Lecture 3 Particle Kinetics OK

8/10/2019 Lecture 3 Particle Kinetics OK

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Chapter 3. Particle Kinetics

Chapter 3

Particle Kinetics

Chapter Objectives

To state Newton's Laws of Motion and Gravitational Attractionand to define mass and weight

To anal!e the accelerated motion of a particle "sing thee#"ation of motion with different coordinate sstems.

To develop the principle of wor$ and energ and appl it tosolve pro%lems that involve force& velocit. and displacement.

To st"d pro%lems that involve power and efficienc.

To introd"ce the concept of a conservative force and appl thetheorem of conservation of energ to solve $inetic pro%lems.

To develop the principle of linear imp"lse and moment"m for aparticle

To st"d the conservation of linear moment"m for particles

To introd"ce the concept of ang"lar imp"lse and moment"m

3.1 Kinetics of a Particle: Force and Acceleration

3.1.1 Basic concepts

The %asic aioms of dnamics were give in Chapter(. Newton)s first and third laws of motion

were "sed etensivel in statics to st"d %odies at rest and the forces acting at them. These

two laws are also "sed in dnamics& in fact& the are s"fficient for the st"d of the motion

which have no acceleration. *owever when %odies are accelerated& i.e.& when the magnit"de

or the direction of their velocit changes& it is necessar to "se the second law of motion in

order to relate the motion of the %od with the forces acting on it.

Meas"rements of force and acceleration can %e recorded in a la%orator so that in accordance

with the second law& if a $nown "n%alanced forceFis applied to a particle& the acceleration a

of the particle ma %e meas"red. +ince the force and acceleration are directl proportional& the

constant of proportionalit& m, ma %e determined from the ratio toF,a. The positive scalar m

is called the mass of the particle. -eing constant d"ring an acceleration& m provides a

#"antitative meas"re of the resistance of the particle to a change in its velocit.

Newton's second law of motion ma %e written in mathematical form as

F = ma 3.(/

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This e#"ation& which is referred to as the e#"ation of motion& is one of the most important

form"lations in mechanics. As previo"sl stated& its validit is %ased solel on eperimental

evidence. 1n (24& however& Al%ert 5instein developed the theor of relativit and placed

limitations on the "se of Newton's second law for descri%ing general particle motion. Thro"gh

eperiments it was proven that timeisnot an a%sol"te #"antit as ass"med % Newton6 as a

res"lt& the e#"ation of motion fails to predict the eact %ehavior of a particle& especiall when

the particle's speed approaches the speed of light .3 Gm,s/. 7evelopments of the theor of

#"ant"m mechanics % 5rwin +chr8dinger and others indicate f"rther that concl"sions drawn

from "sing this e#"ation are also invalid when particles are the si!e of an atom and move

close to one another. 9or the most part& however& these re#"irements regarding particle speed

and si!e are not enco"ntered in engineering pro%lems& so their effects will not %e considered

in this co"rse.

1t is worth to note that the sstem of aes with respect to which the acceleration a is

determined is not ar%itrar. These aes m"st have a constant orientation with respect to the

stars& and their origin m"st either %e attached to the s"n or move with constant velocit with

respect to the s"n. +"ch a sstem of aes is called a Newtonian frame of reference or inertial

sstem. 1n most engineering pro%lems& the acceleration a ma %e determined with respect to

aes attached to the earth and 5#. 3.(/ "sed witho"t an apprecia%le error. :n the other hand&

this e#"ation does not hold if arepresents a relative acceleration meas"red with respect to

moving aes& s"ch as aes attached to an accelerated car or to a rotating piece of machiner.

;e can consider a inertial sstem as the one which does not rotate and is either fied or

translates in a given direction with a constant velocit !ero acceleration/. This definition

ens"res that the particle's acceleration meas"red % o%servers in two different inertial frames

of reference will alwas %e the same.

+ince m is constant& we can also write

= d( )dm

tvF 3.


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To ill"strate application of this e#"ation& consider the particlePshown in 9ig. 3.(a& which has

a mass mand is s"%=ected to the action of two forces& F(and F of the forces acting on the particle is !ero& it follows

from 5#. 3.3/ th"s the acceleration is also !ero& so that the particle will either remain at rest

or move along a straight line path with constant velocit. +"ch are the conditions of static

e#"ili%ri"m,Newton's first law of motion.

