- 1 -

The presented text is prepared on the basis of numerous resources which are recommended for student self studies as well:

1. D.Halliday, R.Resnick, K.S.Krane, Physics, vol.1,2, J.Wiley & Sons, Inc., New York 1992. 2. P.A.Tipler, Physics for Scientists and Engineers, W.H.Freeman & Co, 4th ed. 1999 3. H.D.Young, Physics, Addison-Wesley Publ.Co., 8th ed. 1992. 4. H.D.Young, R.A. Freedman, University Physics with Modern Physics, Addison-Wesley Publ.Co 12 ed.2007 5. E. Hecht, Physics: Calculus, Brooks/Cole, 2nd ed. 2000. 6. D.C.Giancoli, Physics, Principles with Applications, 6th ed. Pearson education Inc, 6th ed. 2005. 7. M.Mansfield, C.O’Sullivan, Understanding Physics, J.Wiley & Sons Ltd, 1998. 8. R.Resnick, D.Halliday Fizyka, 14th ed, PWN 1999. 9. D.Halliday, R.Resnick, J.Walker, Fizyka, t. 1-5, PWN 2003. 10. A.K. Wróblewski, J.A. Zakrzewski, Wstęp do fizyki, PWN 1976. 11. B.Jaworski, A.Dietłaf, L.Miłkowska, Kurs fizyki, PWN 1984. 12. C.Kittel, W.D. Knight, M.A.Ruderman, Mechanika, PWN 1973. 13. E.M. Purcell, Berkeley Physics Course, Elektryczność i Magnetyzm. 14. A.K.Wróblewski, Historia Fizyki, PWN 2007.

Internet resources: http://en.wikipedia.org http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html http://physicsweb.org/resources/- simulations http://www.pintarmedia.com/-virtual lab http://www.whfreeman.com/tipler/content/instructor/index.htm http://info.fuw.edu.pl/~akw/historia.html http://ptf.fuw.edu.pl/fizyka.html- physics links http://ocw.mit.edu/OcwWeb/Physics/8-01Physics-IFall1999/VideoLectures/index.htm http://www.animations.physics.unsw.edu.au http://www.howstuffworks.com/ http://www.educypedia.be/electronics/javaelectricity.htm http://www.alcyone.com/max/physics/laws/

- 2 -

Lecture 1 VECTORS

Notes:

Lecture 1- 1

„ For we do not t hi nk t hat we k now a t hi ng un t i l we are acquai n t ed wi t h i t s pri mary condi t i ons or f i rst pri n ci pl es, an d have carri ed ou r ana l ysi s asf a r as i t s si mpl est el emen t s.

Aristotle, Physics, written 350 B.C.E

Observations

Analysis

Model

Application

Experimental verification

quantities measures

standardsunits

Scientific methodLanguage of physics

„ "To measure i s t o k now.„" If y ou can not measure i t , y ou can not i mprove i t .„

Lord Kelvin

experiments

„ "Mat hemat i cs i s t he on l y good met aphysi cs."„

" I hav e no sat i sfact i on i n f ormu l as un l essI f eel t hei r nu meri cal magn i t ude."

Lord Kelvin

Maths

Scientific Method

Key terms: Scientific method, feedback, measures, units, standards

Scientific Method is a procedure involving the observation of phenomena, the formulation of a hypothesis, experimentation to evaluate the hypothesis, and a conclusion that validates or modifies the hypothesis. The final stage involves applications.

Learn more:

http://en.wikipedia.org/wiki/Scientific_method

Concepts: Mediocrity principle The principle that there is nothing particularly interesting about our place in space or time, or about ourselves. This principle probably first made its real appearance in the scientific community when Shapley discovered that the globular clusters center around the center of the Galaxy, not around the solar system. The principle can be considered a stronger form of the uniformity principle; instead of no place being significantly different than any other, the mediocrity principle indicates that, indeed, where you are is not any more special than any other. Source: http://www.alcyone.com/max/physics/laws/ Copyright © 2009 Erik Max Francis

- 3 -

Problem Solving General Strategy

Lecture 1- 2 Vectors & scalars

Vector’ features: - additional directional quality• independent on the system of coordinates:

value, direction.• obeys vectors’ addition laws:

parallelogram construction, commutative law,(rotation does not obey commutative law)

Scalar’ features: - quantity•independent on the system of coordinates.•obeys algebraic rules.

