units, physical quantities and vectors - physics at penn...
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Phys211C1 p1
Units, Physical Quantities and Vectors
What is Physics?
Natural Philosophy
science of matter and energy
fundamental principles of engineering and technology
an experimental science: theory↔experiment
simplified (idealized) models
range of validity
Terminology Alert: A Theory
•is not an unproven concept
•is an explanation of phenomena
•is based on observation and accepted fundamental principles
size
spee
d
Classical
Mechanic
s
Quantum
Mechanics
Relativistic
Mechanics
Quantum
Field
Theory
Phys211C1 p2
Quantifying predictions and observations
physical quantities: numbers used to describe physical phenomena
• height, weight e.g.
• operational definition: a quantity defined in terms of how it is measured
standard units: International System (SI aka Metric)
• defined units established in terms of a physical quantity
• derived units established as algebraic combinations of other units
Quantity Unit
Length
Time
Mass
Temperature
Electric Current
meter (m)
second (s)
kilogram (kg)
kelvin (K)
ampere (A)
Phys211C1 p4
Prefix Abbre-
viation
Power
of Ten
femto
pico
nano
micro
milli
centi
kilo
mega
giga
f
p
n
µ
m
c
k
M
G
10-15
10-12
10-9
10-6
10-3
10-2
103
106
109
1/1,000,000,000,000,000
1/1,000,000,000,000
1/1,000,000,000
1/1,000,000
1/1,000
1/100
1,000
1,000,000
1,000,000,000
Common prefixes(know these!)
Phys211C1 p5
Dimensional Analysis: consistency of units
Algebraic equations must always be dimensionally consistent.
You can’t add apples and oranges!
Carry units with numbers through calculation
provides check on calculations
provides correct units for answer
also see google calculator!
( )s5s
m2m10
time speed distance
/
/=
×== vtd
Phys211C1 p6
cminch
cm
ft
inchesftft
inch
cmcminch
48.30540.212
11
1540.2540.21
=
=
=→=
converting units
treat units as algebraic quantities
multiplying or dividing a quantity by 1 does not affect its value
Problem Solving Strategy (ISEE)
Identify relevant concepts
Set up the problem
Execute the solution
Evaluate your answer
Phys211C1 p7
Units Conversion Examples
Example The world speed record, set in 1997 is 763.0 mi/h. Express this speed in m/s
Example how man cubic inches are there in a 1.100 liter engine?
Phys211C1 p8
Significant Figures and Uncertainty
Every measurement of a physical quantity involves some error
error = uncertainty, not “mistake”
random error
averages out
small random error → precise measurement
systematic error
does not average out
small systematic error → accurate measurement
0
5
10
15
20
number 0 0 0 0 0 0 0 0 0 4 8 15 13 5 3 0 0
7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55
0
5
10
15
20
number 0 0 0 0 0 0 2 6 10 18 11 1 0 0 0 0 0
7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55
0
5
10
15
number 0 0 0 0 0 0 1 0 4 4 12 10 6 4 3 2 1
7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55
less accurate less precise
Precise and accurate
Precise
Accurate
Phys211C1 p9
Indicating the accuracy of a number: x ± ∆x or x± δx sometimes x(δx)
example: 20.3±.5 cm or 20.3(.5)cm. A measured length of 20.3 cm ± .5 cm means that the actual length is expected to lie between 19.8 cm and 20.8 cm.
nominal value: the indicated result of the measurement, 20.3 cm in the example
numerical uncertainty: how much the “actual value” might be expected to differ from the nominal value, .5 cm in the example
sometimes called the numerical error
fractional uncertainty: the fraction of the nominal value corresponding to the numerical uncertainty
percentage uncertainty: the percentage of the nominal value corresponding to the numerical uncertainty
025.cm3.20
cm5.==
∆
x
x
%5.2cm3.20
%5.2%100cm3.20
cm5.%100
±
=×=×∆
x
x
Phys211C1 p10
Significant Figures: common way of implicitly indicating uncertainty
number is only expressed using meaningful digits (sig. figs.)
last digit (the least significant digit = lsd) is uncertain3 one digit
3.0 two digits (two significant figures = 2 sig. figs.)
