltmeter or ammeter are introducing a systematic error
I
that you cannot see, then plot a voltage versus current
graph. Such a graph will give you the resistance through
its slope. According to Ohm’s Law, it will pass through
(0,0). However, if there is a systematic error, the graph
will be shifted so that it will not pass through (0,0), but
the slope will not be changed. If you simply determined
resistance by dividing V by I, you would not get the correct result. However, the slope of the graph will still give
the correct result in a case like this.
2
SPH3U1
Lesson 03
Introduction
Random Errors
Another type of error might make your measurement either too large or too small in a random, unpredictable fashion.
For example, using a stop-watch to time a 100 metre race requires you to judge when to start and when to stop the stopwatch. In any given race, your result might be a bit too high or a bit too small, but you cannot tell which. This is called a
random error.
In a case where you are experiencing random errors, these errors can be reduced.
• Repeat the measurement several times and average the results. If the errors are truly random, then some will
be too high and some will be too small. The average should therefore cause some of these to cancel out and
give a more accurate result. Repeating can take several forms. In the case of the 100 m race, you cannot have
the race re-run, so you use several timers for each runner and average their measurements. In the case of
measuring the period of a pendulum, repetition is achieved by measuring the period of say 10 periods and
dividing the result by 10. Since the majority of the error in such a measurement comes at the start (first period)
and at the end (last period), you are effectively spreading out this error over 10 periods and thus reducing it.
• Another method of repetition is to create a table of data an plot it. Using the voltage and current example
again, random errors will cause the data points to vary sometimes above and sometimes below the trend.
Analyzing the data with a best-fit line will produce a result that minimizes the random error.
Overall, graphing is a powerful tool for analyzing data. It incorporates repetition with techniques for dealing with both
random errors and systematic errors. If at all possible, data analysis should be done graphically. Using a graph has one
further advantage. When you plot your data, you can usually see the trend it follows. Occasionally, you will have one or
more data points that do not follow the trend. These data points are called outliers (they lie outside the trend). In the VI example above, if you were to calculate resistance from each V-I pair and then average your results, you give the outlier
an equal weight as the other data points in determining your average. The best fit line technique actually reduces the
weight given to outliers and gives you potentially a more accurate result.
PRACTICE
1.
Use a ruler calibrated in centimetres to measure the distance from the centre of each plus sign to the next.
Include the error in your measurement.
2.
The following voltage and current data was gathered to determine the resistance of a resistor.
a)
First, fill the third column by dividing V by I. Once
Current (A)
Voltage (V)
you have completed the column, average the
0.024
1.0
values and record your result.
0.046
2.0
b)
Plot a graph of the values. Use the slope of a bestfit line to get your resistance. Record your result.
c)
Explain why your result in b) is more accurate than
your result in a).
0.065
0.078
0.097
0.119
V/I
3.0
4.0
5.0
6.0
3
SPH3U1
Lesson 03
Introduction
8 TRIGONOMETRY
8.1
Right Triangles
Trigonometry deals with the relationships between the sides and angles in
right triangles. For a given acute angle in a right triangle, there are three
important ratios. These are called the primary trigonometric ratios. The
primary trigonometric ratios can be used to find the measures of unknown
sides and angles in right triangles.
sin ! =
"
&
"
cos ! =
tan ! =
ℎ#"
ℎ#"
&
(SOH CAH TOA)
N O T E: The values of the trigonometric ratios depend on the angle to which
the opposite side, adjacent side, and hypotenuse correspond.
If the value of a trigonometric ratio is known, its corresponding angle can be found on a scientific calculator using the
inverse of that ratio.
-1
-1
-1
(i.e., use 2nd SIN for sin , 2nd COS for cos , and 2nd TAN for tan ).
PRACTICE
1.
2.
3.
4.
5.
Determine the value of each ratio to four decimal places.
(a) sin 35°
(b) cos 60°
(c) tan 45°
(d) cos 75°
(e) sin 18°
(f) tan 38°
(g) cos 88°
(h) sin 7°
Determine the size of θ to the nearest degree.
(a) sin θ = 0.5299
(b) cos θ = 0.4226
(c) tan θ = 4.3315
(d) cos θ = 0.5000
(e) sin θ = 0.2419
(f) tan θ = 0.0875
(g) cos θ = 0.7071
(h) sin θ = 0.8829
Solve for x to one decimal place.