Vector forms of physics laws do not depend on the coordinates system

BFV

;;

iET ;;

Tensors

scalar quantities are tensors of rank 0; Vector quantities are tensors of rank I (1 dim array)

Tensor of rank II (3x3 – 9 components)

Anisotropy: tension,moment of inertia,electric susceptibility,

Maths:multi-dimensional array relative to a choice of basis of the particular space on which it is definedPhys:tensorial quantities vary from point to point (generalised directional properties)

Tensors’ rank: number of array indices required to describe such a quantity

Key terms: Vector, scalar, tensor, rank, isotropy, anisotropy

IDENTIFY the relevant concepts: First, decide which physics ideas are relevant to the problem. Although this step doesn’t involve any calcuIations, it's sometimes the most challenging part of solving the problem. Don't skip over this step, though; choosing the wrong approach at the beginning can make the problem more difficult than it has to be, or even lead you to an incorrect answer. At this stage you must also identify the target variable of the problem-that is, is the quantity whose value you're trying to find. Sometimes the goal will be to find a mathematical expression rather than a numerical value. Sometimes, too, the problem will have more than one target variable. The target variable is the goal of the problem-solving process; don't lose sight of this goal as you work through the solution. SET UP the problem: Based on the concepts you selected in the Identify step, choose the equations that you'll use to solve the problem and decide how you'll use them. If appropriate, draw a sketch of the situation described in the problem. EXECUTE the solution: In this step, you "do the math." Before you launch into a flurry of calculations, make a list of all known and unknown quantities, and note which are the target variable or variables. Then solve the equations for the unknowns. EVALUATE your answer: The goal of physics problem solving isn't just to get a number or a formula; it's to achieve better understanding. That means you must examine your answer to see what it's telling you. Be sure to ask yourself, "Does this answer make sense?" If your target variable was the radius of the earth and your answer is 6.38 centimeters (or if your answer is a negative number!), something went wrong in your problem-solving process. Go back and check your work, and revise your solution as necessary. Source: H.D.Young, R.A.Freedman, University Physics, Pearson, Addison-Wesley, 12th ed. 2007

- 4 -

Notes:

Lecture 1- 3

ABBA

)( CBACBA

BkAkBAk

)(

B

A

B

C

cos2222

ABBAC

)( BABAC

Vectors

b

b A

BC

B

A

B C

commutative law

resultant vector

vector addition

associative law

vector subtraction

CBA

Key terms: Vector algebra, associative law, commutative law, parallelogram method, tail-to-head method, vector sum (resultant)

Scalar quantity is a number and it obeys the usual rules of arithmetic. It changes when its magnitude changes. Vector quantity has direction and sense as well as magnitude and follows vector algebra rules, in particular the rules of vector addition. The change of any of the vector’s features (direction, sense, magnitude) results in a change of vector quantity. Commutative = it is independent of the order of addition The addition of rotation is non-commutative. Rotations have well-defined magnitudes and directions but they are not, in general, vector quantities.

- 5 -

Notes:

Lecture 1- 4

w

A

z

wA

zA

A

xA

Components of vectors

)ˆ,(cosˆˆ xAAxxAA xx

unit vector - versor

)ˆ,(cos xAAAx

A

xxA

defines direction in space

x

r

a

r

a

raa ˆ

raa ˆ

1ˆ r

antiparallel

Key terms: Unit vector, versor, vector component, parallel vectors, antiparallel vectors, negative of a vector

Unit vector describes direction in space. A unit vector has a magnitude of one, with no units.

- 6 -

Notes:

Lecture 1- 5

yx

z

A

yA

xA

zA

Vectors in a Cartesian coordinates

i

kj

zyx AAAA

kAjAiAA zyxˆˆˆ

],,[ zyx AAAA

222zyx AAAA

kA

jA

iA

ˆ,

ˆ,

ˆ,

AA

AA

AA

z

y

x

cos

cos

cos

Cartesian notation

Key terms: System of coordinates, orthogonal system, Cartesian system,

- 7 -

Notes:

Lecture 1- 6

],,[ zyx AAAA

],,[ zyx BBBB

kBAjBAiBA zzyyxxˆ)(ˆ)(ˆ)(

AnC

Vectors’ operations

kAjAiAA zyxˆˆˆ

kBjBiBB zyxˆˆˆ

kCjCiCC zyxˆˆˆ

zzzyyyxxx BACBACBAC ;;

],,[ zyx CCCC

BAC

Vector products

AnAnC

knAjnAinAAnC zyxˆ)(ˆ)(ˆ)(

Key terms: Cartesian notation, Component method for adding vectors

Learn more:

http://en.wikipedia.org/wiki/Euclidean_vector

- 8 -

Notes:

