AP Physics The Right Stuff

Units and ProcessLinda Summitt

Systems of Units, Trigonometry & Vectors

Physical Units Mechanics is the branch of

physics in which the basic physical units are developed. The logical sequence is from the description of motion to the causes of motion (forces and torques) and then to the action of forces and torques. The basic mechanical units are those of

Mass Length Time All mechanical quantities can be

expressed in terms of these three quantities. The standard units are the Systeme Internationale or SI units. The primary SI units for mechanics are the kilogram (mass), the meter (length) and the second (time). However if the units for these quantities in any consistent set of units are denoted by M, L, and T, then the scheme of mechanical relationships can be sketched out.

Dimensional Analysis Having the same units on both sides of an

equation does not guarantee that the equation is correct, but having different units on the two sides of an equation certainly guarantees that it is wrong! So it is good practice to reconcile units in problem solving as one check on the consistency of the work. Units obey the same algebraic rules as numbers, so they can serve as one diagnostic tool to check your problem solutions.

For example, in the solution for distance in constant acceleration motion, the distance is set equal to an expression involving combinations of distance, time, velocity and acceleration. But the combination of the units in each of the terms must yield just the unit of distance, since the left hand side of the equation has the dimension of distance.

Combinations of units pervade all of physics,

and doing some analysis of the units is common practice. For example, in the case of centripetal force, it is not immediately evident that the quantity on the right has the dimensions of force, but it must. Checking it out:

2221

00 2 statsvmyymsm

sm

r

vmF

2

Science Process & Metrics

Linda Summitt

State Objective

To develop basic laboratory skills emphasizing safety as well as problem solving

Teaching Objectives

Explain the importance of a standardized system of measure

List the SI prefixes and their numerical equivalents

Write numbers in standard and scientific notation

Convert from one metric system to another

Convert between the metric and English units given conversion factors

What is That?

Each student will be given an odd object in a bag or a box and five minutes to write a description of the object.

They will trade the description to another student who will draw the object from their description.

The partners will then be allowed to discuss description, objects, and drawings verbally.

Focus

Have you ever tried to describe something to someone who has never seen the thing you are trying to describe? The importance of communication skills and a common knowledge becomes apparent very quickly. Scientist have developed a standard system for communicating data in order to enhance global scientific communication. Learning to use the international system is becoming more important as our society adopts more metric measurements.

See the space program mars project.

How Can Science Help?

Science is the systematized, orderly, organized acquisition of information, knowledge and general truths and laws.

Much of this organization leads to naming and communication of knowledge.

Science

Scientific study or science process enables us to pose questions, investigate natural phenomena and solve problems.

The skills learned in science help improve daily problem solving abilities.

Science and Technology Pure science is the acquisition

of knowledge for the sake of the knowledge itself. Ex. Collecting information about baseball statistics

Applied science is the practical application of knowledge and is also known as technology ex. Computers

Limitations in ScienceLimitations are barriers

that prevent science from advancing.

Materials, skills, equipment, interpretations, prejudices, etc.

Scientists Scientists must be curious, observant,

organized, and willing to change. Science is full of uncertainties and exceptions. Through communication and collaboration scientists often find that there are different explanations for the same phenomena and different method of solving problems.

Scientific Method Inductive Method Define the problem Gather information Form a Hypothesis Design an

experiment Carry out the

experiment and collect data

Analyze the data Draw conclusions Publish

Deductive Method Define the problem Form a Hypothesis Gather information Design an

experiment Carry out the

experiment and collect data

Analyze the data Draw conclusions Publish

Becoming A Good Problem Solver Understanding

and using science process will help you to become a better problem solver in many ways!

Steps to Good Problem Solving

1. Understand the problem: read and reread, decide what is unknown or asked for, list all the information and laws needed, Break the work up into smaller problems when necessary

2. Analyze the data: check for trends and patterns

3. Make a sketch and note any additional information and think about the big picture (diagrams help with signs and principles)

4. Solve the problem (calculus and algebra are applied after substitution)

5. Check the problem does it make sense (Check dimensions)

Physics

Physics studies the fundamental laws of nature and motion of matter.

The technology developed through physics overlaps into biological and chemical fields as microcircuits, high-speed computers and imaging are used in medical fields and analytical fields.

