SPH 3U0
Physics Prerequisite Skills
PHYSICS PREREQUISITE SKILLS #1
Order of Operations for Computing Numbers
Measurements
SPH 3U0
Metric Prefixes
Rules for Significant Figures/Digits
Physics Prerequisite Skills
SPH 3U0
Calculations with Measurements
Scientific Notation
Physics Prerequisite Skills
SPH 3U0
Physics Prerequisite Skills
Rules for using Scientific Notation
1) S.N. need not be used for numbers between 0.1 and 1000.
2) Move decimal after the first significant digit.
3) Use the power base 10 to indicate the number of places the decimal has moved. Remember, the
exponent is (+) if the number is one or greater, and (–) if the number is less than one.
When adding or subtracting numbers in scientific notation, it is necessary to ensure that both
measurements have the same power of 10 and both have the same units. If they do not, you must convert
one of the quantities to make it so.
e.g.
4
2
4.32 x 10 kg + 2.2 x 10 kg
Convert the smaller power of 10 to the larger one and now you have:
4
4
4
4.32 x 10 kg + 0.022 x 10 kg = 4.34 x 10 kg
When multiplying or dividing numbers in scientific notation, multiply or divide the coefficients as you
normally would and add (if multiplying) or subtract (if dividing) the powers of 10.
e.g.
4
4
2
2
2.2 x 10 m / 2 x 10 m = 1.1 x 10 m or, if sig. fig., 1 x 10 m
Rounding
Accuracy
How close a measurement is to an accept/actual value. (How "correct" the answer is, or how closed to an
accepted know value a measurement is.)
e.g. Which number is more accurate for the height of your desk?
A) 120 cm
B) 1.97342 km
SPH 3U0
Physics Prerequisite Ski
Precision
Depends on the size of the smallest gradation on a measuring device. Refers to the number of digits in a
measurement. Can also refer to how close a group of measurements are to each other.
e.g. Which number is more precise?
A) 13 cm
sig figs
B) 12.9723 cm
sig figs
e.g. In the table below, measurement set
close to each other.
is
because all the measurement are
Set 1
1April 13.5 g
1Bhupi 10.3 g
1Chan 19.4 g
Ave 14.4 g
Set 2
2Dan 13.3 g
2Eve
13.7 g
2Farah 13.5 g
Ave
13.5 g
Practice: Metric Conversions
a.
1 kilogram =
c.
1 millimetre =
e.
1 centigram =
g.
1 centimetre =
i.
1 centilitre =
k.
1 metre =
m.
1 decagram =
o.
1 hectometre =
q.
24 litres =
s.
66 metres =
u.
gram
b.
1 decilitre =
metre
d.
1 decametre =
gram
f.
1 litre =
millimetres
h.
1 kilogram =
grams
kilolitres
j.
1 millimetre =
decimetres
l.
1 centigram =
kilograms
centigrams
n.
1 decilitre =
kilometres
p.
88 millimetres =
centimetres
r.
683 kilograms =
grams
t.
9 decalitres =
1000 metres = 1
v.
0.001 gram = 1
w.
10 litres = 1
x.
0.01 metres = 1
y.
0.1 metre = 1
z.
100000 grams = 1
centimetres
millilitres
centimetres
litres
metres
millilitres
decalitres
kilolitres
SPH 3U0
Physics Prerequisite Ski
1. Practice: More Metric Conversions
a.
5m=
cm
b.
7g=
c.
2L=
d.
6 cm =
mm
e.
4 hg =
f.
8 daL =
dL
g.
45 mm =
h.
30 cL =
L
i.
76 dg =
kg
j.
83 mL =
daL
k.
39 cg =
g
l.
57.9 g =
mg
m.
257 mL =
L
n.
0.2 L =
o.
6.2 mm =
m
p.
965 cm =
m
q.
0.75 kg =
g
r.
50.6 mg =
g
s.
19.47 L =
kL
t.
