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 p3

British Units

•See http://en.wikipedia.org/wiki/British_units

•Pounds (force) versus Pounds (mass)

Metrication Dates

Image source: Wikipedia

User:Canuckguy

license:Public Domain

1 inch = 2.54 cm (exactly)

1 mi = 1.6 km

1 yd = .91 m

1 kg = 2.2 lb (equivalent!)

1 N = .225 lb

1 mph = .447 m/s

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 +Ay 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