Science Measurement Measurement has been important ever since man settled from his

nomadic lifestyle and started using building materials; occupying land and trading with his neighbors. As society has become more technologically orientated much higher accuracies of measurement are required in an increasingly diverse set of fields, from micro-electronics to interplanetary ranging.

Body PartsOne of the oldest units of length measurement used in the ancient world was the 'cubit' which was the length of the arm from the tip of the finger to the elbow. This could then be subdivided into shorter units like the foot, hand (which at 4 inches is still used today for expressing the height of horses) or finger, or added together to make longer units like the stride. The cubit could vary considerably due to the different sizes of people. As early as the middle of the tenth century it is believed that the Saxon king Edgar kept a "yardstick" at Winchester as the official standard of measurement. A traditional tale tells the story of Henry I (1100-1135) who decreed that the yard should be "the distance from the tip of the King's nose to the end of his outstretched thumb".

In November 1900 Queen Victoria handed Bushy House to the Commission of Works for the establishment of a national standards laboratory.

NPL was officially opened by the Prince of Wales on 19th March 1902 giving NPL's mission as:

"To bring scientific knowledge to bear practically upon our everyday industrial and commercial life, to break down the barrier between theory and practice and to affect a union between science and commerce"

Measurement

Measurements, graphs, and models help to bridge the gap between abstract concepts and the concrete. The scales that are chosen in graphing measurements or constructing models allows comparisons to be made and helps to determine the effectiveness of the graph or model

Using the Body to Measure

The “hand”, The “cubit”, The “span If the book is 2 hands wide, then it

is: 2 hands x (1cubit /5 hands) = .4 cubits wide because 1 cubit = 5 hands

The books and desks are the same but hands vary.

Standard Units of Measurement What do the following terms mean?

DECade CENTury MILLenium Why is it important to have standard

units of measurement?

In order to communicate with others standard units of measurement are necessary

Measurements Mass : balance Weight: a scale Length: a meter stick Volume: a graduated cylinder or

meter stick Temperature: thermometer Time: a stopwatch Current: ammeter Voltage: voltmeter

Measurements

Mass : balance Platinum iridium alloy 100kg/human 1000 kg/ horse .1 kg/ frog 1-1.5 kg/ text book

Weight: a scale Length: a meter stick length of light in vacuum

1/299 792 458 second 91m football field approx length finger tips to mid back 1 m

Volume: a graduated cylinder or meter stick Time: a stopwatch the second is 9.19 x 109 the

period of 133Cs time between normal heart beat is about 0.8s

Reviewing Metric and Scientific Notation

Metric Kilo Hecta Deka Base

(meter,liter,gram)

Centi Deci Milli

Micro

Englishtrillionbillionmillionthousandhundred

103

102

101.110-2

10-3

10-6

10 12

10 9

10 6

10 3

10 2

SI prefixes prefix symbol factor exponential giga G 1,000,000,000 10E9 mega M 1,000,000 10E6 kilo k 1,000 10E3 hecto h 100 10E2 deka Dam 10 !0 E-2 deci d .1 10E1 centi c .01 10E-2 milli m .001 10E-3 micro u .000 001 10E-6 nano n .000 000 001 10E-9 pico p .000 000 000 001 10E-12

Metric Unit Activity Prepare a Mind Map of basic metric units. Begin your tree with four types of measurement:

length, volume, mass, temperature. Place the metric bas unit under each name with an

object that is approximately one base unit. Place the metric prefixes in order under each base

unit. Title: Metric Units Prefixes: kilo, hecto, deka, base unit, deci, centi, milli Write a paragraph explaining why we need the SI

measurements.

M etric un its

K HDMDCM

M eters

Length

K H D LD C M

Liters

V olum e

K HDG DCM

G ram s

M ass

K H DC DCM

C els iusK elv in

T em perature

B asic m etric units

Metric Measurements Activity

From

To K H D Meter

LiterGram

d c m

50 Km

m 5 0

50,000 m

2105 m

km 2

1 0 5 2.105 km

.045 dl

ml .0 4 5 45.Dl

989 mg

Kg 9 8 9 .000989 kg

22 m mm

2 2 22,000 mm

Moving the Decimal

Uncertainty When a measurement is given, it is also

good to state the estimated degree of certainty. If the meter stick measures millimeters, the measurement can be estimated to +/- 0.0005 meters or +/- 0.5 millimeters.

