Document 6536775

Math Skills
Physics Introductory Unit
• There are several skills, some of which
you have already learned, that you will
need to use extensively in Physics.
• These include the following:
d
– Algebra (manipulation of formulas) v = t
– Scientific Notation (very lg/sm numbers)
Math
– Significant Digits 0.07034
Rules!
– Unit Conversions 1m = 100cm
~The Mathematical Background~
Algebra
Algebra (Sample)
• Numerous times while studying Physics, you will
be required to use algebra to solve equations.
• This involves the manipulation (moving around)
of variables with the goal being to solve the
equation for the unknown variable (i.e. to isolate
the variable).
• Isolating the variable involves the use of inverse
(opposite) operations to move the other
variables.
– Addition(+) and Subtraction(-) are inverse operations.
– Multiplication(× or ·) and Division(÷) are inverse operations.
– Squaring(2) and square rooting(√) are inverse operations.
Other Algebra Samples
• Given the equation:
d
v=
t
• Solve for t.
vt = d
d
t=
v
• Given the equation:
v2 2 = v12 + 2a ( d 2 − d1 )
• Solve for v2.
v2 = ±
5.2 × 10−4
(v
2
1
+ 2a ( d 2 − d1 ) )
Note: When you take the square
root, a ± symbol must be included
in front of the radical.
•
•
•
•
Consider the formula shown.
Solve the equation in terms of d.
To do this, we must move t.
What operation is t associated with?
Division
• What is the inverse operation?
Multiplication
• Perform the operation to solve for d.
• Some other problems may involve
more than one step.
v=
v ⋅t =
d
t
d
⋅t
t
d = v ⋅t
Scientific Notation
• Scientific notation relies on exponential powers
of ten (10x) to simplify extremely large and small
numbers.
Standard Notation
Scientific Notation
4, 673, 000 = 4.673 × 106
Power of Ten
Coefficient
• In all cases, numbers written in scientific
notation have a single digit in the ones place
followed by the remaining digits placed to the
right of the decimal point. This is called the
coefficient.
• A multiplied power of ten is indicated afterwards.
Scientific Notation (Cont.)
Scientific Notation (Multiplication)
• Large numbers correspond to positive
powers of ten.
• At times, numbers in scientific notation will be
multiplied as shown below.
14, 000 = 1.4 ×104
( 4.2 ×10 )( 3.1×10 )
6
• Small numbers correspond to negative
powers of ten.
0.00034 = 3.4 ×10 −4
• Figuring out the power on the ten relates to
how many places you need to move the
decimal point from its initial position.
12080 = 1.208 × 10
4
0.037 = 3.7 ×10
−2
2 Moves
4 Moves
Scientific Notation (Division)
• At times, numbers in scientific notation will be
divided as shown below.
(8.4 ×10 )
(1.4 ×10 )
3
2
• As before, you need to combine terms. The
exponent rule changes to subtraction when
division is involved.
3
 8.4   10 

