MATH SKILLSMATH SKILLSFOR PHYSICSFOR PHYSICS

Units / Unit systemsUnits / Unit systemsScientific notation/ Scientific notation/

Significant figuresSignificant figuresAlgebraic manipulationAlgebraic manipulation

Geometry / Trig identitiesGeometry / Trig identitiesGraphingGraphing

Dimensional analysis Dimensional analysis

MATHEMATICS MATHEMATICS

• This unit is a review of most of the skills that This unit is a review of most of the skills that are necessary for understanding and are necessary for understanding and applying physics. A thorough review is applying physics. A thorough review is critical.critical.

Basic geometry, algebra, formula Basic geometry, algebra, formula rearrangement, graphing, trigonometry, rearrangement, graphing, trigonometry, scientific notation, and such are normally scientific notation, and such are normally assumedassumed for beginning physics. for beginning physics.

Please use this for review and preparation Please use this for review and preparation for classwork.for classwork.

Dimensions / UnitsDimensions / Units The raw material of science is measurement. Every The raw material of science is measurement. Every

measurement is a comparison to some standard.measurement is a comparison to some standard. Every measurement contains errorEvery measurement contains error ““The length of the football field is 100 yds.”The length of the football field is 100 yds.”

Dimension – the physical characteristic being Dimension – the physical characteristic being measured – “length”measured – “length”

Unit – we are using the “yard” which is a unit of Unit – we are using the “yard” which is a unit of length in the common or “British” system.length in the common or “British” system.

Measurement – How many of these units? 100 Measurement – How many of these units? 100 (yds)(yds)

Fundamental or basic Fundamental or basic DimensionsDimensions

We recognize We recognize sevenseven fundamental or basic fundamental or basic physical dimensions – the SI dimensions.physical dimensions – the SI dimensions. Know the five SI units in the table on page Know the five SI units in the table on page

958 of your text.958 of your text. 2 more not listed there:2 more not listed there:

Mole – amount of substanceMole – amount of substance Candela – luminous intensityCandela – luminous intensity

These seven basic dimensions can be These seven basic dimensions can be combined to derive other physical combined to derive other physical characteristics.characteristics.

Derived DimensionsDerived Dimensions Use of the basic dimensions (with correct units) Use of the basic dimensions (with correct units)

to describe many different physical to describe many different physical characteristics –characteristics –

Example –Example – From Houston to Austin is a measurement From Houston to Austin is a measurement

of about 180 miles.of about 180 miles. If I cover that distance in 3 hours, I can If I cover that distance in 3 hours, I can

find my average speed as 180 miles / 3 find my average speed as 180 miles / 3 hours = 60 mi/hr. hours = 60 mi/hr.

I have “derived” a new measurement I have “derived” a new measurement called speed.called speed.

See other commonly used units See other commonly used units (derived) on p. 959 of your text.(derived) on p. 959 of your text.

Take note of the following derived Take note of the following derived units: know the quantity measured units: know the quantity measured and the “conversion” (basic units)and the “conversion” (basic units) HertzHertz JouleJoule NewtonNewton voltvolt

Unit TableUnit TableDimensionDimension SI unit(MKS) SI unit(MKS) cgs Common (B/E) cgs Common (B/E) unitunit

Mass (M)Mass (M) _______ _______ _______ _______ ________ ________

______ s _____________ s _______ ________ ________

______ _______ _______ ft______ _______ _______ ft

______ _______ cm______ _______ cm33 ________ ________

Velocity (L/T) m/s ______Velocity (L/T) m/s ______ ________ ________

Dimension MapDimension MapCan you use the correct units?Can you use the correct units?

UnitsUnits The SI system uses the metric system which is The SI system uses the metric system which is

base 10. (Sometimes referred to as the MKS base 10. (Sometimes referred to as the MKS system - meter, kilogram, second)system - meter, kilogram, second) The cgs (centimeter, gram, second) system is The cgs (centimeter, gram, second) system is

more convenient for smaller quantities. That is more convenient for smaller quantities. That is why it is frequently used in chemistry – you why it is frequently used in chemistry – you don’t use a kilogram of a compound very often!don’t use a kilogram of a compound very often!

We use the “common” or “British” system of We use the “common” or “British” system of units. You can’t just multiply or divide by ten to units. You can’t just multiply or divide by ten to change the size. You have to memorize the silly change the size. You have to memorize the silly things: things: examples: examples: 12 inches in 1 foot 12 inches in 1 foot

3 feet in 1 3 feet in 1 yardyard

1760 yards in 1 mile 1760 yards in 1 mile 5280 feet in 1 mile5280 feet in 1 mile

Working with unitsWorking with unitsSimilar dimensions can be added or subtracted – nothing Similar dimensions can be added or subtracted – nothing

changes.changes.

