MATH SKILLS FOR PHYSICS

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

Dimensions / Units “The length of the football field is 100 yds.”
Dimension – the physical characteristic being measured – “length” Unit – the standard we are using Measurement – How many of these units (standards)? 100 yards Our standard for this measurement is the yard. How would the number (magnitude) change if our standard for this measurement was the centimeter?

Fundamental 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, temperature These 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.

Derived Dimensions I 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 = m2 Volume = m x m x m = m3 Speed = m / s Density = mass/volume

Units The 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 yard 1760 yards in 1 mile

Dimension Map

Unit 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___

Working with units Similar 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 dimensions 3 m + 12 kg = no answer You can’t add a distance to a mass. Similar dimensions can be multiplied or divided 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 (unit cancel) Note: 6m2 / 3 m = 2 m (only one “m” is cancelled)

CAREFUL! 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.

Working 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 m 4 km cg =cannot 9 km / 3s = 3 km/s add different dimensions

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. ns = 10-9 s millimeter = 10-3m Gg = 109 g Mm = 106 m microgram = 10-6 g ks = 103 s

Unit 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 s 1.0 m2 = (cm)2 = 1.0 x 104 cm2 (10-2 m)2

Practice Convert to base unit 20 mm = 2.0 x 101(10-3)m = 2.0 x 10-2 m
1st step – write number in SN 2nd step – prefix substitution 20 mm = 2.0 x 101(10-3)m = 2.0 x 10-2 m 13 mm = 1.3 x 101(10-6)m = 1.3 x 10-5 m 0.027 Mg = 2.7 x 10-2(106)g = 2.7 x 104 g

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 102) (2 x 103) = 6 x 105 (4 x 102) (1 x 10-4) = 4 x 10-2 8 x 103 / 2 x 105 = 4 x 10-2 12 x 10-2 / 2 x 10-4 = 6 x 102

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 1 oC 5 K 4 1.004 J 4 MHz 6

Solve the problems: Find the sum of: 756g, 37.2g, 0.83g, and 2.5g
calculator: g round to zero decimals = 797 g Divide: 3.2m / s calculator: m/s round to 2 sig figs = .90 m/s Multiply: mm x p calculator: mm round to 3 sig figs = mm  does not count for sig figs

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