Fundamental Tools Chapter 1. Fundamental Tools Expectations After this chapter, students will: ...

Fundamental Tools

Chapter 1

Fundamental Tools

Expectations

After this chapter, students will: understand the basis of the SI system of units distinguish between units and dimensions be able to perform dimensional analyses distinguish between fundamental and derived units be able to convert a quantity to different units know standard powers-of-ten prefixes be able to solve right triangles

Fundamental Tools

Expectations (continued)

After this chapter, students will: distinguish between vector and scalar quantities be able to resolve vectors into orthogonal components be able to add and subtract vectors know how vectors can be multiplied by scalars know how vectors can be multiplied by vectors know how many significant figures are in a given

number

What is Physics?

The mapping of mathematics onto the material world.

A mathematical description of the interactions of space, time, matter, and energy.

An experimental science: theory is judged by how well it predicts the results of experiments.

Numbers and Units

“I often say that when you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot express it in numbers, your knowledge is of a meager and unsatisfactory kind; it may be the beginning of knowledge, but you have scarcely, in your thoughts, advanced to the stage of Science, whatever the matter may be.”

--- William Thomson (Lord Kelvin) 1824 - 1907

Numbers and Units

Quantity: a characteristic of an object or material that can be expressed quantitatively (in numbers)

examples: height, weight, volume, density

Dimension: the name of the class or category of units that express a physical quantity

examples: length, mass, time, velocity

Numbers and Units

Unit: a reference standard with an agreed-upon definition that allows quantities to be specified by comparison to it.

examples: meter, second, pound

Types of Units

Base unit: the units of the quantities length, mass, and time.

examples: meter, kilogram, second

Derived units: units made by combining other units

examples: meters / second, kilogram·meters / (second)2

Systems of Units

SI (Système International d’Unités)

Base units: length: meter (m)

mass: kilogram (kg)

time: second (s)

CGS (“small metric”)

Base units: length: centimeter (cm)

mass: gram (g)

time: second (s)

Systems of Units

BE (“British engineering”)

Base units: length: foot (ft)

mass: slug (sl)

time: second (s)

In this class, we’ll use SI units almost exclusively.

Significant Figures

The uncertainty in our knowledge of the numerical value of a physical quantity is indicated by the number of significant figures we use to express that value.

To determine how many significant figures are in a number (example: 0.0000149 m)

Write the number in proper scientific notation: 1.49×10-5 m.

Count the digits in this part of the number: 3 digits

Note: for “proper” scientific notation: a × 10n,

1 ≤ a < 10

Unit Conversions

Basic principles: 1 is the multiplicative identity. Multiplying any quantity

by one does not change the value of that quantity. If a fraction’s numerator and denominator are equal, the

fraction is equal to one.

Example: convert 3.45 years to seconds

s 101.09 min 1

s 60

hr 1

min 60

day 1

hr 24

year 1

days 365.24years 45.3 8

fractions equal to one

Dimensional Analysis

A consistency check for mathematical relationships in physics.

A formula or equation that “passes” a dimensional analysis may or may not be correct.

However …

A formula or equation that “fails” a dimensional analysis cannot be correct.


What is a dimensional analysis? How can I do one?

Substitute the dimensions represented by each variable for that variable in the equation to be analyzed.

Algebraically simplify the equation: exponentiate, multiply, divide, add, subtract, cancel.

In simplest terms, both sides of the equation should have the same dimensions.


(Simple) example: your geometry is a little fuzzy. But you’re pretty sure that the surface area of a sphere is given by:

Check it:

… and the analysis fails. That formula can’t be correct.

3

3

4rA

... No ??? lengthlength

length3

4length

32

32

Common Powers-of-Ten

power of ten

prefix symbol example

109 giga - G GHz

106 mega - M MW

103 kilo - k km

10-2 centi - c cm

10-3 milli - m mm

10-6 micro - m

10-9 nano - n nm

Right-Triangle Trigonometry

Basic relationships

a

b

c

a

b

222

tan

cos sin

bac

b

a

c

b

c

a

a

aa

Right-Triangle Trigonometry

Basic relationships

a

b

c

a

b

a

b

c

a

c

b

b

bb

tan

cos sin

Vector and Scalar Quantities

Scalar: completely specified by a magnitude (size)

Vector: completely specified by both a magnitude and a direction

Examples:

Distance (scalar): the airport is 15 km away from here.