2.38 cm =
hm
cL
dg
dam
cg
mL
2. Practice: State the number of significant figures in each of the following:
a. 3570 
b. 17.505 
c.
41.400 
e.
0.000 572 
-4
g.
4.150
i.
1.234 00
10 
8
10 
d.
0.51 
f.
0.009 00 
h.
0.007 160
j.
4.100
5
10 
7
10 
3. Practice: Perform the following operations and give the answer to the correct number
of significant digits.
a.
15.1 + 75.32 =
c.
4.55
e.
1.805
10 + 5.89
g.
8.166
10 – 7.819
i.
5.677
10 + 7.785
k.
1.99
m.
5.32 x 10
o.
0.024 00
q.
(5.50 x 10 )
s.
4.75
-5
-5
10 – 3.1 x 10 =
4
4
10 =
5
5
10 =
-6
-6
10 =
3.1 =
-4
4.218
-8
10 =
6.000 =
8
5=
5
(4 x 10 ) =
b.
178.904 56 – 125.805 5 =
d.
0.000 159 + 4.0074 =
f.
0.000 817 - 0.000 048 1 =
h.
45.128 + 8.501 87 – 42.18 =
j.
8.75
l.
1200.0
n.
45.32 x 2.3 =
p.
12.4 x 0.30 =
r.
7.4
3=
t.
2.5
6.700
-9
10 + 6.1157
3.0 =
0.891 =
-9
10 =
SPH 3U0
Physics Prerequisite Ski
4. Practice: Convert these numbers, from scientific notation, to ordinary expanded notation.
2
-4
a. -2.2 x 10 =
b. 5.63 x 10 =
-3
c.
-8.66 x 10 =
e.
1.01 x 10 =
3
5
d.
6.7 x 10 =
f.
9.899 x 10 =
-8
5. Practice: Convert the following numbers into scientific notation.
a. 2370 =
b. 985,000,000 =
c.
0.03 =
d.
0.000000274 =
e.
15.045 =
f.
0.00000707 =
6. Practice: State the number of significant figures in each of the following:
a. 192 
b. 5400 
c.
100.0 
d.
7.29 
e.
0.000004 
f.
8,000,000 
g.
0.010060 
h.
10.02 
i.
22 
j.
357 
k.
5.0 x 10 
l.
500 
m.
607 
n.
2.01 x 10 
o.
432.000 
p.
81 
q.
80 
r.
1.00 x 10 
s.
65 
t.
201 
2
4
3
7. Practice: Perform the following calculations and round accordingly.
a. 4.60 + 3 =
b. 0.008 + 0.05 =
c.
22.4420 + 55.981 =
d.
200 – 87.3 =
e.
67.5 – 0.009 =
f.
71.86 – 13.1
g.
3.14 x 5.6 =
h.
300 x 10.6 =
i.
0.059 x 6.95 =
j.
80/0.675 =
k.
0.003/106 =
l.
8.5/0.356 =
m.
7.6 x 21.9 =
n.
2.15 x 3.1 x 100 =
o.
5.00009 x 0.06 =
p.
38/7 =
q.
500009/17.000 =
r.
500,000/5.002 =
Trigonometry
8. Use a calculator to solve. Round to the nearest hundredth.
cos 73
cos 74.5
sin 59
sin 33
tan 81
tan 75
9. Use a calculator to solve. Round to the nearest degree.
-1
-1
tan 0.47
sin 0.99
-1
-1
tan 3.31
cos 0.44
cos 1
-1
sin 0.47
10. Complete
Find tan Z
Find tan Y
5.7
KM =
ZM =
2.2
6.11
KZ =
Find sin M
MP =
PJ
10.5
= 9.25
MJ =
13.99
Find cos B
BZ =
CB =
CZ =
Find tan R
8.5
RC =
UR
= 11.14
7.2
UC =
Find sin D
DF =
-1
YL =
YM =
SG =
9.7
JG
5
FN =
Find cos Y
6
YM =
7.68
LM =
4.8
YL =
7.2
= 3.8
= 8.14
SJ
Find cos K
3.8
XK =
3.1
JK
4.9
0.37
3.6
LM =
Find sin S
10.91
DN =
3.62
JX
2.9
= 5.29
= 4.43
Trigonometry
Right Triangles
Trigonometry deals with the relationships between the sides and angles in right triangles. For a
given angle 0 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.