The percent uncertainty is the ratio of the uncertainty to the measured value.

0.0005.%17%100

29.0

0005.0x

Uncertainty cont’ If a value is 53.5 cm the uncertainty

is implied +/- 0.1 cm. When using uncertain measurements

in calculations (like radius, area, etc.) one can compare the stated value with the “extreme” value. Ex A square has sides 2.5 cm long with

uncertainty 0.1cm. Uncertainty of the area equal (2.6 cm2 - 2.5 cm2).

Propagation of Error

Each measurement has an error associated with it determined by the precision of the instrument.

These errors introduce small errors into the calculations.

Ignoring Significant digits introduces even more error because it creates a false sense of accuracy that does not exist.

Propagation of Error Addition & Subtraction

What is the perimeter of a triangle with the measurements:

22.4 0.2cm; 45.35 0.02cm; 4.45 0.02cm

adding 0.02 to all the average values gives the greatest

possible number

subtracting 0.02 gives the lowes

t possible number

22.4 22.2 22.6

45.35 45.33 45.37 72.2 - 0.24 = 71.96

4.45 4.43 4.47 72.2 + 0.24 = 72.24

72.2 71.96 72.44

0.02 + 0.02 + 0.2 = 0.24 error

Propagation of Error Multiplication and Division

Calculate the volume of a block with the following length, width, and height:

10.3 0.2cm; 5.45 0.02cm; 15.2 0.2cm

10.3 10.1 10.5

5.45 5.43 5.47

15.2

3 3 3

15.0 15.4

853.252 822.645 884.499

853 823 884

subtracting the lowest value from the highest and assuming measurements

above and below the

cm cm cm

3

3

average are equally likely

884-823=61/2=30.5 error so

volume = 853 30.5 watching significant figures

cm

cm

Errors in Measure

Errors in measurements can be calculated by comparing the observed value to the true or expected value

If you boil water at 95 F and it should boil at 100 F the absolute error = 100 - 95 = 5

The percent error = 5 / 100 x 100% = 5% Percent error = abs. err. / true value x

100%

Significant figures The last digit in a measurement: usually

an estimate When adding or subtracting

measurements, round the answer to the same decimal place as the measurement with the fewest decimal points

When multiplying or dividing, the result should have the same number of significant figures as the factor with the fewest

Significant Digits Nonzero digits are always significant.

Leading zeros that appear at the start of a number are never significant because they act only to fix the position of the decimal point in a number less than 1.

Confined zeros that appear between nonzero numbers are always significant.

Trailing zeros at the end of a number are significant only if the number contains a decimal point or contains an over-bar.

Significant figures

127: 3 320: 2 18000. : 5 0.03000: 4 It is easiest to convert to scientific

notation 1st and then disregard any zeros that are not place holders

Significant figures are important because when adding, subtracting, multiplying, or dividing measurements the final answer can not be more accurate than the least accurate measurement

Scientific Notation

The measurement is expressed as the product of 2 numbers, the numerical value expressed as a number between 0 & 10 and a power of 10

Scientific Notation 520 = 5.2 x 102

0.0037 = 3.7 x 10-3

0.223 = 5301 = 53.756 = 0.0564 = 102.36 =

Scientific Notation

520 = 5.2 x 102

0.0037 = 3.7 x 10-3

0.223 = 5301 = 53.756 = 0.0564 = 102.36 =

2

2

1

3

1

100236.1

1064.5

103756.5

10301.5

1023.2

x

x

x

x

x

Scientific Notation Writing

numbers as a product of a number between 1 thru 9 and powers of ten

When multiplying or dividing in scientific notation add or subtract the exponents respectively.

5

2

1

734800 7.348 10

0.03690 3.689 10

10.52 1.052 10

3 4

7

7

6 3

3

3 2

1.2 10 3.5 10

1.2 3.5 10

4.2 10

1.35 10 5.4 10

1.35 5.4 10

0.25 10 2.5 10

Adding and Subtracting

When adding or subtracting numbers in scientific notation the numbers must have the same power.