Lecture 1- 7

),(cos BABABA

B

A

cosA

ABBA

ABBA

0

ABBABA

//

],,[ zyx AAAA

],,[ zyx BBBB zzyyxx BABABABA

1ˆˆ ii

0ˆˆ ji

Scalar product

dot product -> numberBA

commutative law

C

1ˆˆ kk 1ˆˆ jj 1ˆˆˆ 2 iii

0ˆˆ kj 0ˆˆ ik

lFW

work

Key terms: Dot product, square of the unit vector, vector coordinates

- 9 -

Notes:

Lecture 1- 8

],,[ zyx AAAA

],,[ zyx BBBB CBA

),(sin BABABA

B

A

C

ABBA

ABBA

//0

ABBABA

kBABAjBABAiBABABA xyyxzxxzyzzyˆ)(ˆ)(ˆ)(

zyx

zyx

BBB

AAAkji

BA

ˆˆˆ

Vector productcross product -> vector

in the „right-handed system”

0ˆˆ ii

kji ˆˆˆ

0ˆˆ kk0ˆˆ jjikj ˆˆˆ jik ˆˆˆ jki ˆˆˆ © W.H.Freeman & Co

Key terms: Right hand screw rule,

Learn more:

http://en.wikipedia.org/wiki/Right-hand_rule

- 10 -

Notes:

Lecture 1- 9

kji ˆˆˆ ikj ˆˆˆ jik ˆˆˆ

0ˆˆ ii0ˆˆ jj0ˆˆ kk

Cartesian (orthogonal) system

kji ˆˆˆ ikj ˆˆˆ jik ˆˆˆ

iy

x

z

kj

ikji ˆˆˆˆ

Right-handed

iy

x

z

k

j

Left-handed

Key terms: right-handed system, left-handed coordinate system

- 11 -

Notes:

Lecture 1- 10

yx

z

i

kj

r

yr

xr

zr

),,( zyxP

Position in a Cartesian system

zyx rrrr

krjrirr zyxˆˆˆ

],,[ zyx rrrr

222zyx rrrr

kr

jr

ir

ˆ,

ˆ,

ˆ,

rrrrrr

z

y

x

cos

cos

cos

position vector

Key terms: orthogonal system, position vector, vector module, vector coordinates

Coordinate system: A system for specifying points using coordinates measured in some specified way. The simplest coordinate system consists of coordinate axes oriented perpendicularly to each other, known as Cartesian coordinates.

- 12 -

Notes:

Lecture 1- 11

),,( zrP

r

z

z

x

y

Cylindrical coordinate system

- radiusr -azimuthal anglez -elevation

zzryrx

sincos

zzyxrxyarctg

22

conversion

Key terms: coordinate conversion

Learn more:

http://en.wikipedia.org/wiki/Cylindrical_coordinate_system

- 13 -

Notes:

Lecture 1- 12

),,( rP

r

x

y

z

Spherical coordinate system

-radiusr -polar angle -elevation angle

cossinsincossin

rzryrx

rz

xyarctg

zyxr

arccos

222

conversion

Key terms: global positioning system GPS

Learn more:

http://en.wikipedia.org/wiki/Spherical_coordinates

- 14 -

Notes:

Lecture 1- 13

),( rP

y

x

Polar coordinate system, natural systems

r

Polar coordinate system

-radiusr -polar angle

o90

sincos

ryrx

xyarctg

yxr

22

conversion

)(r polar equation

Circle: centre , radius),( or a222 )cos(2 arrrr oo

)cos()( kar polar rose

0A

S Ss - path coordinate

Natural system

Key terms: reference axis

Learn more: :

Polar equation of a curve http://en.wikipedia.org/wiki/Polar_coordinate_system#Polar_equation_of_a_curve

- 15 -

Lecture 2 DESCRIPTION OF MOTION Notes:

Lecture 2- 1 Examining the rate of change

)()( xfxxf

x

x xx

xxfxxf

xf

)()(

)(xf

differential quotient

xxx

fxxfxf )()(

)(xf

xxf

tanaverage change

instantenous rate of change ?

variable changes

function responds

Key terms: secant line, tangent line

Average change does not mean algebraic average (mean value.)

