CHAPTER 11
Kinematics of Particles
11.1 INTRODUCTION TO
DYNAMICS




Galileo and Newton (Galileo’s
experiments led to Newton’s laws)
Kinematics – study of motion
Kinetics – the study of what causes
changes in motion
Dynamics is composed of kinematics
and kinetics
RECTILINEAR MOTION OF
PARTICLES
11.2 POSITION, VELOCITY, AND
ACCELERATION
For linear motion x marks the position of an
object. Position units would be m, ft, etc.
Average velocity is
x
v
t
Velocity units would be in m/s, ft/s, etc.
The instantaneous velocity is
 x dx
v  lim

t 0 t
dt
The average acceleration is
v
a
t
The units of acceleration would be m/s2, ft/s2, etc.
The instantaneous acceleration is
v dv d dx d x

 2

a  lim
t  0  t
dt dt dt dt
2
Notice
If v is a function of x, then
dv dv dx
dv
a

v
dt dx dt
dx
One more derivative
da
 Jerk
dt
Consider the function
x(m)
Plotted
32
x  t  6t
3
2
16
0
2
4
6
2
4
6
2
4
6
t(s)
v(m/s)
12
0
t(s)
-12
v  3t  12t
2
-24
-36
a(m/s2)
12
0
a  6 t  12
-12
-24
t(s)
11.3 DETERMINATION OF THE
MOTION OF A PARTICLE
Three common classes of motion
dv
1. a  f ( t ) 
dt
dv  adt  f ( t )dt
t
dx
 v0
v  v0   f ( t )dt 
dt
0
t
dx  v  f ( t )dt
0

dt
0
t
dx  v  f ( t )dt
0

dt
0


dx  v0 dt    f ( t )dt  dt
0

t


x  x0  v0 t     f ( t )dt  dt
0 0

t
t


x  x0  v0 t     f ( t )dt dt
0 0

t
t
dv
2. a  f ( x )  v
dx
vdv  adx  f ( x )dx
x
1
2
with
(v  v ) 
2
dx
v
dt
2
0

xo
then get
f ( x)dx
x  x(t )
dv
dv
v
3. a  f ( v ) 
dt
dx
v
t
dv

dt

t
v f ( v ) 0
0
x
or
v
dx

 
x0
v0
vdv
f(v)
Both can lead to
x  x( t )
11.4 UNIFORM RECTILINEAR
MOTION
v  constant
a 0
dx
v
dt
x  x0   vdt  vt
x  x0  vt
11.5 UNIFORMLY ACCELERATED
RECTILINEAR MOTION
Also
a  constant
v  v0  at
2
1
x  xo  v0 t  2 at
dv
v
a
dx
v  v  2 a( x  x0 )
2
2
0
11.6 MOTION OF SEVERAL
PARTICLES
When independent particles move along the same
line, independent equations exist for each.
Then one should use the same origin and time.
Relative motion of two particles.
The relative position of B with respect to A
xB A  xB  x A
The relative velocity of B with respect to A
vB A  vB  v A
The relative acceleration of B with respect to A
aB  aB  a A
A
Let’s look at some dependent motions.
G
xA
C
D
A
xB
E
F
B
System has one degree of
freedom since only one
coordinate can be chosen
independently.
Let’s look at the relationships.
x A  2 xB  cons tan t
v A  2v B  0
a A  2 aB  0
xC
xA
C
xB
A
B
System has 2 degrees of freedom.
2 x A  2 xB  xC  cons tan t
Let’s look at the relationships.
2v A  2vB  vC  0
2 a A  2 a B  aC  0
11.7 GRAPHICAL SOLUTIONS OF
RECTILINEAR-MOTION

