Introduction, Significant Figures and Uncertainty

8/6/2019 Introduction, Significant Figures and Uncertainty

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1

Why do Physics?

To understand nature through experimentalobservations and measurements

Establish limited number of fundamental laws,usually with mathematical expressions

Predict natures courseTheory and Experiment work hand-in-hand

Theory works generally under restricted conditions

Understanding laws enables us to control outcomesof natural processes: applied physics, engineering,biology, chemistry, geology, astronomy, etc.

Exp.{

Theory

{


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Fundamental Quantities and Their

Dimension Mechanics uses three fundamental quantities

Length [L]

Mass [M] Time [T]

Other physical quantities can be constructed

from these three

Introduction


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How do we convert quantities from

one unit to another?

Unit 1 = Unit 2Conversion factor X1 inch 2.54 cm

1 inch 0.0254 m

1 inch 2.54x10-5 km

1 ft 30.3 cm

1 ft 0.303 M

1 ft 3.03x10-4 km

1 hr 60 minutes

1 hr 3600 seconds

And many More Here.


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Examples for Unit Conversions

Ex: A silicon chip has an

area of 1.25in2. Express

this in cm2.

2

1.25 in !

22

cm06.8cm45.625.1 !v!

v!

2

22

in1

cm45.6in25.1

Ex 1.4: Where the posted speed limit is 65 miles per hour (mi/h or mph), whatis this speed (a) in meters per second (m/s) and (b) kilometers per hour (km/h)?

1 mi=

65 mi/h !

65 mi/h !

(a)

(b)

2

1.25 in v

What do we need to

know?

22.54 cm

1 i n

5280 ft 1609 m 1.609 km12 in1 ft

2.54 cm

1 in

1 m

100cm

65 mi 29.1 m/s1609 m

1 mi

1

1 h

1 h

3600 s

65 mi 104 km/h1.609 km1 mi

1

1 h


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Prefixes, expressions and their

meanings deci (d): 10-1

centi (c): 10-2

milli (m): 10-3

micro (Q): 10-6

nano (n): 10-9

pico (p): 10-12

femto (f): 10-15

atto (a): 10-18

deca (da): 101

hecto (h): 102

kilo (k): 103

mega (M): 106

giga (G): 109

tera (T): 1012

peta (P): 1015

exa (E): 1018


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Rules for Significant Figures

6

Read from the left and start counting sig figs when you

encounter the first non-zero digit

1. All non zero numbers aresignificant(meaning they count as sig

igs)

123 has three sig igs

123456 has six sig igs

2. Zeros located between non-zero digits aresignificant(they count)

5004 has our sig igs

602 has three sig igs

6000000000000002 has 16 sig igs!

3. Trailing zeros (those at the end) aresignificantonly i the numbercontains a decimal point; otherwise they are insignificant(they dont

count)

5.640 has our sig igs

120000. has six sig igs

120000 has two sig igs unless youre given additional in ormation inthe problem


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7

Rules for Significant Figures

4. Zeros to le t o the irst nonzero digit are insignificant(they dont count); they areonly placeholders!

0.000456 has three sig igs

0.052 has two sig igs

0.000000000000000000000000000000000052 also has two sig igs!


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Significant Figures Significant figures denote the precision of

the measured value

Significant figures: non-zero numbers or zeros that

are not place-holders

How many significant figures? 34, 34.2, 0.001, 34.001 34 has two significant digits

34.2 has 3

0.001 has one because the 0s before 1 are place holders

34.100 has 5, because the 0s after 1 indicates that the numbers

in these digits are indeed 0s.

When there are many 0s, use scientific notation:

31400000=3.14x107

0.00012=1.2x10-4


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9

Significant Figures

Operational rules: Addition or subtraction: Keep the smallest number

of decimal places in the result, independent of the

number of significant digits: 34.001+120.1=154.1

Multiplication or Division: Keep the smallest

significant figures in the result: 34.001x120.1 =

4083, because the smallest significant figures is 4.


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10

Uncertainties

Physical measurements have limitedprecision, no matter how carefully done,

due to:

Number of measurements with different results Quality of instruments (meter stick vs micro-meter)

Experience of the person doing measurements

Etc.

In many cases, uncertainties are more important

and difficult to estimate than the central (or mean)

values

Stat.{

{Syst.


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11

Uncertainties contd Estimated Uncertainty

Suppose a result of a measurement is expressed as

The estimated uncertainty is 0.1cm.

