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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