3.00 three digits,etc. (300 how many digits?)
Combining numbers with significant digits
Addition and Subtraction: least significant digit determined by decimal places (result is rounded)
.57 + .3 = .87 =.9 11.2 - 17.63 = −6.43 = −6.4
Multiplication and Division: number of significant figures is the number of sig. figs. of the factor with the fewest sig. figs.
1.3x7.24 = 9.412 = 9.4 17.5/.3794 = 46.12546 = 46.1
Integer factors and geometric factors (such as π) have infinite precision
π x 3.762 = 44.4145803 = 44.4
Phys211C1 p11
Estimates and Order of magnitude calculations
an order of magnitude is a (rounded) 1 sig fig calculation, whose answer is expressed as the nearest power of 10.
Estimates should be done “in your head”
check against calculator mistakes!
Comparing Two numbers: Percent Difference
%100%
%100%
×−
=
×−
=
A
BAdifference
sometimes
A
BAdifference
Phys211C1 p12
Vectors
Scalars: a physical quantity described by a single number
Vector: a physical quantity which has a magnitude (size) and
direction.
•Examples: velocity, acceleration, force, displacement.
•A vector quantity is indicated by bold face and/or an arrow.
notation
•The magnitude of a vector is the “length” or size (in appropriate units).
The magnitude of a vector is always positive.
•The negative of a vector is a vector of the same magnitude put opposite direction (i.e. antiparallel)
etcoraoror aarr
a
) of magnitude (the aarr
a=
Phys211C1 p13
Combining scalars and vectors
scalars and vectors cannot be added or subtracted.
the product of a vector by a scalar is a vector
x = c a x = |c| a (note combination of units)
if c is positive, x is parallel to a
if c is negative, x is antiparallel to a
Pottsville is about 5 miles north
Frackville is about 3 times further,in the same direction
Hamburg is about 3 times further, in the opposite direction
Phys211C1 p14
Vector addition
most easily visualized in terms of displacements
Let X = A + B + C ···
graphical addition: place A and B tip to tail;
X is drawn from the tail of the first to the tip of the last
A + B = B + A
A
B
X
A
B
X
Phys211C1 p15
Vector Addition: Graphical Method of R = A + B
•Shift B parallel to itself until its tail is at the head of A, retaining its original length and direction.
•Draw R (the resultant) from the tail of A to the head of B.
A
B
+ = A
B
=
R
the order of addition of several vectors does not matter
A
C
B
D
A
B
CD
D
BA
C
Phys211C1 p16
Vector Subtraction: the negative of a vector points in the opposite direction, but retains its size (magnitude)
• A− B = A +( −ΒΒΒΒ)
A
B
− = A
−B
+
R
= A
−B
Phys211C1 p17
Resolving a Vector (2-d)
replacing a vector with two or more (mutually perpendicular) vectors => components
directions of components determined by coordinates or geometry.
A
Ay
Ax
A = Ax + Ay
Ax = x-component
Ay = y-component θ
θθθ sincostan
22
AAAAA
A
AAA
yx
x
y
yx
===
+=
A
Ay
Ax
θBe careful in 3rd , 4th quadrants when using inverse
trig functions to find θ.
Component directions do not have to be horizontal-vertical!
Phys211C1 p18
Vector Addition by components
R = A + B + C
Resolve vectors into components(Ax, Ay etc. )
Add like components
Ax + Bx + Cx = Rx
Ay + By + Cy = Ry
The magnitude and direction of the resultant R can be determined from its components.
in general R ≠ A + B + C
Phys211C1 p19
Example: Add the three displacements:
72.4 m, 32.0° east of north
57.3 m, 36.0° south of west
72.4 m, straight south
Example: A cross-country skier skies 1.00 km north and then 2.00 km east on a horizontal snowfield. How far and in what direction is she from her starting point?