3 m + 3 m = 6 m 52 kg - 12 kg = 40 kg3 m + 3 m = 6 m 52 kg - 12 kg = 40 kg. . BUT ----You cannot add or subtract different BUT ----You cannot add or subtract different

dimensionsdimensions

3 m + 12 kg = no answer 3 m + 12 kg = no answer

You can’t add a distance to a mass – just common sense.You can’t add a distance to a mass – just common sense.

All dimensions can be multiplied All dimensions can be multiplied or dividedor divided

Similar dimensionsSimilar dimensionsIf multiplied then they become squared or cubed.If multiplied then they become squared or cubed.

3 m x 3 m = 9 m3 m x 3 m = 9 m22

If divided, then they cancelIf divided, then they cancel 6 m / 3 m = 2 (no unit – it cancels out)6 m / 3 m = 2 (no unit – it cancels out)

Note: 6mNote: 6m22 / 3 m = 2 m (only one “m” is / 3 m = 2 m (only one “m” is cancelled)cancelled)

Different dimensionsDifferent dimensionsMultiplied: 3 m x 2 s = 6 mMultiplied: 3 m x 2 s = 6 m·s·s

Divided (a ratio): 88 km / 4 s = 22 km/sDivided (a ratio): 88 km / 4 s = 22 km/s

CAREFUL! CAREFUL!CAREFUL! CAREFUL! Even if working in the same dimension (like Even if working in the same dimension (like

mass) I cannot work in different SIZES (this is mass) I cannot work in different SIZES (this is what prefixes mean – like kilo, milli, Mega, etc)!what prefixes mean – like kilo, milli, Mega, etc)!

THE PREFIXES MUST BE THE SAME !!!!!THE PREFIXES MUST BE THE SAME !!!!! 5 kg – 2 kg = 3 kg All is good.5 kg – 2 kg = 3 kg All is good. 5 kg – 2 g = DISASTROUS CATASTROPHY!5 kg – 2 g = DISASTROUS CATASTROPHY!

Gotta be the same - so, 5 kg - .002 kg is OK.Gotta be the same - so, 5 kg - .002 kg is OK. OR - 5000 g - 2 g is OK.OR - 5000 g - 2 g is OK.

Scientific notationScientific notation provides a short-hand provides a short-hand method for expressing very small and very method for expressing very small and very

large numbers.large numbers.

sx

mx

s

mv

3

2

1020.4

1082.5

0042.0

582

0 000000001 10

0 000001 10

0 001 10

1 10

1000 10

1 000 000 10

1 000 000 000 10

9

6

3

0

3

6

9

.

.

.

, ,

, , ,

Examples: 93 000 000 mi = 9.30 x 107 mi

0.000 042 kg = 4.20 x 10-4 kg

sx

mx

s

mv

3

2

1020.4

1082.5

0042.0

582

V = 1.39 x 10 5 m/s

SCIENTIFIC (EXPONENTIAL)SCIENTIFIC (EXPONENTIAL)NOTATIONNOTATION

Since the metric system is base 10, this makes Since the metric system is base 10, this makes multiplying and dividing easy. Exponential multiplying and dividing easy. Exponential notation is a shorthand for writing notation is a shorthand for writing exceptionally large or small values – but it is exceptionally large or small values – but it is also very helpful for controlling significant also very helpful for controlling significant figures.figures.

Using exponents can make the work much Using exponents can make the work much easier. easier.

Learn the metric prefixes from Table 1-3 on Learn the metric prefixes from Table 1-3 on page 12. Study Sample Problem on p. 14 page 12. Study Sample Problem on p. 14

Metric prefixes are used to express Metric prefixes are used to express very small or very large numbers.very small or very large numbers.

Learn the metric prefixes from Table Learn the metric prefixes from Table 1-3 on page 12. Study Sample 1-3 on page 12. Study Sample Problem on p. 14 Problem on p. 14

PREFIX SUBSTITUTIONPREFIX SUBSTITUTION

You MUST learn the value of each You MUST learn the value of each prefix.prefix.

Substitute the value for the prefix. This converts Substitute the value for the prefix. This converts to the base unit.to the base unit.

3.5 x 103.5 x 10-8-8 Tm = 3.5 x 10 Tm = 3.5 x 10-8-8 (10 (101212) m = 3.5 x 10) m = 3.5 x 1044 m m From there you can convert to the needed value.From there you can convert to the needed value.

