Physics Introductory Unit ~The Mathematical Background~

Physics Introductory Unit

~The Mathematical Background~

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.

Unit Conversions

Unit Conversions

Unit Conversions

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.