PHYSICS

Unit 1: Mathematical Toolkit

Physics

• Fundamental Science▫ foundation of other

physical sciences• Divided into major areas

▫ Mechanics Waves

▫ Thermodynamics▫ Electromagnetism▫ Modern Physics

Relativity Quantum Mechanics Nuclear

Scientific Notation

• Placing numbers in scientific (exponential) notation has several advantages.

• For very large numbers and extremely small ones, these numbers can be placed in scientific notation in order to express them in a more concise form.

• In addition, numbers placed in this notation can be used in a computation with far greater ease. This last advantage was more practical before the advent of calculators and their abundance.

Scientific Notation• Placing numbers in scientific notation from standard

form with the following rules:▫ Move the decimal so that there is only 1 non-zero number to

the left of the decimal.▫ If you move the decimal to the left add the number of moves

you made to the exponent.▫ If you move the decimal to the right subtract the number of

move you made from the exponent

• Placing number in standard notation from scientific notation▫ If the power is positive move the decimal to the right an equal

number of spaces as the exponent.▫ If the power is negative move the decimal the left an equal

number of spaces as the exponent.

Try these Numbers• 123,876.3 • 1,236,840. • 4.22 • 0.000,000,000,000,211 • 0.000238 • 0.910

Multiplying in Scientific Notation• (N X 10x) (M X 10y) = (N) (M) X 10x+y

• First multiply the N and M numbers together and express an answer.

• Secondly multiply the exponential parts together by ADDING the exponents together.

• Finally multiply the two results for your final answer.

Dividing in Scientific Notation• N X 10x / M X 10y = N/M X 10x-y

• Make sure to watch your signs.• (+) * (+) = +• (+) * (-) = -• (-) * (+) = -• (+) / (+) = +• (+) / (-) = -• (-) / (+) = -

• Addition and subtraction the largest number dominates

Try These• (3 x 105) (3 x 106) =• (2 x 107) (3 x 10-9) =• (4 x 10-6) (4 x 10-4) =• 9 x 108 / 3 x 10-2 =• 8 x 107 / 4 x 103 = • 6 x 10-2 / 2 x 10-6 =

Scientific Notation Raised to a Power• (N X 10x)y= (Ny) X 10x*y

• Raise N to the Y power• Multiply x * y

Try These• (6.5 * 103)2=

• (7 * 105)3=

• (4.9 * 106)(1/2)=

Adding and Subtracting in Scientific Notation

• (N X 10x) + (M X 10x) = (N + M) X 10x • (N X 10y) - (M X 10y) = (N-M) X 10y

• Convert so that 10 is raised to the same power• Add or subtract M and N• Don’t change if there is only 1 non-zero number to the left

of the decimal (IE Standard form)

Try These• (8 X 103) + (9 X 104) =

• (5 X 102) - (4 X 102) =

• (8 X 102) + (4 X 103) =

• (6 X 10-2) - (2 X 10-1)=

Measurements• Basis of testing theories in science• Need to have consistent systems of units for the measurements

• 7 Basic Measurements all others • Need rules for dealing with the uncertainties

Basic Quantities and Their Dimension• Length [L]• Mass [M]• Time [T]

Systems of Measurement• Standardized systems

• agreed upon by some authority, usually a governmental body

• SI -- Systéme International• agreed to in 1960 by an international committee• main system used in this text• also called mks for the first letters in the units of the

fundamental quantities

Metric Prefixes• Prefixes correspond to powers of 10• Each prefix has a specific name• Each prefix has a specific abbreviation

SI PREFIXES

Abbreviation Prefix Power

G Giga 109

M Mega 106

k kilo 1 000 or 103

h hecta 100or 102

da deka 10 or 101

THE BASE ----- 0

d deci 0.1 or 10-1

c centi 0.01or 10-2

m milli 0.001 or 10-3

m micro 10-6

n nano 10-9

SI Units

• All other SI units are derived from the 7 basic dimensions mentioned.