3.5 x 103.5 x 1044 m x m x kmkm = 3.5 x 10 = 3.5 x 1011 km or 35 km or 35 kmkm

101033 m m Remember your dimensional analysis techniques !!!!Remember your dimensional analysis techniques !!!!

Solve the problem:Solve the problem:Use the fact that the speed of light in a vacuum is Use the fact that the speed of light in a vacuum is

aboutabout

3.00 x 103.00 x 1088 m/s to determine how many kilometers a m/s to determine how many kilometers a pulse from a laser beam travels in exactly one hour. pulse from a laser beam travels in exactly one hour.

(ans: 1.08 x 10(ans: 1.08 x 1099 km) km)

How many meters?How many meters? Easy, substitute 10 Easy, substitute 1033 for “k” for “k” 1.08 x 10 1.08 x 1099 (10 (1033) m = 1.08 x 10 ) m = 1.08 x 10 1212 m m

or 1.08 Tmor 1.08 Tm

Solve the problem:Solve the problem:The largest building in the world by volume is the The largest building in the world by volume is the

Boeing 747 plant in Everett, Washington. It Boeing 747 plant in Everett, Washington. It measures approximately 0.631 km long, 1433 measures approximately 0.631 km long, 1433 m wide and m wide and

7400 cm high. What is its volume in cubic 7400 cm high. What is its volume in cubic meters?meters?

(ans: 6.70 x 10(ans: 6.70 x 1077 m m33))

SIGNIFICANT FIGURES SIGNIFICANT FIGURES (SF)(SF)

Why is this concept so important in science?Why is this concept so important in science? Every measurement is limited in terms of accuracy. Every measurement is limited in terms of accuracy.

This is due to both the instrument and human ability This is due to both the instrument and human ability to read the instrument.to read the instrument.

The number of sig figs in a measurement includes the The number of sig figs in a measurement includes the figures that are certain and the first “doubtful” digit. figures that are certain and the first “doubtful” digit.

With a metric ruler a desk can be measured to 65.2 cm With a metric ruler a desk can be measured to 65.2 cm – but not 65.0002 cm. It just ain’t that good !– but not 65.0002 cm. It just ain’t that good !

The final answer must have the same number of sig The final answer must have the same number of sig figs as the least reliable instrument.figs as the least reliable instrument.

The rules for sig figs and rounding can be found on pages 17- 19 of the text.

Use Tables 1-4, 1-5 and 1-6 for SF rules

How many sig figs (SF) in each of the following measurements?

a. 3000 000 000 m/s

b. 25.030 oC

c. 0.006 070 K

d. 1.004 J

e. 1.305 20 MHz

Solve the problems:Solve the problems:

Find the sum of: 756g, 37.2g, 0.83g, and 2.5gFind the sum of: 756g, 37.2g, 0.83g, and 2.5g

Divide: 3.2m / 3.563 sDivide: 3.2m / 3.563 s

Multiply: 5.67 mm x Multiply: 5.67 mm x (Ooohhh, sneaky. There’s a pi in there.)(Ooohhh, sneaky. There’s a pi in there.)

Working With Formulas:Working With Formulas:Many applications of physics require one to Many applications of physics require one to solve and evaluate mathematical solve and evaluate mathematical expressions called expressions called formulasformulas..

LW

H

Consider Consider Volume VVolume V, for , for example:example:

V = LWHV = LWH

Applying Applying laws of algebralaws of algebra, we can solve for , we can solve for LL, , WW, , or or HH::

VL

WHV

LWH

VW

LHV

WLH

VH

LWV

HLW

Formula RearrangementFormula RearrangementSolve for Solve for AA

Consider the following formulaConsider the following formula::A C

B D

Multiply by B to solve for AMultiply by B to solve for A::BA BC

B D

Notice that Notice that BB has moved has moved up to the rightup to the right..

1

A BC

D

BCA

DBC

AD

Thus, the solution for Thus, the solution for AA becomes:becomes:

Geometry and Trig ReviewGeometry and Trig Review

See pages 946 through 948 to review See pages 946 through 948 to review the basic geometry and trig needed the basic geometry and trig needed for this course.for this course.

Trigonometry ReviewTrigonometry Review You are expected to know the following: You are expected to know the following:

y

x

R

y = R sin y = R sin

x = R cos x = R cos

siny

R

cosx

R

tany

x

R2 = x2 + y2

R2 = x2 + y2

TrigonometryTrigonometry

y = x tan y = x tan

Θ = tan-1(y/x)Θ = tan-1(y/x)