9ig. 3.(

*ere it is worth to note that the e#"ation of motion can also %e rewritten in the form

.m =F a The vector @ma is referred to as the inertia force vector. 1fit is treated in thesame wa as a force vector6 then the state of e#"ili%ri"m created is referred to as dnamic

e#"ili%ri"m. This method for application is often referred to as the d'Alem%ert principle&

named after the 9rench mathematician Bean Le >ond d'Alem%ert.

;e can now etend the e#"ation of motion to incl"de a sstem of nparticles isolated within

an enclosed region in space& as shown in 9ig. 3.


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%o"ndar. The res"ltant eternal force Firepresents. for eample. the effect of gravitational&

electrical. magnetic. or contact forces %etween the ith particle and ad=acent %odies or particles

notincl"ded within the sstem.

The free?%od and $inetic diagrams for the ith particle are shown in 9ig. 3.


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*ence& the s"m of the eternal forces acting on the sstem of particles is e#"al to the total

mass of the particles times the acceleration of its center of mass G.

Note that when appling the e#"ation of motion of the particle we can disting"ish two

pro%lems of dnamics. 1n the first tpe the acceleration is either specified or can %e

determined directl from $nown $inematic conditions. The corresponding forces which act on

the particle are the determined % direct s"%stit"tion into 5#. 3.3/. This pro%lem is generall

#"ite straightforward.

1n the second tpe of pro%lem the forces are specified and the res"ltant motion is to %e

determined. 1f the forces are constant the acceleration is constant and is easil fo"nd from 5#.

3.3/. ;hen the forces are f"nctions of time& position& velocit or acceleration& 5#. 3.3/

%ecomes a differential e#"ation which m"st %e integrated to determine the velocit and

displacement. Pro%lems of this second tpe are often more formida%le& as the integration ma

%e diffic"lt to carr o"t& partic"larl& when the force is a f"nction of two or more motion

varia%les. The n"merical methods co"ld %e "sed for these cases.

Important Points

The e#"ation of motion is %ased on eperimental evidence and is valid onl when

applied from an inertial frame of reference.

The e#"ation of motion states that the "n%alanced force on a particle ca"ses it to

accelerate.

The force acting on the particle is e#"al to the rate of change of the linear moment"m of

the particle.

An inertial frame of reference has aes that either translate with constant velocit or are

at rest.

The s"m of the eternal forces acting on the sstem of particles is e#"al to the total

mass of the particles times the acceleration of its center of mass.

The vector @ma is referred to as the inertia force vector and the state of dnamic

e#"ili%ri"m of the particle is created. This method for application is often referred to as

the 7'Alem%ert principle

;hen appling the e#"ation of motion of the particle we enco"nter two pro%lems of

dnamics.

3H


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3.1. !"#ations of $otion: %ectanlar Coordinates

Consider a particle of mass mmoving relativel to an inertialx&y&zframe of reference. The

forces acting on the particle& as well as its acceleration& ma %e epressed in terms of their i&j&

kcomponents. Appling the e#"ation of motion& we have

+ +( )= ( + + )x y z x y zF F F m a a ai j k i j k

9or this e#"ation to %e satisfied& the respective i&j& 'components on the left side m"st e#"al

the corresponding components on the right side. Conse#"entl& we ma write the following

three scalar e#"ationsI

=x xF ma

=y yF ma 3./

=z zF ma1n partic"lar& if the particle is constrained to move onl in thex-yplane& then the first two of

these e#"ations are "sed to specif the motion.

>ecalling from Chap. < that the components of the acceleration are e#"al to the second

derivatives of the coordinates of the particle& we have

&&=xF mx

&&=yF my 3.H/

&&=zF mz

Keep in mind that all accelerations sho"ld %e meas"red with respect to a newtonian frame of

reference.

Proced#re for Anal(sis

The e#"ations of motion are "sed to solve pro%lems which re#"ire a relationship %etween the

forces acting on a particle and the accelerated motion the ca"se.

Free-Body Diagram

+elect the inertial coordinate sstem. Most often& rectang"lar orx& y& zcoordinates are

chosen to anal!e pro%lems for which the particle has rectilinear motion.

:nce the coordinates are esta%lished& draw the particle's free?%od diagram. 7rawing

this diagram is ver important since it provides a graphical representation that acco"nts

for all the forces which act on the particle& and there% ma$es it possi%le to resolve

these forces into theirx&y&zcomponents.

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The direction and sense of the particle's acceleration a sho"ld also %e esta%lished. 1f the

senses of its components are "n$nown& for mathematical convenience ass"me that the

are in the same direction as the positive inertial coordinate aes.

The acceleration ma %e represented as the mavector on the $inetic diagram.

1dentif the "n$nowns in the pro%lem.