(a) sin 35° = x/8
(b) cos 70° = x/15
(c) tan 20° = x/19
(d) tan 55° = 8/x
(e) sin 10° = 12/x
(f) sin 75° = 5/x
Solve for θ to the nearest degree.
(a) cos θ = 3/8
(b) sin θ = 7/8
(c) tan θ = 15/9
(d) cos θ = 16.8/21.5
(e) tan θ = 25/12
(f) sin θ = ½
Find the value of the unknown in each triangle. Use trigonometry to solve and then check your answer with the
pythagorean theorem. Round your answer to one decimal place.
4
SPH3U1
9
Lesson 03
Introduction
EXTRA PRACTICE QUESTIONS
Recall that every measurement has a degree of uncertainty. It is general practice in physics to
take the uncertainty to be half the smallest division of the measuring device.
Examine the following examples:
the uncertainty
length = 5.8 cm ± 0.5 cm
length = 3.38 cm ± 0.05 cm
Notice that the uncertainty corresponds to the last decimal of the measurement.
1. Record each measurement with the correct number of decimal places and the correct
uncertainty. Assume the units are cm unless otherwise noted.
a.
b.
c.
2. For each measurement, state the uncertainty.
a. 2.3 m
b. 34.8 km
c. 0.0056 mL
d. 23.54 W
e. 12 cm
3. For each of the measurements above, state which digit is the estimate.
5
SPH3U1
Lesson 03
Introduction
4. Three lab groups measured the time taken for a ball to roll down a ramp. They did the
experiment five times each with the same ramp. Their results are shown below.
Time for
Group 1
(s)
Time for
Group 2
(s)
Time for
Group 1
(s)
2.23
2.25
2.55
2.34
2.22
2.64
1.98
2.27
2.49
a. Calculate the percent difference for each group.
b. Which group had the most precise results? How does this relate to your answer
from a?
c. A photo-sensor timed the ball to have actually taken 2.22 s to go down the ramp.
Which group would you say is the most accurate? Explain.
d. Which group probably has a systematic error in their measurements? Explain.
e. Calculate the percent deviation (percent error) of each group (use their average as
the experimental value and the photo-sensor as the accepted value). How do your
values relate to your answer for c?
6
SPH3U1
Lesson 03
ANSWERS
6 Error
1.
(a) 0.5 cm
(b) 0.25 cm
2.
-8.96 %
3.
10.7 %
7 Measurements
2.
(a) average resistance is 47 Ω
(b)
slope of graph is 54 Ω
3.
there are 10 divisions so each mm is divided into 0.1 mm subdivisions
(a) 6.2 mm
(b) 10.8 mm
(d) 26.0 mm
(e) 20.1 mm
8 Trigonometry
1.
(a) 0.5736
(b) 0.5000
(c) 1.0000
(d) 0.2588
(e) 0.3090
(f) 0.7813
(g) 0.0349
(h) 0.1219
0
0
0
0
(b) 65
(c) 77
(d) 60
2.
(a) 32
0
0
0
0
(e) 14
(f) 5
(g) 45
(h) 62
3.
(a) 4.6
(b) 5.1
(d) 5.6
(e) 69.1
4.
(a) 68°
(b) 61°
(d) 39°
(e) 64°
5.
(a) 12.2 mm
(b) 8.0 km
9 Extra Practice Questions
1a) 3.3 cm ± 0.5 cm (or 3.4 cm ± 0.5 cm)
b)3.41 cm ± 0.05 cm
2a) ±0.5 m
b) ±0.5 km c) ±0.0005mL d) ±0.05 W
3a)3
c)6
d)4
5.87%
3.58%
b)8
4a) 16.5%
Introduction
(c) 0.05 cm
(c) 0.6 mm
(f) 7.5 mm
(c)
(f)
(c)
(f)
(c)
6.9
5.2
59°
0
30
54.1 cm
c)43.0 mm ±0.5 mm
e) ±5 cm
e)2
b) group 2 – values are closer together. Percent difference is lowest
c)also group 2 – average is closest to actual value
d) group 3 – answers are consistently too large. Maybe someone was always starting the stopwatch too early.
e)1.44% 0.541%
15.0%
7
SPH3U1
Lesson 04
Introduction
MANIPULATING FORMULAS
LEARNING GOALS
Students will:
•
Review how to manipulate formulas for isolating specific variables.