4.1 x 10-6 kg - 3.0 x 10-7 kg = 4.02 x 106 m + 1.89 x 102 m =

6

222

666

10020189.4

1089.402011089.11040200

108.31030.0101.4

x

xxx

xxx

Scientific Notation Exponents must

agree to add or subtract in scientific notation.3 2

3 3

3

3 0

1.782 10 5.39 10

1.782 10 0.539 10

2.328 10

8.7 10 4.51 10

8700 4.51 8695.49

8700 rounding makes this insignificant!!

4 2

3 3

3

3

1.945 10 7.89 10

19.45 10 0.789 10

18.661 10

18.66 10 though it is a small difference

it is significant

(to the tens unit of the dimension)

Multiplying & Dividing

When multiplying or dividing in scientific notation, the number itself can simply be multiplied or divided. If the operation is multiplication the powers should be added. If the operation is division, the powers should be subtracted.

(4 x 103 kg) (5 x 1011 m) = 8 x 106m3 / 2 x 10-3m2 =

14

9

20 10

4 10

x kg m

x m

Converting Metric to Metric Write the measurement and units as a

quotient Find a conversion factor with the same

units Multiply the measurement by the

conversion factor being sure that the like measurement cancels dekag

g

dekag

mg

gmg 5.1

10

1

1000

115000

Converting metric to metric 2300ml = _____ L 27056 ml = _____ dekaliters 0.0683m = ______ mm

Converting metric to metric 2300ml = _____ L

27056 ml = _____ dekaliters

0.0683m = ______ mm

2.3

2.7056

68.3

L

dekaL

mm

Conversion Factors 2.54 cm = 1 in 1 m = 39.37 in 28.35 g = 1 oz 454 g = 1 lb 1 kg = 2.2 lb 0.946 L = 1 qt 1 L = 1.06 qt 1000cm3 = 1L H2O at 40C

1 mL = 1 cc = 1 cm3 = 1 g H2O at 40C K = 0C + 273 0C = 5/9 (0F – 320) 0F = 9/5 0C + 32

Converting Metric to English

Converting between English and metric is very similar to converting metric to metric

Write the measurement as a quotient Multiply the measurement by the proper

conversion factor(s) being sure the units cancel

Cancel units, and complete math

Converting between English and metric 1 in = 2.54cm 1 lb = 0.4536 kg 20 kg = ________lb 20kg x 1 lb = kg

1 0.4536kg .0563m = ________ in

Converting between English and metric 1 in = 2.54cm 1 lb = 0.4536 kg 20 kg = ________lb

.0563m = ________ in

20 144.1

1 0.4536

kg lblb

kg

0.0563 100 12.21

1 1 2.54

m cm inin

m cm

Accuracy, Precision & Parallax

Accuracy shows the difference between the “true value” and he average of the measurements.

Precision shows how much a measurement differs from the average measurement.

Parallax- displacement of a reading due to different viewing angle

Watch out for these when making measurements

accurate

Accuracy in measurement

The rule below is marked off at every 1/8th of an inch (.125 inch). When measuring with this ruler one could estimate +or- to the 16th of an inch with this ruler.

How long would You estimate the bar to be?

94

16in

Are the numbers 8.79, 8.77, 8.78, 8.78, 8.79, & 8.8 accurate or precise if they are experimental values for the acceleration of gravity which is

9.81 m/s2? a. Accurate b. Precisec. Neitherd. Both

Order of Magnitude Estimations Calculations that are made using

estimated data can be very useful. The calculations are also guesses and

are called order of magnitude estimates.

Many times you can make an estimate of tens, hundreds, thousands etc. this is the order of magnitude.

Order of Magnitude

Estimate the gallons of gas used in the US per year.

300 mill people family of 4 1 car per family 3 x 108 people / 4people/car x 10,000 mi/year = 7.5 x 1011 mi/year/car

If each car gets 20 mi per gal 7.5 x 1011

mi/year/ 20 mi/gal = 3.8 x 1010 gal/year

Matter and Measurement

Matter is anything that has mass and volume.

Mass is a specific quantity of a substance. Volume is the space the substance occupies

(l x w x h). Atoms are the particles that make up

matter; the characteristics or properties of the atoms in a substance determine the properties of the substance and many of theses properties can be measured using both simple and complicated techniques

Matter & Atoms

Early studies in Alchemy led to discovery of the “non-sliceable” atom by Greeks Leucippus and Democritus.

