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