Learn more: http://ocw.mit.edu/ans7870/resources/Strang/strangtext.htm

Calculus, Gilbert Strang: e-book

- 16 -

Notes:

Lecture 2- 2Instantenous rate of changederivative concept

x

)(xf

x

dxxx dfdxxfxf )()(dxx 0

instantenous rate of change

)(')()(lim

0xf

dxdf

xxfxxf

x

derivative)(' xfdxdf

slope of the tangent line

x

)(' xf

x

)(xf

Key terms: rise over the run,

People Standing on the shoulders of giants Infinitesimal calculus

Gottfried Wilhelm von Leibniz http://en.wikipedia.org/wiki/Leibniz

- 17 -

Notes:

Lecture 2- 3 Calculation of derivatives (1)

)(')()(lim

0xf

dxdf

xxfxxf

x

0)( dxdfCxf

dxdfCxCf

dxd

))((

1)( nn nxdxdfxxf x

dxdfxxf 2)( 2

33)( dxdfxxf

aaadxd xx ln

xx eedxd

xx

dxd 1ln

axx

dxd

a ln1log

xxdxd cossin

xxdxd sincos

xx

dxd

2cos1tan

xx

dxd

2sin1cot

properties:

‘power rule’

from def.

Key terms: limiting case, power rule,

- 18 -

Notes:

Lecture 2- 4 Calculation of derivatives (2)

dxdg

dxdfxgxf

dxd

))()(( 323)( 2 xdxdfxxxf

fdxdgg

dxdfxgxf

dxd

))()(( xxxxdxdfxxxf cossin2sin)( 22

2'')

)()((

gfggf

xgxf

dxd

dxdu

dudf

dxdfxuf )]'(([

xxxx

dxdf

xxxf 2sin

cos2sin2sin2)(

)cos(2)sin()( 22 xxdxdfxxf

compound function

Key terms: chain rule

- 19 -

Notes:

Lecture 2- 5 Derivatives (3)

),( yxf

2

2

)())('(dx

fddxdf

dxdxf

dxd

66333)( 3

3

2

223

dxfdx

dxfdx

dxdfxxxf

Multivariable functions

higher order derivatives

3

3

2

2

dxfd

dxfd

dxd

xy yf

xf

partial derivatives

2

2

2

2

yf

xf

xyf

yf

xyxf

xf

y

22

)(xfy dxxfdy )(')(' xfdxdy

differential

dfdgxfxg

)()(

differentiation

- 20 -

Notes:

Lecture 2- 6 Kinematics

Spacetime

Frame of reference

Ideal frame of reference

Isotropic Euclid space

Material point

Rigid body

Concept of motion

Learn more: Spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space being three-dimensional and time playing the role of a fourth dimension that is of a different sort than the spatial dimensions. According to certain Euclidean space perceptions, the universe has three dimensions of space and one dimension of time. more at:

http://en.wikipedia.org/wiki/Spacetime

- 21 -

Notes:

Lecture 2- 7

yx

z

i

kj

r

yr

xr

zr

),,( zyxP

Position in a cartesian system

krjrirr zyxˆˆˆ

],,,[ trrrr zyx

time: 4th dimension

Euclid spaceclassical mechanics

],,[ zyx rrrr

How many dimensions are neededto describe the universe???

Speculative theories such as string theory predict 10 or even 26 dimensions!

Relativistic contexts:time cannot be separated from space(time depends on an object's velocity)

- 22 -

Notes:

Lecture 2- 8 Equations of motion

Cartesian system

)();();()(

],,[

tztytxtr

zyxr

Cylindrical system)();();(

),,(tzttr

zrP

Spherical system)();();(

),,(tttr

rP

equation for path coordinates vs time

0A

S )(tS - path coordinate

Describing motion: equation of motion

Positional vector vs timeCoordinates vs timeTrajectory equation

- 23 -

Notes:

Lecture 2- 9

)()()(

tzztyytxx

),( yxfz

x

y

zTrajectory

Key terms: trajectory of motion

Motion Capture Data MOCAP Capturing marker-trajectories In order to biomechanically analyze the motion of a person or in order to map real world performances onto virtual characters, captured marker-trajectories, e.g. 3D trajectories of optical beacons attached to the body, have to be transformed into the motion parameters of a kinematic skeleton model.

Learn more: http://cat.wiki.softimage.com/index.php/Mapping_Motion_from_Mocap_Point_Clouds

- 24 -

Notes:

Lecture 2- 10

y

x1r

2r

r

s

1r

2r

r

1v

2vO

trv

t

0lim

dtrdv

Displacement, velocity

rrr 12

Average velocitytrvav

instantaneous velocity

Key terms: speed, velocity vector, global change, instantaneous change

Average velocity is not the mean value of the velocity.

