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

Dimensions / Units The raw material of physics is measurement. Every measurement is a comparison to some standard. “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

Fundamental or basic Dimensions We recognize seven fundamental or basic physical dimensions – the SI dimensions. These seven basic dimensions can be combined to describe other 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 of speed.

Derived Dimensions I can use the basic dimensions (and the correct units) to describe many different physical characteristics – How many can you name – and what basic dimensions are used?

Units There are many different systems of units used today. The SI system was established over many years. It uses the metric system which is base 10. The SI system is sometimes referred to as the MKS system (“meter, kilogram, second”)  The cgs (centimeter, gram, second) system is pretty much the same but is more convenient for smaller quantities. That is why it is frequently used in chemistry – you don’t use a kilogram of something very often! Here in the USA 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:  example: 12 inches in 1 foot 3 feet in 1 yard 1760 yards in 1 mile 5280 feet in 1 mile

Dimension Map Units?

Unit Table Dimension SI unit(MKS) cgs Common (B/E) unit Mass (M) _______ _______ ________ ______ s _______ ________ ______ _______ _______ ft ______ _______ cm 3 ________ Velocity (L/T) m/s ______ ________

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 m 2 If divided, then they cancel 6 m / 3 m = 2 (unit cancel) Note: 6m 2 / 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 m/s

CAREFUL! Even if working in the same dimension (like mass) I cannot work in different SIZES! 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 kg is OK. OR g - 2 g is OK.

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.

Practice Multiplying - add the exponents Dividing – subtract the exponents (3 x 10 2 ) (2 x 10 3 ) = (4 x 10 2 ) (1 x ) = 8 x 10 3 / 2 x 10 5 = 12 x / 2 x =

Conversions within the metric system Moving between prefixes is easy. You can always move one decimal place for each power of ten. For more complex changes, use “prefix substitution”

PREFIX SUBSTITUTION You MUST learn the value of each prefix. See page 12 of the text. Substitute the value for the prefix. This converts to the base unit. 3.5 x Tm = 3.5 x (10 12 ) m = 3.5 x 10 4 m From there you can move to any other prefix. 3.5 x 10 4 m x km = 3.5 x 10 1 km or 35 km 10 3 m

Practice Convert to base unit H* 4.37 x 10 6 pg = _______________ = ___________ mg 1.66 x Mm = ______________ = ____________ km 5.0 x Gg = ______________ = ____________  m

Solve the problem: Use the fact that the speed of light in a vacuum is about 3.00 x 10 8 m/s to determine how many kilometers a pulse from a laser beam travels in exactly one hour.

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

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 cm. It just ain’t that good !  The final answer must have the same number of sig figs as the least reliable instrument.

SIGNIFICANT FIGURES Calculations The rules for sig figs and rounding can be found on pps of the text. How many sig figs (SF) in each of the following measurements? a m/s b o C c K d J e MHz

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