Displacement (vector): the airport is 15 km southwest from here.

Vector and Scalar Quantities

Scalar: speed, temperature, time, mass, energy, volume, area, length

Vector: velocity, acceleration, momentum, force

Note that the ability to take on + or – values does not make a quantity a vector. Example: Celsius or Fahrenheit temperature.

Vector Properties

Symbol: an arrow (line segment with a point) arrow length shows vector magnitude arrow points in vector direction

Mathematical notation: bold-font letter A arrow on top of letter “hat” on top of letter (usually a unit vector)

A

i

A

Vector Mathematics

Vectors can be: Added Multiplied

by a scalar by another vector (in two different ways)

Subtracted Divided

Vector Addition

Here’s a graphical look at vector addition: we want to add A and B.

+x

+y

A

B

Vector Addition

First, we note that we can translate a vector to any other location without changing it (either magnitude or direction).

+x

+y

A

A

A

A

Vector Addition

So, we translate B so that its “tail” coincides with A’s “point.”

+x

+y

A

B

Vector Addition

Now, we draw a third vector from the beginning point of A (its “tail”) to the ending point of B (its “point”).

That third vector is the sum: A + B.

+x

+y

A

B

A+B

Vector Multiplication

There are three kinds of multiplication that can be done with vectors.

First: multiplication by a scalar.

Magnitude of the product vector: magnitude of the factor vector times the scalar.

Product vector direction: same or opposite the factor vector direction

VbVb



Second: scalar product of two vectors (“dot product”).

The scalar product is zero

if the vectors are perpendicular;

a maximum value when they

are parallel +x

+y

A

B

cosABBA This kind of vector

multiplication is commutative



Third: vector product of two vectors (“cross product”).

Direction: perpendicular to bothA and B, and in accordancewith the right-hand rule

The vector product is zeroif the vectors are parallel;a maximum value when theyare perpendicular

)(magnitude sinABBAC

+x

+y

A

B

C

This kind of vector multiplication is NOT commutative

Vector Subtraction

This time, we wantA – B.

Graphically:

+x

+y

A

B

Vector Subtraction

Our first step is to muliplyB by the scalar -1,producing – B:

+x

+y

A

-B

Vector Subtraction

And now we move – B to the point of A, just aswe did before:

+x

+y

A

-B

Vector Subtraction

And we draw in thesum: A + (-B)=A – B.

+x

+y

A

-BA-B

Vector Addition by Components

Any vector can be expressed as the sum of two vectors, both orthogonal to the coordinate axes.

One is the X component, and one is the Y component.

+x

+y

A

AX

AY

YX AAA


Simple right-triangle trigonometry allows us to calculate the magnitudes of these components:

sin

cos

AA

AA

Y

X

+x

+y

A

AX

AY


Example: we want to add vectors A and B.

BAC

+x

+y

A

B

A

B


First: resolve A and B into components.

(Replace A and B with component vectors AX, AY, BX, and BY, all orthogonal to the coordinate system.)

+x

+y

AB

BY

BX

AY

AX

A

B


The components of the sum, C, are the sums of the components of A and B.

Since the X components are either parallel or antiparallel, their magnitudes add algebraically.

The same is true of the Y components.

+x

+y

AB

BY

BX

AY

AX

A

B

YYY

XXX

BAC

BAC


BAY

YYY

BAX

XXX

BAC

BAC

BAC

BAC

sinsin

coscos

+x

+y

BY

BX

AY

AX

(magnitude)

(magnitude)


BAY

YYY

BAX

XXX

BAC

BAC

BAC

BAC

sinsin

coscos

(magnitude)

(magnitude)

+x

+y

CY = AY + BY

CX = AX + BX


Pythagoras’ theorem yields the magnitude of C:

The direction of C:

22YX CCC

X

YC C

Carctan

+x

+y

CY =

AY +

BY

CX = AX + BX

C

C


A couple of things to remember: You are free to define your coordinate system so

that it makes your life easier. These are always correct:

as long as you measure counterclockwise from the +X direction.

sin

cos

AA

AA

Y

X

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