physics introductory unit ~the mathematical background~
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Math Skills• There are several skills, some of which
you have already learned, that you will need to use extensively in Physics.
• These include the following:– Algebra (manipulation of formulas)– Scientific Notation (very lg/sm numbers)– Significant Digits– Unit Conversions
Math Rules!
dv
t
45.2 10
1 100m cm
0.07034
Scientific Notation• Scientific notation relies on exponential powers
of ten (10x) to simplify extremely large and small numbers.
• 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.
64,673,000 4.673 10 Standard Notation Scientific Notation
CoefficientPower of Ten
Scientific Notation (Cont.)• Large numbers correspond to positive
powers of ten.
• Small numbers correspond to negative powers of ten.
• Figuring out the power on the ten relates to how many places you need to move the decimal point from its initial position.
414,000 1.4 10
40.00034 3.4 10
412080 1.208 10 20.037 3.7 10 4 Moves 2 Moves
Scientific Notation (Multiplication)• At times, numbers in scientific notation will be
multiplied as shown below.
• The trick is to combine the powers of ten with each other and the non-exponent terms with each other. Then simplify.
6 24.2 10 3.1 10
6 24.2 3.1 10 10 813.02 1091.302 10
Note: Remember that exponents add when like bases are multiplied.
Scientific Notation (Division)• At times, numbers in scientific notation will be
divided as shown below.
• As before, you need to combine terms. The exponent rule changes to subtraction when division is involved.
3
2
8.4 10
1.4 10
3
2
8.4 10
1.4 10
16.0 10
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.
• In simplified form it can be written as follows:
• The same holds true for small numbers.
5100,000 1.0 10
5100,000 10
30.001 10
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.– All nonzero digits are significant.
– Final zeros after the decimal point are significant.
– Zeros between other significant digits are significant.
– Zeros used solely for spacing are not significant.
0 or 37 0.056
. or 0.43 0 0560
or 306 0.705
,000 or 024 .007
Significant Digits (Special Cases)• A bar can be placed over zeros that are not
normally significant in order to make them significant.
• This usually occurs after some instances of rounding. Here a problem would specify to how many digits you must round.
400 vs. 4001 Significant Digit 3 Significant Digits
0.003 vs. 0.0031 Significant Digit 2 Significant Digits
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.– How many significant digits does the number have? – Which digit must be rounded?
– Round up or off?
• Round the number 6798 to three significant digits.
4
the 3
45.645.63 Round Off!
6800
• Keeping correct significant digits while multiplying and dividing relies on the same process.– 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.
Significant Digits (Mult/Div)
0.54 6.333.4182
3.4
2 3
Multiplying
Dividing
7.261 0.236.305
40
4 1
Significant Digits (Add/Sub)• Adding and subtracting rely on the
same process when significant digits are being kept.– Align the addends (for addition) or the
minuends and subtrahends (for subtraction) vertically.
– Draw a vertical line down the least precise number (the one with least decimal places).
– Add or subtract the values.
– Round to the left of the vertical line.
– Addition problems can have more than two numbers.
Addition
Subtraction
363.7 14.374363.7
14.734
378.434
378.4
Units and Unit Conversion
• Anthony jumped in his car and drove 10 to the grocery store, where he bought 5. He returned within 30.
WARNING: You will lose points for any answer that does not have proper units!!!
Units and Unit Conversion
• In this class we will use the MKS system. M meter (m) … unit for length
K kilogram (kg) … unit for mass
S second (s) … unit for time
All other units are derived units … they come from the 3 above.
Standard Units
Unit Conversions We can multiply any number by 1 and not change its value.
1 100m cm
.
1 1001
100 100
m cm
cm cm
How many m are there in 5783cm?
15783 * 57.83
100
mcm m
cm
Practice Problem
.
6.3 ?hr s
1 60minhr
1min 60sec
60min1
1hr
60s1
1min
60min 60s6.3 * * 22680
1 1minhr s
hr
Practice Compound Problem
.
55 ?mile mhr s
1 1.61mile km1min 60s
1 1000km m
1 60minhr
1.61 1000 1 1min55 * * * *
1 1 60min 60
miles km m hr
hr miles km s
24.6 ms
Algebra• Numerous times while studying Physics, you will
be required to use algebra to solve equations.
• Isolating the variable involves the use of inverse order of operations to manipulate the variables.– Addition(+) and Subtraction(-) are inverse operations.– Multiplication(× or ·) and Division(÷) are inverse operations.– Squaring(2) and square rooting(√) are inverse operations.
Find the value of x2
(9 7)*4
(4 3) 17x
When solving for the value of an equation, you must use
ORDER OF OPERATIONS
Parenthesis (Grouping)
Exponents / PowersMultiplicationDivisionAdditionSubtraction
When solving for a variable in an algebraic equation, you must use
INVERSE ORDER OF OPERATIONS
1) Collect like terms2) Addition / Subtraction3) Move variable from denominator to the
numeratora) Cross multiplyb) Reciprocalc) Multiply both sides by the variable
4) Multiplication / Division5) Exponents6) Parenthesis (Grouping)
Algebra (Sample)• 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.
dv
t
dv t t
t
d v t
Other Algebra Samples• Given the equation:
• Solve for t.
• Given the equation:
• Solve for v2.
dv
t 2 2
2 1 2 12v v a d d
vt dd
tv
22 1 2 12v v a d d
Note: When you take the square root, a symbol must be included in front of the radical.
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