• Example• Distance/ Time = [meter/second]

ORDER OF MAGNITUDES

• An order of magnitude calculation is a rough estimate designed to be accurate to within a factor of about 10

• To get ideas and feeling for what size of numbers are involved in situation where a precise count is not possible or important

ORDER OF MAGNITUDE TYPICAL DISTANCES

• Diameter of the Milky Way 2x1020 m• One light year 4x1016 m• Distance from Earth to Sun 1.5x1011m• Radius of Earth 6.37x106m• Length of a football field 102m• Height of a person 2x100 m• Diameter of a CD 1.2x10-1m• Diameter of the aorta 1.8x10-2 m• Diameter of a red blood cell 8x10-6 m• Diameter of the hydrogen atom 10-10 m• Diameter of the proton 2x10-15 m

ORDER OF MAGNITUDE

EXAMPLEEstimate the number of seconds in a human"lifetime."You can choose the definition of "lifetime."Do all reasonable choices of "lifetime" give answersthat have the same order of magnitude?

The order of magnitude estimate: 109 seconds• 70 yr = 2.2 x 109 s• 100 yr = 3.1 x 109 s• 50 yr = 1.6 x 109 s

Summary for Range of Magnitudes

• You will need to be able to state (express) quantities to the nearest order of magnitude, that is to say to the nearest 10x

Range of magnitudes of quantities in our universe • Sizes

• From 10-15 m (subnuclear particles)• To 10+25 m (extent of the visible universe)

• masses• From 10-30 kg (electron mass)• To 10+50 kg (mass of the universe)

• Times• From 10-23 s (passage of light across a nucleus)• To 10+18 s (age of the universe)

• You will also be required to state (express) ratios of quantities as differences of order of magnitude.Example:• the hydrogen atom has a diameter of 10-10 m• whereas the nucleus is 10-15 m• The difference is 105

• A difference of 5 orders of magnitude

Experimental Error• When in an experiment, it is important to analyze and

present data that is as correct.• Keeping in mind, that neither the measuring instrument or the act

of measuring is ever perfect.• Every experiment is going to have some type of experimental error.

• Experimental Error affects the accuracy and precision of data.

Accuracy • Describes how close a measurement is to a known or

accepted value.• Example: Suppose that the mass of a sample is known to be 5.85

grams. A measurement of 5.81 grams would be more accurate than a measurement of 6.05 grams.

Precision• Describes how close several measurements are to each

other• The closer the measurements are to each other, the

higher their precision

Accuracy & Precision• Measurements can be precise even if they are not

accurate.• Example: A group of measurements can be close to one another

but not be close to the actual, true value.

Systematic Errors• These are errors that occur every time you make a certain

measurement.• Examples: Errors due to the calibration of measuring instruments

or errors due to faulty procedures/assumptions.

• This type of error will make your measurements higher or lower than they would be if these problems did not exist.• Example: A balance that is not zeroed out (calibrated properly)

• Measurements cannot be accurate if there are systematic errors.

Random Errors• These are errors that cannot be predicted.

• Examples: errors of judgment in reading a meter or a scale, error due to fluctuating experimental conditions.

• If the random errors in an experiment are small, then the experiment is said to be precise.• Example: You measure the temperature in a classroom over a

period of several days. If there is a large difference in classroom temperature could result in random errors when measuring the experimental temperature change.

Significant Digits• The data recorded in a lab/experiment, should include

only significant digits (significant figures)• Significant figures are the digits that are meaningful in a

measurement or calculation.• The measurement device determines how many

significant digits should be recorded.• If using a digital device, record the measurement value exactly as it

is shown on the screen.• If reading a result from a ruled scale, the value should include each

digit that is 100% certain and one uncertain (estimated) digit.

Example of Determining Result

Rules for SigFigs

Uncertainties• Uncertainties in measurements should always be rounded

to one significant digit.• When measurements are made with devices that have a

ruled scale, the uncertainty is HALF the value of the precision of the scale. (The markings of the scale show the precision)• For example, the markings on the scale below are every 0.1

centimeter, so the uncertainty would be half of this (0.05 centimeter).

• So when recording this measurement, it should be recorded as 8.42 ± .05 cm

Adding/Subtracting Measurements• When adding and subtracting quantities, the result should

have the same number of decimal places as the least number of decimal places in any of the numbers you are adding or subtracting.

Multiplying/Dividing Measurements• When multiplying and dividing, the result should have the

same number of significant digits as the number in the calculation with the least number of significant digits.

Mean• The mean of a set of data is the sum of all the

measurement values divided by the number of measurements.• If your data is a sample of a population, then the mean you

calculate is an estimate of the mean of a population.

• The mean can be calculated using the following formula

Standard Deviation• The measure of how spread out data values are.

• If you have similar values, then the standard deviation is small.• If you have a wide range of values, then the standard deviation is

high.

