CHAPTER 11CHAPTER 11

Kinematics of Kinematics of ParticlesParticles

11.1 INTRODUCTION TO11.1 INTRODUCTION TO DYNAMICS DYNAMICS

Galileo and Newton (Galileo’s Galileo and Newton (Galileo’s

experiments led to Newton’s experiments led to Newton’s

laws)laws) Kinematics – study of motionKinematics – study of motion Kinetics – the study of what Kinetics – the study of what

causes changes in motioncauses changes in motion Dynamics is composed of Dynamics is composed of

kinematics and kineticskinematics and kinetics

RECTILINEAR MOTION OF RECTILINEAR MOTION OF PARTICLESPARTICLES

Velocity units would be in m/s, ft/s, Velocity units would be in m/s, ft/s, etc.etc.The instantaneous velocity isThe instantaneous velocity is

11.2 POSITION, VELOCITY, AND11.2 POSITION, VELOCITY, AND ACCELERATION ACCELERATION

For linear motion x marks the position of an For linear motion x marks the position of an object. Position units would be m, ft, etc.object. Position units would be m, ft, etc.Average velocity is Average velocity is

xv

t

t 0

xv lim

t

dxdt

The average acceleration isThe average acceleration is

t

va

The units of acceleration would be m/sThe units of acceleration would be m/s22, ft/s, ft/s22, , etc.etc.

The instantaneous acceleration isThe instantaneous acceleration is

t

vlima

0t

dt

dv

dt

dx

dt

d

2

2

dt

xd

dt

dva

dt

dx

dx

dv

dx

dvv

NoticeNotice

One more One more derivativederivative

dt

daJerk

If If vv is a function of is a function of xx, then, then

Consider the functionConsider the function

23 6ttx

t12t3v 2

12t6a

x(m)

0

16

32

2 4 6

t(s)

v(m/s)

a(m/s2)

t(s)

PlottedPlotted

12

0

-12

-24

2 4 6

2 40 6

12

-12

-24

-36

t(s)

11.3 DETERMINATION OF THE11.3 DETERMINATION OF THEMOTION OF A PARTICLEMOTION OF A PARTICLE

Three common classes of motionThree common classes of motion

)t(fa.1

adtdv

t

0

0 dt)t(fvv

dt)t(fdtdv

0vdtdx

t

0

0 dt)t(fvdtdx

t

0

0 dt)t(fvdtdx

dtdt)t(ftvxxt t

0 0

00

dtdt)t(fdtvdxt

0

0

dtdt)t(ftvxxt

0

t

0

00

)x(fa.2

adxvdv

x

xo

dxxfvv )()( 20

221

dt

dxv withwith then getthen get )(txx

dx

dvv

dx)x(f

)v(fa.3

t

0

v

v

dt)v(f

dv

0

v

v

x

x 00)v(f

vdvdx Both can lead Both can lead

to to

)t(xx

oror

dx

dvv

dt

dv

t

11.4 UNIFORM RECTILINEAR11.4 UNIFORM RECTILINEARMOTIONMOTION

constantv 0a

vdtxx 0

vtxx 0 vt

dxv

dt

11.5 UNIFORMLY ACCELERATED11.5 UNIFORMLY ACCELERATEDRECTILINEAR MOTIONRECTILINEAR MOTION

AlsAlso o a

dx

dvv

constanta atvv 0

221

0 attvxx o

)xx(a2vv 020

2

11.6 MOTION OF SEVERAL11.6 MOTION OF SEVERAL PARTICLES PARTICLES

When independent particles move along the When independent particles move along the same line, same line, independent equations exist for each.independent equations exist for each.Then one should use the same origin and Then one should use the same origin and time.time.

The relative velocity of B with respect to The relative velocity of B with respect to A A

AB vvvA

B

The relative position of B with respect to AThe relative position of B with respect to A

AB xxxA

B

Relative motion of two particles.Relative motion of two particles.

The relative acceleration of B with respect to The relative acceleration of B with respect to AA

ABA

Baaa

Let’s look at some dependent motions.Let’s look at some dependent motions.

A

C D

B

E F

G

System has one degree of System has one degree of freedom since only one freedom since only one coordinate can be chosen coordinate can be chosen independently.independently.

xA

xB

ttanconsx2xBA

0v2vBA

0a2aBA

Let’s look at the relationships.Let’s look at the relationships.

B

System has 2 degrees of freedom.System has 2 degrees of freedom.

C

A

xA

xC

xB

ttanconsxx2x2CBA

0vv2v2CBA

0aa2a2CBA

Let’s look at the relationships.Let’s look at the relationships.

Skip this section.Skip this section.

11.7 GRAPHICAL SOLUTIONS OF 11.7 GRAPHICAL SOLUTIONS OF RECTILINEAR-MOTIONRECTILINEAR-MOTION

Skip this section.Skip this section.

