MATH SKILLS FOR PHYSICS Units / Unit systems Scientific notation/ Significant figures Algebraic manipulation Geometry / Trig identities Graphing Dimensional analysis

MATHEMATICS This unit is a review of most of the skills that are necessary for understanding and applying physics. A thorough review is critical. Basic geometry, algebra, formula rearrangement, graphing, trigonometry, scientific notation, and such are normally assumed for beginning physics. Please use this for review and preparation for classwork.

Dimensions / Units The raw material of science is measurement. Every measurement is a comparison to some standard. Every measurement contains error “The length of the football field is 100 yds.” Dimension – the physical characteristic being measured – “length” Unit – we are using the “yard” which is a unit of length in the common or “British” system. Measurement – How many of these units? 100 (yds)

Fundamental or basic Dimensions We recognize seven fundamental or basic physical dimensions – the SI dimensions. Know the five SI units in the table on page 958 of your text. 2 more not listed there: Mole – amount of substance Candela – luminous intensity These seven basic dimensions can be combined to derive other physical characteristics.

Derived Dimensions Use of the basic dimensions (with correct units) to describe many different physical characteristics – Example – From Houston to Austin is a measurement of about 180 miles. If I cover that distance in 3 hours, I can find my average speed as 180 miles / 3 hours = 60 mi/hr. I have “derived” a new measurement called speed.

See other commonly used units (derived) on p. 959 of your text. Take note of the following derived units: know the quantity measured and the “conversion” (basic units) Hertz Joule Newton volt

Unit Table Dimension SI unit(MKS) cgs Common (B/E) unit Mass (M) _______ _______ ________ ______ s _______ ________ ______ _______ _______ ft ______ _______ cm3 ________ Velocity (L/T) m/s ______ ________

Dimension Map Can you use the correct units?

Units The SI system uses the metric system which is base 10. (Sometimes referred to as the MKS system - meter, kilogram, second) The cgs (centimeter, gram, second) system is more convenient for smaller quantities. That is why it is frequently used in chemistry – you don’t use a kilogram of a compound very often! We use the “common” or “British” system of units. You can’t just multiply or divide by ten to change the size. You have to memorize the silly things: examples: 12 inches in 1 foot 3 feet in 1 yard 1760 yards in 1 mile 5280 feet in 1 mile

Working with units Similar dimensions can be added or subtracted – nothing changes. 3 m + 3 m = 6 m 52 kg - 12 kg = 40 kg . BUT ----You cannot add or subtract different dimensions 3 m + 12 kg = no answer You can’t add a distance to a mass – just common sense.

All dimensions can be multiplied or divided Similar dimensions If multiplied then they become squared or cubed. 3 m x 3 m = 9 m2 If divided, then they cancel 6 m / 3 m = 2 (no unit – it cancels out) Note: 6m2 / 3 m = 2 m (only one “m” is cancelled) Different dimensions Multiplied: 3 m x 2 s = 6 m·s Divided (a ratio): 88 km / 4 s = 22 km/s

CAREFUL! CAREFUL! Even if working in the same dimension (like mass) I cannot work in different SIZES (this is what prefixes mean – like kilo, milli, Mega, etc)! THE PREFIXES MUST BE THE SAME !!!!! 5 kg – 2 kg = 3 kg All is good. 5 kg – 2 g = DISASTROUS CATASTROPHY! Gotta be the same - so, 5 kg - .002 kg is OK. OR - 5000 g - 2 g is OK.

Scientific notation provides a short-hand method for expressing very small and very large numbers. 000000001 10 000001 001 1 1000 000 9 6 3 . , = - Examples: 93 000 000 mi = 9.30 x 107 mi 0.000 042 kg = 4.20 x 10-4 kg V = 1.39 x 10 5 m/s

SCIENTIFIC (EXPONENTIAL) NOTATION Since the metric system is base 10, this makes multiplying and dividing easy. Exponential notation is a shorthand for writing exceptionally large or small values – but it is also very helpful for controlling significant figures. Using exponents can make the work much easier. Learn the metric prefixes from Table 1-3 on page 12. Study Sample Problem on p. 14

Metric prefixes are used to express very small or very large numbers. Learn the metric prefixes from Table 1-3 on page 12. Study Sample Problem on p. 14

PREFIX SUBSTITUTION You MUST learn the value of each prefix. Substitute the value for the prefix. This converts to the base unit. 3.5 x 10-8 Tm = 3.5 x 10-8 (1012) m = 3.5 x 104 m From there you can convert to the needed value. 3.5 x 104 m x km = 3.5 x 101 km or 35 km 103 m Remember your dimensional analysis techniques !!!!

Solve the problem: Use the fact that the speed of light in a vacuum is about 3.00 x 108 m/s to determine how many kilometers a pulse from a laser beam travels in exactly one hour. (ans: 1.08 x 109 km) How many meters? Easy, substitute 103 for “k” 1.08 x 109 (103) m = 1.08 x 10 12 m or 1.08 Tm

Solve the problem: The largest building in the world by volume is the Boeing 747 plant in Everett, Washington. It measures approximately 0.631 km long, 1433 m wide and 7400 cm high. What is its volume in cubic meters? (ans: 6.70 x 107 m3)

SIGNIFICANT FIGURES (SF) Why is this concept so important in science? Every measurement is limited in terms of accuracy. This is due to both the instrument and human ability to read the instrument. The number of sig figs in a measurement includes the figures that are certain and the first “doubtful” digit. With a metric ruler a desk can be measured to 65.2 cm – but not 65.0002 cm. It just ain’t that good ! The final answer must have the same number of sig 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 25.030 oC 0.006 070 K 1.004 J 1.305 20 MHz

Solve the problems: Find the sum of: 756g, 37.2g, 0.83g, and 2.5g Divide: 3.2m / 3.563 s Multiply: 5.67 mm x p (Ooohhh, sneaky. There’s a pi in there.)

Working With Formulas: Many applications of physics require one to solve and evaluate mathematical expressions called formulas. Consider Volume V, for example: L W H V = LWH Applying laws of algebra, we can solve for L, W, or H:

Formula Rearrangement Solve for A Consider the following formula: Multiply by B to solve for A: Notice that B has moved up to the right. Thus, the solution for A becomes:

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

Trigonometry Review y = R sin q R y x = R cos q q y = x tan q x You are expected to know the following: Trigonometry y = R sin q y x R q x = R cos q y = x tan q R2 = x2 + y2 Θ = tan-1(y/x)