P!h!!!"! c!î!

P h ! " î c #$!!"#$$%

P h ! " î c #$!!"#$$%

Worawarong Rakreungdet, Physics Dept., KMUTTVectors

Weekly Goal: Vectors. Adding and multiplying vectors.

Resource: HyperPhysics: Physics concept maps.http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.htmlClass Textbook:D. Halliday, R. Resnick and J. Walker, Fundamental of Physics, John Wiley & Son Inc., New York, USA.

(based on graphics) (based on vector components) (based on polar forms)

Vector Calculus• The “del,” the

collection of partial derivatives

• Gradient:

• Divergence:

• Curl:

• LaPlacian:

Vector Product

Vector Addition

Scalar Product

B will be placed on the x-axis and both A and B in the xy plane

i� e1

j � e2

k � e3

1, if i = j

0, if i �= j�ij =

Extra:ei · ej = �ij

�ijk =+1##if##(i,j,k)#is#(1,2,3),#(3,1,2)#or#(2,3,1)##

.1##if##(i,j,k)#is#(3,2,1),#(1,3,2)#or#(2,1,3)##

0##otherwise:##i = j##or##j = k or k = i#

⇥a�⇥b = ⇥c; ci =3�

j,k=1

�ijkajbk

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P!h!!!"! c!î!

P h ! " î c #$!!"#$$%

P h ! " î c #$!!"#$$%

Worawarong Rakreungdet, Physics Dept., KMUTTNewton’s Laws and the Causes of motion

Important Websites:Class URL: all information about PHY 103 (2/2011)http://webstaff.kmutt.ac.th/~worawarong.rak/classes/2554-2/PHY103/home.htmlHyperPhysics: Physics concept maps, nice illustration.http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html

Standard Newton’s Laws Problems

nX

i=1

�Fi = m�a

Free-Body Diagram

A free-body diagram is a sketch of an object of interest with all the surrounding objects stripped away and all of the forces acting on the body shown. The drawing of a free-body diagram is an important step in the solving of mechanics problems since it helps to visualize all the forces acting on a single object. The net external force acting on the object must be obtained in order to apply Newton's Second Law to the motion of the object.

Newton’s Law1st Law:

nX

i=1

�Fi = 0;! �v = constant

�Faction

= �Freaction

nX

i=1

�Fi = 0;! �v = constant

Newton’s Law2nd Law:

Newton’s Law3rd Law:

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Collision'and'Impulse'

•  From''''''''''''''''''''''','the'net'change'of'the'system'due'to'collision'is''

'the'le8'side'of'the'equa:on''is'a'measure'of'both'the'magnitude'and'dura:on'of'the'collisional'force,'defined'as'the'impulse(of'the'collision.'

d�p = �F (t)dt

� tf

ti

d�p(t) =� tf

ti

�F (t)dt� tf

ti

d�p(t) = �pf � �pi = ��p

Impulse = ��p =� tf

ti

�F (t)dt

P!h!!!"! c!î!

P h ! " î c #$!!"#$$%

P h ! " î c #$!!"#$$%

Worawarong Rakreungdet, Physics Dept., KMUTTSystem of Particles / Center of mass concept

Center of Mass (COM): The point that moves as though:1. all of the system’s mass were concentrated there2. all external forces that create translation were applied there

Two-point system

Point&source&

�F

Finite&source&

H&

L&

�F

�rcom =�n

i=1 mi�ri�ni=1 mi

�rcom =�

�rdm�dm

=1M

⇥�rdm

General definition

Note: For simplicity, we will always assume that an object is uniform in this course

The$mo'on$of$the$c.o.m.$of$any$system$of$par'cles$is$governed$by$

$$

�Fnet = M�acom

All$external$force.$Forces$on$one$part$of$the$system$from$another$part$of$the$systems$(internal$forces)$are$not$included$here.$

Total$mass.$No$mass$enters$or$leaves$the$system$as$it$moves.$(M$=$constant).$This$is$referred$to$as$a$closed$system$$

Accelera'on$of$the$c.o.m.$of$the$system.$There$is$no$informa'on$regarding$any$other$point$of$the$system$$

Newton'2nd'Law'of'mo.on'

Linear'momentum' �p = m�vFor a particle �P = M�vcom

�Fnet =d�P

dt

For a system of particles

same area under the curve

pfx � pix =� tf

ti

Fx(t)dt

e.g. along the x-direction

�p = Favg�tWe can simplify the impulse using

Conserva)on*of*linear*momentum*�Fnet =

d�P

dt= 0 �P = constant �Pi = �Pf (closed,)isolated)system))

Momentum(and(Kine-c(Energy(in(Collisions(•  Conserva-on(of(linear(momentum(

•  Conserva-on(of(total(energy(

•  Considering(the(kine-c(energy(of(the(system,(–  If(the(kine-c(energy(is(conserved,(then(the(collision(is(elas%c.(–  If(the(kine-c(energy(is(not(conserved,(then(the(collision(is(inelas%c.(