Skip this section.
11.8 OTHER GRAPHICAL METHODS

Skip this section.
CURVILINEAR MOTION OF PARTICLES
11.9 POSITION VECTOR, VELOCITY,
AND ACCELERATION
 r
v
 t
y


v  lim r  dr
t  0  t
dt
P’

r

r

r s
P
s  s
t
x
ds
v
dt
Let’s find the instantaneous velocity.
z

v
'
v
y

v
P’

r

r
z
P
x

 v
a
t



a  lim v  dv
t  0  t
dt
y

v
'
v
x
y

v
P’

r

r
z
z

 v
a
t
Note that the acceleration is not
necessarily along the direction of
the velocity.
P
x
11.10 DERIVATIVES OF VECTOR
FUNCTIONS




dP  lim P  lim  P( u  u )  P( u ) 

u 0 

u

0
u
du
u



 

d ( P  Q ) dP dQ


du
du
du


d ( fP ) df 
d
P

P f
du
du
du
 
   
dQ
d ( P  Q ) dP

Q  P 
du
du
du
 

  
d ( P  Q ) dP
dQ

 Q P 
du
du
du

dP  dPx î  dPy ˆj  dPz k̂
du du
du
du
Rate of Change of a Vector

P  Px î  Py ˆj  Pz k̂
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
OF VELOCITY AND
ACCELERATION

ˆ
r  xî  yj  zk̂

ˆ
v  xî  y j  zk̂

ˆ
ˆ


a  xi  yj  zk̂

a
v y ˆj
y
v x î
v z k̂
y

v
yˆj
P
z
zk̂
z
x

r
xî
x

a
y
a y ˆj
x
a z k̂
z
a x î
Velocity Components in Projectile Motion
a x  x  0
vx  x  vxo
x  vxot
a y  y   g
az  z  0
vz  z  vzo  0
v y  y  v yo  gt
z 0
y  v yot  gt
1
2
2
11.12 MOTION RELATIVE TO A
FRAME IN TRANSLATION
y’
y
O
z

rB / A

rB

rA
B
A
z’
x
x’
  
rB  rA  rB / A
  
rB  rA  rB / A
  
rB  rA  rB / A
  
vB  v A  vB / A
  
vB  v A  vB / A
  
aB  a A  aB / A
  
aB  a A  aB / A
  
rB  rA  rB / A
11.13 TANGENTIAL AND NORMAL
COMPONENTS
Velocity is tangent to the path of a particle.
Acceleration is not necessarily in the same
direction.
It is often convenient to express the
acceleration in terms of components tangent
and normal to the path of the particle.
Plane Motion of a Particle
êt'
y
ê
t
ên
ên'
P’

v  vêt
êt
P
O
x

ê
t
'
êt

êt
êt
êt
 2 sin  2  
 ên lim
lim
 ên lim 

 0 
 0 
 0 

 sin  2 
 ên lim 
 ên

 0
  2 
dêt
ên 
d
dêt
ên 
d

v  vêt

 dv dv
dêt
a
 êt  v
dt dt
dt
 dv
dêt
a  êt  v
dt
dt
êt'
y
s  


êt
P’
s ds
  lim

 0 
d
s
P
O
x
dêt dêt d ds dêt v
v

 ên

dt d ds dt d  
 dv
v2
a  êt  ên

dt
2
 dv
v
a  êt  ên

dt

a  at êt  an ên
dv
at 
dt
an 
v
2

Discuss changing radius of curvature for highway curves
Motion of a Particle in Space
êt'
y
ên
ên'
P’
êt
P
O
z
The equations are the same.
x
11.14 RADIAL AND TRANSVERSE
COMPONENTS
Plane Motion
ê
y
êr

r
P

x
ê ê
ê
dêr
 ê
d
êr êr
êr
dê
 êr
d
dêr dêr d 
  ê

dt
d dt
dê dê d

  êr
dt
d dt

 dr d
v
 ( rêr )  rêr  rêr
dt dt

v  rêr  r ê  vr êr  v ê
vr  r
v  r
ê
y
êr


r

x
êr  î cos   ˆj sin 
dêr
 î sin   ˆj cos   ê
d
dê
 î cos   ˆj sin   êr
d

v  rêr  r ê

a  rêr  r êr  r ê  rê  r ê

2





a  rêr  r ê  r ê  r ê  r êr

a  ( r  r 2 )êr  ( r  2 r ) ê
ar  r  r 2
Note
dvr
ar 
dt
a  r  2 r
dv
a 
dt
Extension to the Motion of a Particle in Space:
Cylindrical Coordinates

r  Rêr  zk̂
 

v  RêR  R ê  zk̂

2







a  ( R  R )êR  ( R  2 R )ê  zk̂