Note - Chapter 01

8/11/2019 Note - Chapter 01

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Chapter 1

Units, Physical Quantities, and Vectors

1.3 Standards and Units

The metric system is also known as the S I system

of units. (S I ! Systme International).

A. Length

The unit of length in the metric system is the meter.

A meter is the distance that light travels in vacuum in

(1 / 299,792,457) seconds.

B. Mass

The unit of mass in the metric system is the kilogram.

A kilogram is the mass of a particular cylinder of

platinum-iridium alloy kept at the International Bureau of

Weights and Measures near Paris, France.

C. Time

The unit of time in the metric system is thesecond. A

second is the time required for 9,192,631,770 cycles of a


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particular microwave radiation, which causes cesium

atoms to undergo a transition between its two lowest

energy states.

The metric system is also known as the mks system of

units.

1.4 Conversion of Units

Units can be treated as algebraic quantities that can

cancel each other.

1.6 Estimates and Order of Magnitude

Fermi Questions Order of Magnitude:The point of such

questions is that reasonable assumptionslinked with


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simple calculationscan often narrow down the range of

values within which an answer must lie. The order of

magnituderefers to the power of 10 of the number that

fits the value. To increase by an order of magnitude

means to increase by a power of 10.

Thus based on some reasonable assumptions, you

approximatean answer to a given physical problem.

Fermi Question 1: Estimate the number of revolutions

that the tires of your car make when you drive your car

from the parking lot at OCC to your home.

Fermi Question 2: How many piano tuners are there in

New York City?

Fermi Question 3: How many hairs are on human head?

etc


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1.5 Uncertainty and Significant Figures

When a value is not known precisely, the amount of

uncertainty is usually called an error.Errorrepresents

uncertainty and has nothing to do with mistakes or

sloppiness.

Asignificant figureis a reliably known digit. When we

say that a quantity has the value 3, we mean by

convention that the value could actually be anywhere

between 2.5 and 3.5. However, if we say that the value is

3.0, then we mean the value lies between 2.95 and 3.05.

A. Significant Figures in Addition or Subtraction:

The number of decimal places in the result should

equal the smallest number of decimal places of any term

in the sum (or subtraction). That is

(1) 23.45 + 1.345 = 24.795 ! 24.80

(2) 56 34.56 = 21.44 ! 21


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B. Significant Figures in Multiplication or Division:

The number of significant figures in the final product

is the same as the number of significant figures in the

factor having the lowest number of significant figures.

That is,

(1) 123.56 x 7.89 = 974.8884 ! 975

(2) 564 / 0.0034 = 165882.352941!

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1.7 Vectors and Vector Addition

A.AScalarquantity is a physical quantity that has

magnitudeonly. Examples include:

(a) mass m (in kilograms, kg)

(b) time t (in seconds, s or sec)

(c) temperature T (in Kelvin, K)

(d) volume V (in cubic meters, m3)

(e) density !(in kg/m3)

(f) energyE(in Joules, J)

(g) distance d (in meters, m)


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(h) speed v (in meters per second, m/s)

(i) electric charge q (in Coulombs, C)

B.A vectorquantity is a physical quantity that has both

magnitudeand direction. Examples include:

(a) displacement rr

! ( in meters, m)

(b) velocity vr

(in meters per second, m/s)

(c) acceleration ar

(in meters per second square, m/s2

)

(d) force Fr

(in Newtons, N)

(e) linear momentum pr

(in kg m/s)

(f) angular momentum Lr

(in kg m2/ sec)

(g) electric fieldE

r

(in Volts per meter, V/m)(h) magnetic field B

r

(in Teslas, T)

(i) torque !r

(in Newton meter, N m)

The displacementvector rr

! (here they call itr

A)

The displacement vector rr

! of an object is defined as the

vector whose magnitude is the shortest distance between

the initial and final positions of the object, and whose


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direction points from the initial position to the final

position.


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1.8 Components of a Vector

A. The componentsof a vector are theprojectionsof the

vector along the axes of a rectangular coordinate system.

Ax!the x-component of vector Ar

.

Ay !the y-component of vector Ar

.

A

r

or A denote the magnitude of vector Ar

.

"denotes the direction of vector Ar

.

Any vector alone gives you the same physics as its two

rectangular components together.


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1.9 Unit Vectors

A unitvector is a vector of magnitude one.

!i

a vector of magnitude one and in the direction of the positive x-axis.!j a vector of magnitude one and in the direction of the positive y-axis.

!k a vector of magnitude one and in the direction of the positive z-axis.

In general, any vector Ar

can be written in terms of its

rectangular components and unit vectors. That is,

kAjAiAA zyx ++=

r

so that its magnitude may also be written as

222

zyx AAAA ++=r


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Some Properties of Vectors

A.Multiplication (or Division) of Vectors by a Scalar

B.Addition of Vectors Ar

and Br

(i) Ar

and Br

in thesamedirection

(ii) Ar

and Br

in oppositedirections


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(iii) Ar

and Br

in #directions

(iv) Ar

and Br

in arbitrarydirections...

1. Graphical method

2. Algebraic method

3. Head-to-Tail method

C. Subtraction of Vectors Ar

andBr

Think of Ar

- Br

simply as Ar

+ (-Br

) and proceed as in

vector addition

1.10 Products of Vectors

A. The Scalar (Dot) Product of Two Vectors

One important thing to remember about thescalar(or

dot)productbetween two vectors is that you multiply two

vectors and the result is ascalar! There are two formulas

worth remembering. These are:


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(i) If you know the magnitudes of the two vectors (and the

smallest angle $between them) whosescalar productyou

wish to find, then use

r

Ar

B =r

Ar

B cos(")

(ii) If you know the two vectors in component form, that

is, if you have

kAjAiAA zyx ++=r

and

kBjBiBB zyx ++=r

,

then use

zzyyxx BABABABA ++=rr

From the first definition above, note that:


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B. The Vector (cross) Product of Two Vectors

When you multiply any two vectors Ar

and Br

via the

cross product, the result is a third vector Cr

such that

BACrrr

!= . What are the magnitude and direction of the

cross product Cr

?

(a) The magnitudeof Cr

is given by

r

C =r

Ar

B sin "( )

where "is the smallest angle between the two vectors Ar

and Br

.

(b) The directionof Cr

is

1.perpendicularto both vectors Ar

and Br

. This narrows

down the choices for the direction of vector Cr

to two

possible directions.2.Use the right-hand-rule to choose between these two

possible choices for the direction of Cr

. Simply point

the 4 fingers of your right hand in the direction of the


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firstvector in the cross product, namely Ar

, and aim

thepalmof your right hand in the direction of the

secondvector in the cross product, namely Br

. The

thumb of your right hand then points in the direction

of the cross product Cr

.

The cross product can also be expressed in determinant

form as

r

A"r

B =

i j k

Ax Ay Az

Bx By Bz

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Note - Chapter 01