Percent Uncertainty: Simply the ratio of theuncertainty to the measured value multiplied by

100:

If uncertainties are not specified, it is assumed tobe one or two units of the last digit specified:

For length given as 5.2cm, the uncertainty is assumed to

be about 0.1cm

5.2 0.1cms

0.1100 2%

5.2v !


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Dimension and Dimensional Analysis

An extremely useful concept in solving physicalproblems

Good to write physical laws in mathematicalexpressions

No matter what units are used the base quantities arethe same Length(distance) is length whether meter or inch is used to

express the size: Usually denoted as [L]

The same is true forMass ([M])and Time ([T]) One can say Dimension of Length, Mass or Time

Dimensions are used as algebraic quantities: Can performtwo algebraic operations; multiplication or division


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Dimension and Dimensional Analysis

One can use dimensions only to check thevalidity of ones expression: Dimensionalanalysis Eg: Speed [v] = [L]/[T]=[L][T-1]

Distance (L) traveled by a car running at the speedV in time T

L = V*T = [L/T]*[T]=[L]

More general expression of dimensionalanalysis is using exponents: eg. [v]=[LnTm]=[L]{T-1] where n = 1 and m = -1


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Examples Show that the expression [v] = [at]is dimensionally correct

Speed: [v]=L/T Acceleration: [a]=L/T2

Thus, [at]= (L/T2)xT=LT(-2+1) =LT-1 =L/T= [v]

2m !

mn

vkra !Dimensionless

constantLength Speed

1 2L T

!

r

vvkra

221 !!

r va

n m !

Suppose the acceleration aof a circularly moving particle

with speed vand radius ris proportional to rnand vm. What

are nandm?

m

n LL

T

!

n m mL T

!m 2

2n !1! ! n 1


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1.3 The Role of Units in Problem Solving

DIMENSIONAL ANALYSIS

[L] = length [M] = mass [T] = time

2

21 vtx !

Is the ollowing equation dimensionally correct?

? A ? A ? A? ATLTT

LL

2 !

!


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1.3 The Role of Units in Problem Solving

Is the ollowing equation dimensionally correct?

vtx !

? A ? A ? ALTT

LL !

!


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1.4 Trigonometry


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1.4 Trigonometry

h

ho!Usin

h

ha!Ucos

a

o

h

h!Utan


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1.4 Trigonometry

m2.6750tan oh!Q

m0.80m2.6750tan !! Qoh

a

o

h

h

!Utan


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1.4 Trigonometry

! h

ho1sinU

!

h

ha1cosU

!

a

o

h

h1tanU


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1.4 Trigonometry

!

a

o

h

h1tanU Q13.9m0.14

m25.2tan 1 !

! U


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1.4 Trigonometry

222

ao hhh !Pythagorean theorem:


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1.5 Scalars and Vectors

Ascalarquantity is one that can be described

by a single number:

temperature, speed, mass

A vectorquantity deals inherently with both

magnitude and direction:

velocity, orce, displacement


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1.5 Scalars and Vectors


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1.6 Vector Addition and Subtraction

Often it is necessary to add one vector to another.


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5 m 3 m

8 m


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2.00 m

6.00 m


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2.00 m

6.00 m

222 m00.6m00.2 !R

R

m32.6m00.6m00.2 22 !!R


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2.00 m

6.00 m

6.32 m

U

00.600.2tan !U

Q4.1800.600.2tan 1 !! U


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When a vector is multiplied

by -1, the magnitude of the

vector remains the same, but

the direction of the vector isreversed.


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A

T

BT

BA TT

A

TB

T

BATT


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1.7The Components of a Vector

.ofcomponentvectortheand

componentvectorthecalledareand

r

yxT

TT

y

x


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.AAA

AA

A

yx

TTT

TT

T

!thatsoyvectorialltogetheraddand

axes,andthetoparallelarethatandvectors

larperpendicut oareofcomponentsvectorThe

yxyx


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It is often easier to work with the scalar components

rather than the vector components.

.of

componentsscalartheareand

A

Tyx AA

1.magnitudeithrsunit vectoareand yx

yxA yx AA !T


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Example

A displacement vector has a magnitude of175 m and points atan angle of50.0 degrees relative to thex axis. Find thex andy

components ofthis vector.

ry!Usin

m1340.50sinm175sin !!! QUry

rx!Ucos

m1120.50cosm175cos !!!QUrx

yxr m134m112 !T


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1.8 Addition of Vectors by Means of Components

BACTTT

!

yA yx !T

yxB yx BB !T


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1.8 Addition of Vectors by Means of Components

yyyC

yyxx

yxyx

!

!T

xxxC ! yyy BA !

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Introduction, Significant Figures and Uncertainty