Equations of Motion

1f the forces can %e resolved directl from the free?%od diagram& appl the e#"ations of

motion in their scalar component form.

1f the geometr of the pro%lem appears complicated& which often occ"rs in three

dimensions& Cartesian vector analsis can %e "sed for the sol"tion.

1f the particle contacts a ro"gh s"rface& it ma %e necessar to "se the fritiona!

equation, which relates the coefficient of $inetic friction "k to the magnit"des of the

frictional and normal forces Ffand )acting at the s"rfaces of contact& i.e.& FfE "k#.

>emem%er that Ff& alwas acts on the free?%od diagram s"ch that it opposes the motion

of the particle relative to the s"rface it contacts.

1f the particle is connected to an elastic springhaving negligi%le mass& the spring force

Fscan %e related to the deformation of the spring % the e#"ationFsE ks. *ere kis the

spring's stiffness meas"red as a force per "nit length& andsis the stretch or compression

defined as the difference %etween the deformed length $and the "ndeformed length !

i.e.&sE !@ !.

%inematis

1f the velocit or position of the particle is to %e fo"nd& it will %e necessar to appl the

proper $inematic e#"ations once the particle's acceleration is determined from FFE ma.

1f acceleration is a f"nction of time, "se a = d&'dt and &= ds'dt which& when integrated&

ield the particle's velocit and position.

1f acceleration is a f"nction of displacement, integrate ads E & d&to o%tain the velocit

as a f"nction of position.

1f acceleration is constant, "se & = &( ) at,1

+2

2

0 0= +s s v t at to determine the

velocit or position of the particle.

1n all cases& ma$e s"re the positive inertial coordinate directions "sed for writing the

$inematic e#"ations are the same as those "sed for writing the e#"ations of motion.

:therwise& sim"ltaneo"s sol"tion of the e#"ations will res"lt in errors.

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1f the sol"tion for an "n$nown vector component ields a negative scalar& it indicates

that the component acts in the direction opposite to that which was ass"med.

!*A$P+! 3.1

The 4?kgcrate shown in 9ig. 3.3a rests on a hori!ontal plane for which the coefficient of

$inetic friction is "k = .3. 1f the crate is s"%=ected to a 0?N towing force as shown&

determine the velocit of the crate in 3sstarting from rest.

9ig. 3.3

,ol#tion

sing the e#"ations of motion& we can relate the crate's acceleration to the force ca"sing the

motion. The crate's velocit can then %e determined "sing $inematics.

Free-Body Diagram.

The weight of the crate is *E mgE 4 $g 2.H( m,s


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&= &() a t E J4.(2 3/ E (4.D m,s.

!*A$P+! 3.

The (?kg%loc$shown in 9ig. 3.0ais released from rest. 1f the masses of the p"lles and

the cord are neglected& determine the speed of the


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The necessar third e#"ation is o%tained % relating a to aB "sing a dependent motion

analsis. +ince the coordinates s andsB meas"re the positions of andB from the fied

dat"m& 9ig. 3.4a& we can write


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>ecall that =tdv

adt

represents the time rate of change in the magnit"de of velocit.

Conse#"entl. if FFtacts in the direction of motion& the particle's speed will increase& whereas

if it acts in the opposite direction& the particle will slow down. Li$ewise.

2

=n va represents

the time rate of change in the velocit's direction. +ince this vector alwas acts in the positive

ndirection& i.e.& toward the path's center of c"rvat"re& then FFnwhich ca"ses analso acts in

this direction. 9or eample. when the particle is constrained to travel in a circ"lar path with a

constant speed& there is a normal force eerted on file particle % the constraint in order to

change the direction of the particle's velocit an/. +ince this force is alwas directed toward

the center of the path& it is often referred to as the centripetal force.


;hen a pro%lem involves the motion of a particle along a $nown c"rved path, normal and

tangential coordinates sho"ld %e considered for the analsis since the acceleration components

can %e readil form"lated. The method for appling the e#"ations of motion& which relate the

forces to the acceleration& has %een o"tlined in the proced"re given in +ec. 3.(.


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7etermine the %an$ing angle 1for the race trac$ so that the wheels of the racing cars shown

in 9ig. 3.Dawillnot have to depend "pon friction to prevent an car from sliding "p or down

the trac$. Ass"me the cars have negligi%le si!e& a mass m& and travel aro"nd the c"rve of

radi"s0with a speed &.

9ig. 3.D

,ol#tion

-efore loo$ing at the following sol"tion. give some tho"ght as to wh it sho"ld %e solved

"sing t& n& /coordinates.