- 25 -

Notes:

Lecture 2- 11

dtrdv

zyxzyx vvvvkvjviv ˆˆˆ

.

.

.

zdtdzv

ydtdyv

xdtdxv

z

y

x

222zyx vvvv

vv

kv

vv

jv

vviv

z

y

x

)ˆ,cos(

)ˆ,cos(

)ˆ,cos(

Instantaneous velocity

dtdr

kdt

drj

dtdri zyx ˆˆˆ

dtdzk

dtdyj

dtdxi ˆˆˆ

r

x

y

zv

xvyv

zv

ij

k

- 26 -

Notes:

Lecture 2- 12

1r

2r

a

1v

2v

O

vtv

dtvd

tva

t

0

lim

dtrdv

2

2

dtrd

dtvda

2

2

2

2

2

2

dtzda

dtyda

dtxda

z

y

x

222zyx aaaa

2

2

2

2

2

2

2

2ˆˆˆ

dtzdk

dtydj

dtxdi

dtrda

zyx aaaa

zyx akajaia ˆˆˆ

zayaxa

zvyvxv

zyx

zyx

,,

,,

Acceleration

tvvaav

12

jerk (jolt ): 3

3

dtrd

dtad

Key terms: Newton notation, Leibnitz notation

Any change of any of the velocity characteristics (e.g. magnitude, direction, sense) produces acceleration

- 27 -

Notes:

Lecture 2- 13

Curvilinear motion

01

P 1

P

0

nV

s

skav

Average curvature

velocity versorradius versorn

local curvature

radius of curvature

kdds 1

dsd

sk

s

lim0

dsd

Key terms: curvature, convex, concave, centre of curvature, radius of curvature,

- 28 -

Notes:

Lecture 2- 14

1

dtdV

dtdV

dtVda ˆˆ)ˆ(

2sinˆ2ˆ

2

2sin

ˆˆ lim0

2

d

versor increment

ˆˆ

dtd

ˆ n

a

VP

P 1 r s

O

n n

Planar curvilinear motion

dtdn

dtd ˆˆ

dtdnV

dtdVa ˆˆ

A

dd 1ˆ

VV

2

2sin

ˆ

Key terms: versor increment

Time derivative of the versor (versor increment) defines another versor which perpendicular to the given one.

- 29 -

Notes:

Lecture 2- 15

Vdtds

nt aanV

dtdVa

ˆˆ

2

22

2

dtdVa

V

22nt aaa

V

s

O

an

Tangent and normal acceleration

dtds

dsdnV

dtdVa ˆˆ

dtdnV

dtdVa ˆˆ

dtdVat

nVan ˆ2

at

an

1

dsd

radius of curvature

tangent normal

- 30 -

Notes:

Lecture 2- 16

rs

rs

22

rs

V

rdtdr

dtds

dtd

dtdVat

rVan

2

4222 raaa nt

fT

const 22

)(t

2

2

dtd

y

xr

Circular motion

s

dtdr

2

2

dtdr

dtdtdd

r

r

r2

angular velocity

angular acceleration

Key terms: angular path,

- 31 -

Notes:

Lecture 2- 17

0

rV

dtVda

dtd

V

ryx

z

tarrdtd

vdtrd

navdtrd

rvan

)()()( baccabcba

rat

Circular motion – acceleration (s)

rran

)()(

rdtd

dtrdr

dtd

r 2

© W.H. Freeman & Co

Key terms: angular velocity, right hand rule, normal acceleration, centripetal acceleration

In a uniform circular motion the direction of velocity vector changes. That directional change is due to the centripetal acceleration.

- 32 -

Notes:

Lecture 2- 18

nt aavra

rvan 2

rat

V

r yx

z

anat

rr

van2

2

rat

Circular motion – vectors

Key terms: tangent (linear) acceleration, radial (normal) acceleration

- 33 -

Notes:

Lecture 2- 19

y

xr

)(t )cos(cos 0 trrx)sin(sin 0 trry

)sin( 0 trdtdxvx

)cos( 0 trdtdyv y

xtrdt

dva xx

20

2 )cos(

ytrdt

dva y

y2

02 )sin(

nyx araaa 222

Uniform circular motion in polar coordinates

Uniform circular motion can be described as result of overlapping of two linear , simple harmonic oscillations along perpendicular directions, shifted in phase by:

2