Phys211C1 p20
Unit Vectors
a unit vector is a vector with magnitude equal to 1 (unit-less and hence dimensionless)
in the Cartesian coordinates:
Right Hand Rule for relative directions: thumb, pointer, middle for
Express any vector in terms of its components:
A=Ax i +A y j+A z k
i unit vector in the +x direction ( i hat )
j unit vector in the + y direction ( j hat )
k unit vector in the +z direction ( k hat )
i , j , k
Phys211C1 p21
Products of vectors (how to multiply a vector by a vector)
Scalar Product (aka the Dot Product)
φ is the angle between the vectors
A.B = Ax Bx +Ay By +Az Bz = B.A
= B cos φ A is the portion of B along A times the magnitude of A
= A cos φ B is the portion of A along B times the magnitude of B
ABBABArrrrrr
⋅===⋅ φφ coscosAB
1800 ≤≤ φ
B A
φ
B cosφ
note: the dot product between perpendicular vectors is zero.
0ˆˆ0ˆˆ0ˆˆ
1ˆˆ1ˆˆ1ˆˆ
=⋅=⋅=⋅
=⋅=⋅=⋅
ikkjji
kkjjii
Phys211C1 p22
Example: Determine the components of, and the scalar product between
A = (4.00m, 53.0°) and B = (5.00m, 130.0°)
Phys211C1 p23
Products of vectors (how to multiply a vector by a vector)
Vector Product (aka the Cross Product) 3-D always!
φis the angle between the vectors
Right hand rule: A×B = C
A – thumb
B – pointer
C – middle
Cartesian Unit vectors
φsinABC =×−=×= ABBACrrrrr
1800 ≤≤ φ
jikikjkji
kkjjii
ˆˆˆˆˆˆˆˆˆ
0ˆˆ0ˆˆ0ˆˆ
=×=×=×
=×=×=×
Phys211C1 p24
Write vectors in terms of components to calculate cross product
kjiBA
kkjkik
kjjjij
kijiii
kjikjiBA
ˆ)(ˆ)(ˆ)(
ˆˆˆˆˆˆ
ˆˆˆˆˆˆ
ˆˆˆˆˆˆ
)ˆˆˆ()ˆˆˆ(
xyyxzxxzyzzy
zzyzxz
zyyyxy
zxyxxx
zyxzyx
BABABABABABA
BABABA
BABABA
BABABA
BBBAAA
−+−+−=×
×+×+×+
×+×+×+
×+×+×=
++×++=×
rr
rr
C = AB sin φ
= B sin φ A is the part of B perpendicular A times A
= A sin φ B is the part of A perpendicular B times B
φsinABC =×−=×= ABBACrrrrr
B A
φ
B sinφ
Phys211C1 p25
Example: A is along the x-axis with a magnitude of 6.00 units, B is in the x-y plane, 30° from the x-axis with a magnitude of 4.00 units. Calculate the cross product of the two
vectors.
Phys211C1 p26
Vectors computers and calculators:
TI 89 representations of
[5,1,3] displays as [5. 1. 3.]
[2,7,-1] displays as [2. 7. -1.]
[5,1,3]+[2,7,-1] produces [7. 8. 2.]
dotP( [5,1,3],[2,7,-1]) produces 14.
CrossP( [5,1,3],[2,7,-1]) produces [-22. 11. 33.]
Vectors in 2-D: polar coordinates, rectangular coordinates and complex numbers
(watch degrees vs. radians mode in calculator, mode->complex->rectangular or polar)
R=20 θ = 37 can be input as (20∠37) which produces 15.9729+12.0363 i (rect. mode)
input 15.9729+12.0363 i input becomes (20.0002∠36.9997) (polar mode)
WATCH FOR HOW ANGLES ARE SPECIFIED IN EACH PROBLEM
Other notions: <5,1,3> (POVRay), variables in arrays r[i] i=0,1,2 etc
Learn techniques, not just calculator shortcuts!
A=5 i +1 j+3 k and B=2 i+7 j+�1 k