Free-Body Diagram

Asshown in 9ig. 3.Da& and as stated in the pro%lem& no frictional force acts on the car. *ere

#+represents the resistantof the gro"nd on all fo"r wheels. +ince ancan %e calc"lated& the

"n$nowns are#+and 1.

Equations of Motion

sing the n, /aes we have

#Ccos1@mgE (/

2

sin =Cv

N m


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Pac$ages& each having a mass of < kg& are delivered from a conveor to a smooth circ"lar

ramp with a velocit of &E ( m,sas shown in 9ig. 3.a.1f the radi"s of the ramp is.4m.

determine the angle 1maxat which each pac$age %egins to leave the s"rface.

9ig. 3.

,ol#tion

Free-Body Diagram

The free?%od diagram for a pac$age& when it is located at the general position1& is shown in

9ig. 3.%. The pac$age m"st have a tangential acceleration at. since its speed is alwas

increasing as it slides downward. The weight is *E


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1 0

=4.905 sinv

vdv d

&


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to the radial coordinate % "sing the angle 2& 9ig. 3.2%, which is defined %etween the etended

radial line and the tangent to the c"rve.

This angle can %e o%tained % noting that when the particle is displaced a distance dsalong

the path 9ig. 3.2c& the component of displacement in the radial direction is dr and the

component of displacement in the transverse direction is rd1.

+ince these two components are m"t"all perpendic"lar& the angle 2can %e determined from

=tan/

r

dr d

9ig. 3.2

1f 2is calc"lated as a positive #"antit& it is meas"red from the etended radial line to the

tangent in a co"ntercloc$wise sense or in the positive direction of 1. 1f it is negative& it is

meas"red in the opposite direction to positive 1.


Clindrical or polar coordinates are a s"ita%le choice for the analsis of a pro%lem for which

data regarding the ang"lar motion of the radial line r are given& or in cases where the path can

%e convenientl epressed in terms of these coordinates. :nce these coordinates have %een

esta%lished& the e#"ations of motion can %e applied in order to relate the forces acting on the

particle to its acceleration components. The method for doing this has %een o"tlined in the

proced"re for analsis given in +ec. 3.(.


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se the methods of +ec.


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;e can calc"late r, r& &r with EDo& E and E. ;e get

r E .0D< & rE?.(33 & &r E.(2earranging the terms

and integrating %etween the limits vE v(& at tE t( &and vE v


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is clear that if each of the vectors in 5#. 3.00/ is resolved into itsx&y& andz components& we

can write sm%olicall the following three scalar e#"ationsI

+ =

+ =

+ =

2

1

2

1

2

1

) 1 2

) 1 2

) 1 2

( ) ( )

( ) ( )

( ) ( )

t

x x x

t

t

y y y

t

t

z z z

t

m v F dt m v

m v F dt m v

m v F dt m v

3.04/

Keep in mind that while $inetic energ and wor$ are scalar #"antities& moment"m and

imp"lse are vector #"antities and 5#s. 3.04/ represent the principle of linear imp"lse and

moment"m for the particle in thex&yandzdirection.


The principle of linear imp"lse and moment"m is "sed to solve pro%lems involving force&

time and velocit& since these terms are involved in the form"lation. 9or application it is

s"ggested that the following proced"re %e "sed.

Free-Body Diagram

5sta%lish thex&y&zinertial frame of reference and draw the particle's free?%od diagram

in order to acco"nt for all the forces that prod"ce imp"lses on the particle.

The direction and sense of the particle's initial and final velocities sho"ld he esta%lished.

1f a vector is "n$nown& ass"me that the sense of its components is in the direction of the

positive inertial coordinates.

Prini7!e of


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-loc$s andBshown in 9ig. 3.


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Now we can etend the principle of linear imp"lse and moment"m to a sstem of particles

moving relative to an inertial reference& 9ig. 3.


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Therefore we can state that the initial linear moment"m of the aggregate particle pl"s the

eternal imp"lses acting on the sstem of particles from t( to t


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weight will have a negligi%le effect on the change in moment"m& and therefore it is

nonimp"lsive. Conse#"entl& it can %e neglected from an imp"lse?moment"m analsis d"ring

this time. 1f an imp"lse?moment"m analsis is considered d"ring the m"ch longer time of

flight after the rac$et?%all interaction& then the imp"lse of the %all's weight is important since

it& along with air resistance& ca"ses the change in the moment"m of the %all.

9ig. 3.


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+onser&ation of >inear Momentum

mA&/(Jm-&B/(EmAJ m-/&

Lecture 3 Particle Kinetics OK

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