11.8 OTHER GRAPHICAL 11.8 OTHER GRAPHICAL METHODSMETHODS

11.9 POSITION VECTOR, VELOCITY, 11.9 POSITION VECTOR, VELOCITY, AND ACCELERATIONAND ACCELERATION

CURVILINEAR MOTION OF PARTICLESCURVILINEAR MOTION OF PARTICLES

x

z

y

P

P’

r

r

trv

sr

tss

dtrd

trlimv

0t

dt

dsv

Let’s find the instantaneous velocity.Let’s find the instantaneous velocity.

x

z

y

P

P’

r

r

v

'v

x

z

y

tva

v

x

z

y

P

P’

r

r

v

'v

x

z

y

x

z

y

tva

vt

vlimat

0 dt

vd

Note that the acceleration is not Note that the acceleration is not necessarily along the direction ofnecessarily along the direction ofthe velocity.the velocity.

11.10 DERIVATIVES OF VECTOR 11.10 DERIVATIVES OF VECTOR FUNCTIONSFUNCTIONS

uPlim

duPd

u

0

u

)u(P)uu(Plim

0u

duQd

du

)QP(d

du

Pd

duPdf

Pdudf

du

)Pf(d

du)QP(d

Qdu

Pd

duQd

P

duQd

P

du)QP(d

Qdu

Pd

kdu

dPzidu

dPx jdu

dPydu

Pd

kPziPx

jPyP

Rate of Change of a Vector

The rate of change of a vector is the same with respect to a fixed frame and with respect to a frame in translation.

11.11 RECTANGULAR COMPONENTS 11.11 RECTANGULAR COMPONENTS OF VELOCITY AND OF VELOCITY AND

ACCELERATIONACCELERATION

r

kzjyix

jyv

ix kz

jya

ixˆ kz

x

z

y

r

jy

kz

ix

x

z

y

P

v

ivx

jvy

kvz

a

x

z

y

jay

kaz

iax

a

Velocity Components in Projectile MotionVelocity Components in Projectile Motion

0xax

xoxvxv

tvxxo

0za

z

0vzvzoz

0z

gyay

gtvyvyoy

2

21

yogttvy

x

z

y

x’

z’

y’

O

A

B

ABAB rrr /

11.12 MOTION RELATIVE TO A 11.12 MOTION RELATIVE TO A FRAME IN TRANSLATIONFRAME IN TRANSLATION

Br A/B

r

Ar

A/BABrrr

A/BABrrr

A/BABvvv

A/BABvvv

A/BABaaa

A/BABrrr

A/BABaaa

Velocity is tangent to the path of a particle.Velocity is tangent to the path of a particle.

Acceleration is not necessarily in the same Acceleration is not necessarily in the same direction.direction.

It is often convenient to express the It is often convenient to express the acceleration in terms of components tangent acceleration in terms of components tangent and normal to the path of the particle.and normal to the path of the particle.

11.13 TANGENTIAL AND NORMAL 11.13 TANGENTIAL AND NORMAL COMPONENTSCOMPONENTS

Plane Motion of a ParticlePlane Motion of a Particle

O x

y

tevv

t

e

'

te

te

ne'

ne

P

P’

t

0

elim

t

0n

elime

2sin2lime

0n

d

ede t

n

ne

2

2sinlime

0n

te

'

te

te

dt

vda

d

ede t

n

tevv

tedt

dv

dt

edv t

nev

O x

y

te

'

te

P

P’

s

s

d

dsslim

0

tedt

dva

dt

edv t

dt

ds

ds

d

d

ed

dt

ed tt

v

d

ed t

tedt

dva

n

2

ev

tedt

dva

n

2

ev

nntt eaeaa

dt

dvat

2

n

va

Discuss changing radius of curvature for highway curvesDiscuss changing radius of curvature for highway curves

Motion of a Particle in SpaceMotion of a Particle in Space

The equations are the same.The equations are the same.

O x

y

te

'

te

ne'

ne

P

P’

z

11.14 RADIAL AND TRANSVERSE 11.14 RADIAL AND TRANSVERSE COMPONENTSCOMPONENTS

Plane MotionPlane Motion

x

y

P

ree

r

ree

ere re

e

e

d

ed r red

ed

dt

d

d

ed

dt

ed rr

e

dt

d

d

ed

dt

ed re

evev rr

rvr rv

dt

rdv

)er(

dt

dr rr erer

ererv r

x

y

ree

r

sinjcosier

ecosjsinid

ed r

resinjcosid

ed

ererv r

ererererera rr

r2

r ererererera

e)r2r(e)rr(a r2

dt

dva r

r dt

dva

2r rra

r2ra

Note

Extension to the Motion of a Particle in Space: Extension to the Motion of a Particle in Space: Cylindrical CoordinatesCylindrical Coordinates

kzeRr r

kzeReRv R

kze)R2R(e)RR(a R2