By the 1900’s the nucleus had been discovered following the model of the solar system.

By the 1930’s atomic number had been defined as the number of protons and was used to identify elements and the atomic mass as the average number of protons and neutrons.

Electrons & Quarks It was determined that there were

smaller particles known as electrons that were negative charge carriers spinning around the nucleus.

It was discovered that smaller particles known as quarks (up, charmed, and top :+2/3 and down, strange and bottom :-1/3) combine to make the protons (+2/3 +2/3 -1/3) and neutrons (-1/3 1/3 +2/3)

Derived Units Derived units are

measurements that are a combination of two or more basic units.

Derived units are defined by a mathematical equation. Ex: Density = mass/ volume

What is Density? Density is how much mass there

is in a specific volume.

Density is the number of particles / amount of space.

Differences in density are related to differences in atomic mass

m mD orV V

Density-A Property of Matter

Density = the mass/ the volume.

Density can be determined by graphing the mass of several different volumes of a substance and finding the slope of the line.

Graphing Density If the mass of

several volumes was collected and plotted to give the following graph, the density could be determined as shown.

60g 50g 40g 30g 20g 10g 0g 5mL 10mL

15mLSlope = y2-y1 / x2-x1

= 30 – 10 / 10 – 5

= 20/5 = 4 g/mL

2

1

Graphs and Tables

Graphs and tables are used to organize information so that trends, changes and parts of a whole can be easily detected and predictions can be made.

Both should be labeled clearly. Their are many different kinds of graphs

and tables. Bar graphs, pie charts, line graphs, curve

graphs, spread sheets, comparison tables ETC.

GraphsGraphs are representations of complex phenomena. They are a type of model and can be used to visualize relationships between variables.

Graphs

All graphs should have the following Title Choose units for the axis Choose the scale 10s 100s etc Label the axis Plot data Analyze data

Analysis-Using your DATA

Look for patterns, relationships, cause & effect, and supporting or contradicting data.

Prepare charts, graphs, or database.

Using Equations! Read Carefully!!! Draw Pictures! Write expressions for knowns

and unknowns Label drawings Write an equation Solve Check to be sure it makes

sense and answers the question

Molar Mass & Avagadro’s #

12C has exactly 6 protons and 6 neutrons. 12 g of 12C has 6.022 x1023 atoms in it.

This is defined as a mole! 1 mole of any substance has 6.022 x 1023

particles and has a mass equal to the atomic mass of the substance .

matom =molar mass/Navagadro

How much mass would 2 moles of iron have? 55.85

2 111.701

gFem molFe g

molFe

Dimensional Analysis Measurements have magnitude and dimensions or

a number and units. These cannot be separated!!! Quantities can be added and subtracted only if

they are the same units or dimensions! Terms on either side of an equation must have the

same units on either side of the equation to be valid.

Ex. l=m v=l/t =m/s a=l/t2=m/s2

Show

tl

tl

ttl

tl

tl

20

0

Common Geometry Formulas

2

2

:

4

Rectangle :

2 2

:

2

1: 2

1: 2

Square A s

P s

A lw

P l w

Circle A r

C r

Triangle A bh

Trapezoid A h a b

a

bb

r

l

ss

w

h

h

ss

sr

h

w

hl

r

hr

3

2

3

2

:

:

Rectangular Solid :

4: 3

1: 3

Cube V s

Cylinder V r h

V lwh

Sphere V r

Cone V r h

Mathematical Notations

sum

xxinchangex

toequalelyapproximat

alityproportion

if )(..

..

Conclusions

The analysis is made more understandable or summarized.

Theories are formed from supporting evidence.

New focus or extension should be suggested.

Summary

The Scientific Method of Problem Solving Can help solve all types of problems and enhance decision making abilities. If the knowledge gained through science is shared systematically all of society can benefit from the knowledge and technology.