• Since standard deviations are a measure of uncertainty, they should be standard using only one significant digit.

• It is represented by, σ, and to calculate you use the following formula

where xi is each measurement, x with a line over it is the mean, n is the number of measurements

Standard Error• Is an estimate of the precision of the data.

• Measures the data’s uncertainty by reducing the standard deviation if a large number of data values are included.

• Represented by SE in an equation.

• You can calculate the standard error using the following formula

Where SE is the standard error, n is the number of measurements you have, σ is the standard deviation

EXAMPLE TIME – YAY • Suppose you have the following values for the

temperature of a substance…

• Calculate the mean• Calculate the standard deviation• Calculate the standard error

ANSWERS !• The mean

• The standard deviation

• The standard error

Confidence Interval• A range of values within which the true value has a

probability of being.• If you measure a single quantity multiple times and get a small

standard deviation, the confidence interval would be narrow.• A large confidence interval would indicate that you have lots of

errors in your measurement

Bell Curves & Confidence Intervals• Confidence intervals can be presented in different ways,

most commonly in the form of a bell curve.• This applied to data that has a normal (bell-shaped) distribution.

• The mean lies at the peak of distribution

• Confidence intervals on either side of the peak describe multiples of the standard deviation from the mean.

• The percentages are determined by finding the area under the curve

Propagation of Error• If you have results that include two or more

measurements, you must state the combined uncertainty of the measurements• Basically if your measurements have different uncertainties, before

you do anything with these measurements.

Combined Uncertainty (Add/Subtract)• When adding or subtracting quantities, you can calculate

the combined uncertainties of measurements by square rooting the sum of the squares of their individual quantities.• For example. you want to calculate a quantity K = F + G + H, where

F, G, H are measured values and their uncertainties are ΔF, ΔG, and ΔH where Δ means “the uncertainty of” … then the uncertainty of K, is…

Example

Combined Uncertainty (Multiply/Divide)

• To calculate the combined uncertainty of quantities that are multiplied or divided, the uncertainties must be divided by the mean values.• For example, suppose that now K = F x G x H where F, G, H are

measured values and their uncertainties are ΔF, ΔG, and ΔH where Δ means “the uncertainty of” … then the uncertainty of K, is…

Example

Percent Difference• If two lab groups measure two different values for an

experimental quantity, you may be interested in how the values compare to each other.

• A comparison of values is often expressed as a percent difference.

• You can calculate the percent difference as follows:

Percent Error• If you were wanting to check the accuracy of your

measurements (compare your result to an actual, true value) you would calculate the percent error of your data.

• To calculate the percent error you would use the following formula:

Graphs• Graphs are an excellent way to present or analyze data.• Here are the guidelines that you should follow when

creating a graph:• Each axis should be labeled with a variable that is plotted with its

correct units• Each axis should include a reasonable number of labeled tick

marks at even intervals • Graphs should be labeled with a meaningful title or caption.

Variables• When you graph data, you MUST plot an independent

variable versus a dependent variable.• An independent variable is a variable that stands alone

and is not changed by other variables you are trying to measure. This variable is always plotted on the x-axis of a graph.

• A dependent variable is something that depends on other variables. This variable is always plotted on the y-axis of a graph

Graphical Data• A useful way to analyze data is to determine whether it

corresponds in a certain mathematical model (equation).• There are three different types of mathematical models

that you should be familiar with.• Linear Functions• Quadratic Functions• Exponential Functions

Linear Functions• A linear function follows the general

equation: y=mx+b• m is the slope• b is the y-intercept

• Two variables that have a linear relationship means the following things:• As x increases, y increases.• As x increases, y decreases.

• When data is graphed and has a linear function there is no need to linearize the data.

Quadratic Equations• Quadratics follow the general form

y=ax2+bx+c.• Where a, b, c are arbitrary numbers

• If two variables have a quadratic (parabolic) relationship then they have one of the following relationships.• As x increases, y2 increases proportionally• As x2 increases, y increases proportionally.

• If data is graphed in this way then they must be linearized in order for you to evaluate better without a calculator or further mathematical analysis (calculus).

Exponential Functions• The general equation for an

exponential function is y=Aebx

• Where A, b are arbitrary constants

• This type of equation is not used as commonly in this course but it is still useful to know its general form.

Linearizing Data• If your data is not graphed in a way that you can

manipulate and utilize without the use of calculus you must linearize it, in other words … change how it is graphed.

Example• KE=1/2mv2