�Pf = �Pi

Ef = Ei

(for(a(closed,(isolated(system)(

(always(true!)(

Conserved/=/Has/the/same/value/both/before/and/a7er/

m1v1i = m1v1f + m2v2f

12m1v

21i =

12m1v

21f +

12m2v

22f

Example: Elastic Collision in 1 dimension

v2f =2m1

m1 + m2v1i

v1f =m1 �m2

m1 + m2v1i

Extra: completely inelastic = largest energy lost in the system. This will result in two bodies stick together

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The$Kine(c$Energy$of$Rolling$must$take$into$account$both$rota(on$and$transla(on$

12Icom�2 1

2Mv2

com+ = (K.E.)rolling

rota%onal(kine(c$energy$due$to$rota(ons$about$its$center$of$mass$

transla%onal(kine(c$energy$due$to$transla(on$of$its$

center$of$mass$Kine(c$Energy$(K.E.)$of$a$rolling$object$

P!h!!!"! c!î!

P h ! " î c #$!!"#$$%

P h ! " î c #$!!"#$$%

Worawarong Rakreungdet, Physics Dept., KMUTTRotation + Rolling

I =�

mir2i I =

�r2dm

Work and Rotational Kinetic Energy

P =dW

dt= �⇥ (Power, rotation about a fixed axis)

�K =12I�2

f �12I�2

i = W. (Work-Kinetic Theory for Rotation)

⇥� = ⇥r � ⇥F

⇤⇥net = I⇤�

�l = �r � �pL = I�

⇥�net =d⇥L

dt= 0

�L = constant

TORQUE

ANGULAR MOMENTUM

If

(conserv. of ang. momentum)

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p =�F

�A

Pascal’s'Principle'and'the'Hydraulic'Lever'

Considering'the'work'done'by'the'output'piston,'

W = Fodo =

�Fi

Ao

Ai

⇥�di

Ai

Ao

⇥= Fidi

Work'done'by'the'output'piston'in'li=ing'the'load'placed'on'it'

Work'done'on'the'input'piston'by'the'applied'force'

Hydraulic*Lever*

Pascal’s*Principle:'A'change'in'the'pressure'applied'to'an'enclosed'incompressible'fluid' is'transmiCed'undiminished'to'every'porDon'of'the'fluid'and'to'the'walls'of'its'container.”'

P!h!!!"! c!î!

P h ! " î c #$!!"#$$%

P h ! " î c #$!!"#$$%

Worawarong Rakreungdet, Physics Dept., KMUTTFluid Dynamics

Av1 = Av2

This%rela*onship%also%apply%to%any%so0called%tube%of%flow.%%

Any%imaginary%flow%whose%boundary%consists%of%streamlines.%

Volume%flow%rate% Mass%flow%rate%

RV = Av = const. Rm = �RV = const.

Equa*on%of%Con*nuity%

Bernoulli’s+Equa/on+A+principle+of+fluid+flow+based+on+conserva/on+of+energy+

p +12�v2 + �gy = constant

� �

Streamline*represents*the*fluid*path*

Flow

*of*Ide

al*Fluids*

“Real*Fluids”*Turbulence)flow)of)a)fluid)around)an)obstacle)

h9p://www.jet.efda.org/pages/focus/modelling/images/turbulence.jpg*

Real*fluids*****!*very*complicated*** ****** *****!*not*well*understood*

Thus*we’ll*only*focus*on*“ideal*fluids”*Ideal*fluids****!**

1.  steady*flow*(***************)*2.  Incompressible*flow*(******************)*

3.  Nonviscous*flow*(no*drag*force)*4.  IrrotaLonal*flow*(no*rotaLon)*

d�vi

dt= 0

� = const.

A"net"upward"buoyant"force"on"whatever"fills"the"hole"

A"net"downward"force"on"the"stone"

|�Fg| > |�Fb|

i.e."accelerate"downward"“SINK”"

A"net"upward"force"on"the"wood"

i.e."accelerate"upward"“RISE”"

|�Fg| < |�Fb|

|�Fb| = mfg

Archimedes’ principle: T h e b u o y a n t f o r c e o n a s u b m e r g e d object is equal to the weight of the fl u i d t h a t i s displaced by the object

p = p0 + �ghwhere%%p0#=%the%pressure%at%the%reference%level,%%ρ%=%fluid%density%%h%=%the%depth%of%a%fluid%sample%below%% % %the%reference%#p%=%pressure%in%the%sample%

Pressure%varia:on%with%height%and%depth:%

Density

(

(uniform)density))

� =M

V

� = lim�V�0

�m

�V=

dm

dV

� =�m

�V

For) a) small) volume)∆V),)measuring)a)mass)∆m,)the)density)is$

For)a) infinitesimal) volume)dV)with)a)mass)of)dm,)we)define)a)density)

In)a)case)that)a)material)is) much) larger) than)atomic)dimensions,))

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