Presentation on theme: "MATH SKILLS FOR PHYSICS"— Presentation transcript:
1MATH SKILLS FOR PHYSICS Units / Unit systemsScientific notation/ Significant figuresAlgebraic manipulationDimensional analysisGeometry / Trig identities
2Dimensions / Units “The length of the football field is 100 yds.” Dimension – the physical characteristic being measured – “length”Unit – the standard we are usingMeasurement – How many of these units (standards)? 100 yardsOur standard for this measurement is the yard. How would the number (magnitude) change if our standard for this measurement was the centimeter?
3Fundamental or basic Dimensions We recognize seven fundamental or basic physical dimensions – the SI dimensions.List: length, mass, temperature, time, amount of substance(mole), electric current, candela, temperatureThese seven basic dimensions can be combined to describe other physical characteristics. These combinations are called “derived” dimensions.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 - speed.
4Derived DimensionsI can use the basic dimensions (and the correct units) to describe many different physical characteristics –How many can you name? (include units)Area = m x m = m2Volume = m x m x m = m3Speed = m / sDensity = mass/volume
5UnitsThe SI system was established over many years. It describes seven fundamental (basic) units of measurement for the seven fundamental physical dimensions.This 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: inches in 1 foot feet in 1 yard1760 yards in 1 mile
7Unit Table Dimension SI unit(MKS) cgs Common (B/E) unit Mass (M) ___kg____ ___g____ __slug_______Time_____ s __s_____ ___s______Length___ __m_____ __cm_____ ft_volume_ _m3____ cm ___ft3__Velocity (L/T) m/s _cm/s___ _ft/s___
8Working with unitsSimilar dimensions can be added or subtracted – nothing changes.3 m + 3 m = 6 m kg kg = 40 kg.BUT ----You can’t add or subtract different dimensions3 m + 12 kg = no answerYou can’t add a distance to a mass.Similar dimensions can be multiplied or dividedIf multiplied then they become squared or cubed.3 m x 3 m = 9 m2If divided, then they cancel6 m / 3 m = 2 (unit cancel)Note: 6m2 / 3 m = 2 m (only one “m” is cancelled)
9CAREFUL! 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, kg kg is OK.OR g g is OK.
10Working with units -If units are being added or subtracted, they must be the same (and that includes any metric prefix). Unlike units can be multiplied or divided.2 m x 4 m = 8 m2 8 m2 / 2 m = 4 m4 km cg =cannot 9 km / 3s = 3 km/sadd differentdimensions
11Conversions 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”
12PREFIX 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.ns = 10-9 s millimeter = 10-3mGg = 109 g Mm = 106 mmicrogram = 10-6 g ks = 103 s
13Unit Conversions - 25 m/s = 25m x km x 3.6 x 103 s = 90 km/hr 103m hr 1.0 yr = 1.0 yr x 365 day x 24 hr x 3.6 x 103s yr day hr= x 107 s1.0 m2 = (cm)2 = 1.0 x 104 cm2(10-2 m)2
14Practice Convert to base unit 20 mm = 2.0 x 101(10-3)m = 2.0 x 10-2 m 1st step – write number in SN2nd step – prefix substitution20 mm = 2.0 x 101(10-3)m = 2.0 x 10-2 m13 mm = 1.3 x 101(10-6)m = 1.3 x 10-5 m0.027 Mg = 2.7 x 10-2(106)g = 2.7 x 104 g
15SCIENTIFIC (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.
16Practice Multiplying - add the exponents Dividing – subtract the exponents(3 x 102) (2 x 103) = 6 x 105(4 x 102) (1 x 10-4) = 4 x 10-28 x 103 / 2 x 105 = 4 x 10-212 x 10-2 / 2 x 10-4 = 6 x 102
17SIGNIFICANT 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.
18SIGNIFICANT FIGURES Calculations The rules for sig figs and rounding can be found on ppsof the text.How many sig figs (SF) in each of the following measurements?a m/s 1oC 5K 41.004 J 4MHz 6
19Solve the problems: Find the sum of: 756g, 37.2g, 0.83g, and 2.5g calculator: g round to zero decimals = 797 gDivide: 3.2m / scalculator: m/s round to 2 sig figs = .90 m/sMultiply: mm x pcalculator: mm round to 3 sig figs = mm does not count for sig figs