NOTE: ›
›
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.
2ndSIN
2ndTAN
-1 2ndCOS
-1
-1
(i.e., use
for sin ,
for cos , and
for tan ).
Practice
1. Determine the value of each ratio to four decimal places.
(a) sin 35°
(c) tan 45°
(b) cos 60°
(d) cos 75°
2.
3.
4.
5.
(e) sin 18°
(f) tan 38°
(g) cos 88°
(h) sin 7°
Determine the size of /A to the nearest d e g r e e .
(a) sin A = 0.5299
(c) tan A = 4.3315
(b) cos A = 0.4226
(d) cos A = 0.5000
(e) sin A = 0.2419
(f) tan A = 0.0875
(g) cos A = 0.7071
(h) sin A = 0.8829
Solve for x to one decimal p l a c e .
(a) sin 35° = x/8
(c) tan 20° = x/19
(b) cos 70° = x/15
(d) tan 55° = 8/x
(e) sin 10° = 12/x
(f) sin 75° = 5/x
Solve for /B to the nearest degree .
(a) cos B = 3/8
(c) tan B = 15/9
(b) sin B = 7/8
(d) cos B = 16.8/21.5
(e) tan B = 25/12
(f) sin B = ½
Use two different methods to find the value of the unknown in each triangle. Round your answer to one decimal
place.
(a)
(b)
(c)
SPH3U
Significant Figures and Sci.
Notation
Mr. Mohideen
Significant Figures and Sci. Notation Practice
Name:
Accuracy
The Accuracy of Measured Quantities
Every measurement has a degree of certainty and uncertainty. As such, there is an international agreement about
the correct way to record measurements · “Record all those digits that are certain plus one uncertain digit,
and no
more!”. These “certain-plus-one” digits are called significant digits. Thus, the certainty or accuracy of a
measurement is indicated by the number of significant digits.
Exact Numbers
› 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 kW ·h = 3600 kJ,
and w = 3.141592654 are all definitions of equalities. w has an infinite number of decimal places as do numbers in
equations such as C = 2wr.
When Digits Are Significant V
›
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 right 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 significant digits can
be assessed by inspection. For example, a reading of 1250 km on a car’s odometer has four significant digits.)
Practice
1. How many significant digits are there in each of the following measured quantities?
(a) 353 g
(g) 30.405 ml
(m) 10.00 m
(s) 46.03 m
(b) 865.7 cm
(h) 40.070 nm
(n) 6 050.00 mm
(t) 0.000 000 000 68 m
(c) 926.663 L
(i) 0.3 MW
(o) 47.2 m
(u) 0.07 m
(d) 35 000 s
(j) 0.006 ns
(p) 401.6 kg
(v) 908 s
(e) 76 600 000 g
(k) 0.003 04 GW
(q) 0.000 067 s
(w) 7.60 L
(f) 7.05 kg
(l) 0.50 km
(r) 6.00 cm
(x) 0.005 0 mm
2. Express each of the numbers above in scientific notation with the correct number of significant digits.
Precision
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.
Practice
1. Use the three centimetre rulers to
measure and record the length of the pen
graphic.
(a) Child’s ruler
(b) Elementary ruler
(c) Ordinary ruler
2.
3.
4.
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?
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 page 2.
State the precision of the measured quantities (a) to (x) in practice question #1.
Error
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 bullseye target:
No Precision
No Accuracy
Precision without
Accuracy
Precision with
Accuracy
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.
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.
Practice
1. Refer to page 3.
State the possible error of measurements (a) to (c) in practice question #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]
2
2
3. Calculate the percentage difference between the two experimental values (8.94 m/s and 9.95 m/s ) in question
#2 above.