Vector Analysis

Pythagorean’s Theorem C2=A2+B2

A

B

C

Trigonometric Functions

Sin = opp / hyp Cos = adj / hyp Tan = opp / adj

opp

adj

hyp

Scalar and Vector

Scalar: quantities with magnitude but no direction 60 miles / hr

Vector: quantities with magnitude and direction 60 miles / hr @ 600 N

Trigonometric Functions Sin = opp / hyp Cos = adj / hyp Tan = opp / adj

Pythagoreans C2 = A2 + B2

opp

adj

hypC

B

A

Scalar and Vector

Scalar: quantities with magnitude but no direction 60 miles / hr

Vector: quantities with magnitude and direction 60 miles / hr @ 600 N

Position Vectors Any point has a set of coordinates

that defines its position. A line can be drawn from the origin to the coordinates. This line is a position vector and is written in bold or as a lower case r with an arrow above it.

3

2

P (2,3)

(x,y)r

Displacement

Displacement is a change in position over time. It is a vector quantity.

Vectors that have the same direction are parallel.

Vectors that have the same magnitude and direction are equal even if they start from different points.

Planar-polar Coordinates

Point P is distance r from the origin and angle q from the reference line

+ is ccw -is cw

r

Vectors, resultants and equilibrants

Vectors are added graphically by placing the tail of one vector at the head of the other vector.

The sum of the vectors is known as the RESULTANT. The resultant is drawn from the tail of the first vector to the head of the last vector.

The EQUILIBRANT is a force that can be applied to a non-zero net force to balance that force.

When the net forces acting on a point are zero the forces are at equilibrium.

Vector Addition in Two Dimensions

Vectors are added by placing the tail of one vector at the head of the other vector.

Graph paper and a protractor may be used to resolve vectors.

Addition of several vectors

Three or more vectors can be added in the same way. The direction and length of the vector must be to scale and must not be changed.

Vector Quantities are Independent

Perpendicular vector quantities are independent.

Ex. the velocity north or south does not change the velocity east or west

If a boat is traveling at 9.4 m/s at 32 N and it crosses a river 80 meters wide and the boat’s velocity is 8 m/s east, then it takes 10 seconds for the boat to cross the river. The boat will drift north 50 meters during that time.

Pythagorean’s Theorem and resultant vectors

The resultant vector of two perpendicular vectors is the hypotenuse of a right triangle, therefore, Pythagorean theorem can be used to determine the resultant

If a 110 N North force and a 55 N East force act on an object and the forces are applied at right angles, then the resultant force is equal to the square root of 1102 and 552 or 123 N.

Trigonometric functions and resultant angles

The angle of the resultant vector can be found by using one of the trigonometric functions such as: sin @ = Opposite side / Hypotenusecos @ = Adjacent side / Hypotenuse tan @ = Opposite side / Adjacent

side The resultant angle in the above

problem is found : 110/55 = tan @ 2.0 = tan @ so @ = 64 degrees

Resolve a vector into its horizontal and vertical components

A single vector can be broken down into its COMPONENTS . Any vector can be thought of as the resultant of two components.

Ex. The boat traveling at 9.4 m/s at 32 N can be RESOLVED into two components: 10 m/s east and 5 m/s N

VECTOR RESOLUTION is the process of finding the magnitude of the components in each direction.

Adding Vectors at Angles through vector resolution.

When adding vectors at angles: 1st resolve each vector into its components, 2nd add all of the vertical components, 3rd add all of the horizontal components

The resultant vector is the resultant of the sum of the vertical vectors and the sum of the horizontal vectors.

Ex. A force of 12N at 10 N and a force of 14N at 310 0 can be broken down to 11.8 E and 2.1N and 9.0 E and 10.7 S. The horizontal sum is 20.8 E and the vertical sum is 8.6 S . The resultant using Pythagorean Theorem is 22.5 N and using the laws of trigonometry the angle is 22.5 SE

Subtraction of Vectors

To subtract vector B from vector A, reverse the direction of vector B and add it to vector A

Vector Products

Two vectors originating at the same point can be multiplied together to get the vector product.

A x B = AB cos t where A & B are the magnitude of A and B

Ex A is 4.0 00 N of E and B is 5.0 at 770 N of E then the vector product is 4.0 x 5.0 x cos 770

Problem Solving

The key to successful problem solving is to ask the right question !

Go back to the simplest thing that you know.

Work forward from that simple knowledge

Summary

Charts and tables will always be available to help you convert measurements in the metric system. Students should know how to use the charts and tables to become familiar with the metric system