ORDER OF OPERATIONS
To isolate a variable in an equation, you must remember your Order of Operations. But while
with “plugging n’ chugging” in a regular equation you can use BEDMAS to remember order of
operations to solve a problem, you have to work backwards for isolating a variable.
S.A.M.D.E.B.
Subtraction
Addition
Multiplication
Division
Exponents
Brackets
RULES
1. Whatever is done to one side of the equation must be done on the other side as well to
remain balanced.
2. Do the opposite operation that is being done (ex: if the multiple or quotient is being
divided in the equation, multiply it on both sides).
3. Add or subtract all quotients not containing the variable to one side of the equation.*
4. If the variable is in a denominator, get it out by multiplying by the reciprocal on both
sides.
5. Multiply or divide numbers not the variable. *
6. Exponents (or radicals) must be done next.
7. If the variable was in brackets and that has been isolated, the brackets disappear and
continue isolating starting at the beginning of SAMDEB all over again.
*Multiplication and division are transitive, so the order does not matter for them. You can divide
and then multiple or vice versa. The same holds true for addition and subtraction.
The key to isolating variables is to get every multiple or quotient with that variable on one side of
the equal sign and everything without that variable on the other side. Finally, combine any
terms that aren't already combined, and get rid of the coefficient of the variable, if there is
one. You may also need to factor out the desired variable.
As some math operations are transitive, there is usually more than one way to isolate a variable
in any given equation. As long as you are following the laws of mathematics correctly, the
answer should eventually be the same.
Simplify your answer as much as possible. There should NEVER be a fraction within a
fraction. There should also never be a radical sign in the denominator only (only when an entire
fraction is under a radical sign is a radical sign acceptable in this case). You should also never
have negative exponents.
This process can be VERY tricky, so practice A LOT!!!
1
SPH3U1
Lesson 04
Introduction
EXAMPLE 1
‫ݖ‬
‫ݕ‬
‫ݖ‬
ሺ‫ݕ‬ሻ‫ = ݔ‬ሺ‫ݕ‬ሻ
‫ݕ‬
‫ݖ = ݕݔ‬
Isolate for z in the following equation:
‫=ݔ‬
EXAMPLE 2
Isolate for a in the following equation:
4ܿ = 2ܽ + 6ܾ
4ܿ − 6ܾ = 2ܽ + 6ܾ − 6ܾ
4ܿ − 6ܾ 2ܽ
=
2
2
4ܿ 6ܾ
−
=ܽ
2
2
2ܿ − 3ܾ = ܽ
EXAMPLE 3
Isolate for x in the following equation:
5‫ ݔ‬+ 8 = 3‫ ݔ‬− 6
5‫ ݔ‬+ 8 − 8 = 3‫ ݔ‬− 6 − 8
5‫ ݔ‬− 3‫ = ݔ‬−14
2‫ = ݔ‬−14
2‫ ݔ‬−14
=
2
2
‫ = ݔ‬−7
EXAMPLE 4
Isolate for m in the following equation:
6݊ + 9݉
2݉ − 5‫݋‬
6݊ + 9݉
ሺ2݉ − 5‫݋‬ሻ
3‫ݏ‬ሺ2݉ − 5‫݋‬ሻ =
2݉ − 5‫݋‬
3‫ݏ‬ሺ2݉ሻ − 3‫ݏ‬ሺ5‫݋‬ሻ = 6݊ + 9݉
6‫ ݉ݏ‬− 15‫ = ݋ݏ‬6݊ + 9݉
6‫ ݉ݏ‬− 15‫ ݋ݏ‬− 9݉ = 6݊ + 9݉ − 9݉
6‫ ݉ݏ‬− 9݉ − 15‫ ݋ݏ‬+ 15‫ = ݋ݏ‬6݊ + 15‫݋ݏ‬
6‫ ݉ݏ‬− 9݉ = 6݊ + 15‫݋ݏ‬
݉ሺ6‫ ݏ‬− 9ሻ = 6݊ + 15‫݋ݏ‬
݉ሺ6‫ ݏ‬− 9ሻ 6݊ + 15‫݋ݏ‬
=
6‫ ݏ‬− 9
6‫ ݏ‬− 9
6݊ + 15‫݋ݏ‬
݉=
6‫ ݏ‬− 9
3ሺ2݊ + 5‫݋ݏ‬ሻ
݉=
3ሺ2‫ ݏ‬− 3ሻ
2݊ + 5‫݋ݏ‬
݉=
2‫ ݏ‬− 3
3‫= ݏ‬
2
SPH3U1
Lesson 04
Introduction
PRACTICE QUESTIONS (ON A SEPARATE PAGE)
Solve each equation for the term indicated.