 2 
 1.4   10 
6.0 × 101
Significant Digits
• Significant digits (sometimes called significant
figures) are those digits that are considered
important in a given number.
• In order to determine which digits are significant,
one must look to the following rules.
2
• The trick is to combine the powers of ten with
each other and the non-exponent terms with
each other. Then simplify.
( 4.2 × 3.1) (106 ×102 )
13.02 ×108
1.302 ×109
Note: Remember
that exponents add
when like bases are
multiplied.
Scientific Notation (10x)
• Numbers that are simply powers of ten can be
written in a shorter form without a coefficient.
• Consider the example dealing with 100,000.
100, 000 = 1.0 ×105
• In simplified form it can be written as follows:
100,000 = 105
• The same holds true for small numbers.
0.001 = 10 −3
Significant Digits (Special Cases)
• A bar can be placed over zeros that are not
normally significant in order to make them
significant.
1 Significant Digit
400 vs. 400
3 Significant Digits
– All nonzero digits are significant.
370 or 0.056
– Final zeros after the decimal point are significant.
43.0 or 0.0560
– Zeros between other significant digits are significant.
306 or 0.705
– Zeros used solely for spacing are not significant.
24, 000 or 0.007
1 Significant Digit
0.003 vs. 0.003
2 Significant Digits
• This usually occurs after some instances of
rounding. Here a problem would specify to how
many digits you must round.
Significant Digits (Rounding)
• Instead of rounding to a place, you round a number
to a specified number of significant digits. This is
done by rounding up or rounding off the number that
would constitute an extra place.
• Round the number 45.63 to 3 significant digits.
Significant Digits (Mult/Div)
• Keeping correct significant digits
while multiplying and dividing relies
on the same process.
45.6
• Round the number 6798 to three significant digits.
6800
Significant Digits (Add/Sub)
– Align the addends (for addition) or the
minuends and subtrahends (for subtraction)
vertically.
– Add or subtract the values.
– Draw a vertical line down the least precise
number (the one with least decimal places).
– Round to the left of the vertical line.
– Addition problems can have more than two
addends.
Addition
363.7 + 14.374
363.7
+14.734
378.434
378.4
×
Multiply By
Conversion
Factor
Fraction
41.2
−0.779
40.421
40.4
=
Equals
Term With
New Units
Dividing
4
1
7.261 ÷ 0.2
36.305
40
Significant Digits (Sci. Notation)
Sig. Digit Counting
Multiplication/Division
4
8.803 × 1014
( 5.91×10 )(1.9 ×10 )
2
4.5 × 10−144
Unit Conversions (Overview)
Term With
Original Units
3
• Only the coefficients in scientific notation
numbers count towards significant digits.
• Aside from this, all normal rules apply.
• See the examples below.
Subtraction
• In physics you will encounter many different types of units.
• At times, you will need to convert from one to another
within the same type of measurement. (e.g. time or speed)
• This is done by multiplying the original number by a fraction
that cancels out the original units and replaces them with
the new ones.
• In order to do this, you need to know the conversion factor.
(e.g. 1ft. = 12in.)
2
0.54 × 6.33
3.4182
3.4
– Count the number of significant digits
in each of the numbers being
multiplied or divided.
– Calculate and round your answer to
the number of significant digits found
in the least significant input.
– It is sometimes easier to write these
problems horizontally.
– How many significant digits does the number have? 4
– Which digit must be rounded? the 3
Round Off!
– Round up or off? 45.63
• Adding and subtracting rely on similar
processes when significant digits are
being kept.
Multiplying
3
2
−4
1.1229 ×10
6
3
1.1× 103
Unit Conversion (Metric Prefixes)
•
•
Knowing metric prefixes is very
important, as they are
represented in many
measurements that you will
encounter. A complete table of
these can be found in your
book on page 9.
The prefixes that occur most
frequently are shown to the
right.
Prefix
Symbol
Multiple
Nano
n
10-9
Micro
µ
10-6
Milli
m
10-3
Centi
c
10-2
Kilo
k
103
Mega
M
106
Giga
G
109
Unit Conversions (Simple)
• Simple conversions involve one unit change.
• Example: Convert 1.34m into cm.
– The conversion factor is as follows:
1m = 100cm
– See the calculation below.
Desired Unit
(to incorporate)
1.34m ×
100cm
= 134cm
1m
Undesired Unit
(to cancel)
A Second Look
Now you try to convert
456cm into m.
456cm ×
1m
= 4.56m
100cm
Unit Conversions (Square/Cubic)
• Square and cubic unit conversions deal mostly with
areas and volumes respectively (e.g. ft2 or cm3).
• Here the linear conversion factor must be known and
then taken to the desired power before being used in a
fraction to make the conversion.
• Convert 500ft2 to m2.
What if it were cubic units?
2
3

2  1m
 1m 
500 ft × 

500 ft 3 × 

 3.28 ft 
 3.28 ft 
1m2
1m3
3
500 ft 2 ×
500 ft ×
10.7584 ft 2
35.2875 ft 3
46.5m2
14.2m3
Unit Conversions (Compound)
• Compound conversions involve fractional units, such
as speed or any other rate (e.g. mi/h).
• Multiple units within the fraction may have to be
converted here.
• Convert 60mi/h into ft/s.
60 mih ×
5280 ft
1h
×
= 88
1mi
3600 s
This fraction
cancels mi.
ft
s
Conversion Factors
1mi = 5280 ft
1h = 3600 s
This fraction
cancels h.
Conclusion
• Physics is a math-based science course.
• All four major skills will come into use
during the course of the year, many as
early as next section.
• Don’t forget how to do:
– Algebraic Manipulation
– Scientific Notation
– Significant Digits
– Unit Conversions