a)
݀ = ‫ݐݒ‬
‫?= ݒ‬
b)
c)
‫ ݒ = ݑ‬+ ܽ‫ݐ‬
‫?= ݒ‬
d)
‫ ݒ = ݑ‬+ ܽ‫ݐ‬
‫?= ݐ‬
e)
ܽ =
v−u
t
‫?= ݑ‬
f)
ܽ =
‫?= ݒ‬
g)
ܽ =
v−u
t
‫?= ݐ‬
h)
F
m
݉ =?
j)
GMm
d2
݉ =?
l)
‫?= ܫ‬
n)
1
‫ ݐݑ = ݏ‬+ ܽ‫ ݐ‬ଶ
2
‫?= ݑ‬
i)
k)
m)
ܽ =
‫= ܨ‬
ܴ =
V
I
‫= ݒ‬
‫?= ݐ‬
d
t
v−u
t
‫ܽ݉ = ܨ‬
‫= ܨ‬
GMm
d2
ܴ =
ܽ =?
݀ =?
ܸ =?
V
I
o)
1
‫ ݐݑ = ݏ‬+ ܽ‫ ݐ‬ଶ
2
ܽ =?
p)
1
‫ ݐݑ = ݏ‬+ ܽ‫ ݐ‬ଶ
2
‫?= ݐ‬
q)
1
‫ܧ‬௧ = ݉݃ℎ + ݉‫ ݒ‬ଶ
2
݉ =?
r)
1
‫ܧ‬௧ = ݉݃ℎ + ݉‫ ݒ‬ଶ
2
ℎ =?
s)
1
1 1
= −
di
f do
݂ =?
t)
1 1
1
= −
di
f do
݀௢ =?
u)
‫ ݒ‬ଶ = ‫ݒ‬଴ଶ + 2ܽሺ‫ ݔ‬− ‫ݔ‬଴ ሻ
‫?= ݔ‬
v)
ܶ = 2ߨඨ
‫?= ܯ‬
x)
w)
‫ ܪ = ܦ‬ቀ1 −
݉
ቁ
‫ܯ‬
ܴ=
‫ܮ‬
݃
1 ሺܵ + ܶሻ
4 ܷଶ
‫?= ܮ‬
ܶ =?
3
SPH3U1
Lesson 05
Introduction
PERIODIC MOTION AND THE RECORDING TIMER
LEARNING GOALS
Students will:
•
•
Perform measurements using some motion equipment.
Practice writing a lab report.
LAB ACTIVITY
Purpose: To determine the period and frequency of the ticker-tape timer.
Apparatus:
- period timer
- ticker tape, carbon disk
- stopwatch
Method:
1.
Set up the timer and with it OFF, practice pulling the tape through the timer at a
constant speed.
2.
Draw a line approximately 5 cm from each end of the tape and using the stop
watch, and with the timer OFF, practice timing the travel from line to line.
3.
When ready, turn on the timer and one person pull the tape while the other times.
START the time when the first line passes the striker and STOP when the second
line passes.
4.
Record the length of tape between the lines, the number of dots printed, and the
elapsed time.
5.
Determine the period and frequency of the timer.
LAB BOOK
Pre-Lab Prep
• What are you trying to discover and why? What is expected?
• What are you going to do?
During The Lab
• What did you discover?
• What might have contributed to differences from what you expected?
Post-Lab
• What new information have you gained and how might you use these results?
HOMEWORK
Complete a formal lab report using the lab report template found on the Website/Moodle and
include sections: Introduction, Experiment, Results & Analysis, Experimental Uncertainties and
Discussion. The title of your lab report should reflect the purpose of the lab.
1