# Kinematics of Particles

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14 Νοε 2013 (πριν από 4 χρόνια και 6 μήνες)

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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

Velocity units would be in m/s,
ft
/s, etc.

The instantaneous velocity is

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

t 0
x
v lim
t

dx
dt

The average acceleration is

t
v
a

The units of acceleration would be m/s
2
,
ft
/s
2
, etc.

The instantaneous acceleration is

t
v
lim
a
0
t

dt
dv

dt
dx
dt
d

2
2
dt
x
d

dt
dv
a

dt
dx
dx
dv

dx
dv
v

Notice

One more derivative

dt
da
Jerk
If
v

is a function of
x
, then

Consider the function

2
3
6
t
t
x

t
12
t
3
v
2

12
t
6
a

x(m)

0

16

32

2

4

6

t(s)

v(m/s)

a(m/s
2
)

t(s)

Plotted

12

0

-
12

-
24

2

4

6

2

4

0

6

12

-
12

-
24

-
36

t(s)

11.3 DETERMINATION OF THE

MOTION OF A PARTICLE

Three common classes of motion

)
t
(
f
a
.
1

dv

t
0
0
dt
)
t
(
f
v
v
dt
)
t
(
f

dt
dv

0
v
dt
dx

t
0
0
dt
)
t
(
f
v
dt
dx

t
0
0
dt
)
t
(
f
v
dt
dx
dt
dt
)
t
(
f
t
v
x
x
t
t

0
0
0
0
dt
dt
)
t
(
f
dt
v
dx
t

0
0
dt
dt
)
t
(
f
t
v
x
x
t
0
t
0
0
0

)
x
(
f
a
.
2

vdv

x
x
o
dx
x
f
v
v
)
(
)
(
2
0
2
2
1
dt
dx
v

with

then get

)
(
t
x
x

dx
dv
v

dx
)
x
(
f

)
v
(
f
a
.
3

t
0
v
v
dt
)
v
(
f
dv
0

v
v
x
x
0
0
)
v
(
f
vdv
dx

)
t
(
x
x

or

dx
dv
v

dt
dv

t

11.4 UNIFORM RECTILINEAR

MOTION

constant
v

0
a

vdt
x
x
0
vt
x
x
0

vt

dx
v
dt

11.5 UNIFORMLY ACCELERATED

RECTILINEAR MOTION

Also

a
dx
dv
v

constant
a

at
v
v

0
2
2
1
0
at
t
v
x
x
o

)
x
x
(
a
2
v
v
0
2
0
2

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.

The relative velocity of B with respect to A

A
B
v
v
v
A
B

The relative position of B with respect to A

A
B
x
x
x
A
B

Relative motion of two particles.

The relative acceleration of B with respect to A

A
B
A
B
a
a
a

Let’s look at some dependent motions.

A

C

D

B

E

F

G

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

x
A

x
B

t
tan
cons
x
2
x
B
A

0
v
2
v
B
A

0
a
2
a
B
A

Let’s look at the relationships.

B

System has 2 degrees of freedom.

C

A

x
A

x
C

x
B

t
tan
cons
x
x
2
x
2
C
B
A

0
v
v
2
v
2
C
B
A

0
a
a
2
a
2
C
B
A

Let’s look at the relationships.

Skip this section.

11.7 GRAPHICAL SOLUTIONS OF
RECTILINEAR
-
MOTION

Skip this section.

11.8 OTHER GRAPHICAL METHODS

11.9 POSITION VECTOR, VELOCITY,
AND ACCELERATION

CURVILINEAR MOTION OF PARTICLES

x

z

y

P

P

r

r

t
r
v

s

r

t
s
s

dt
r
d
t
r
lim
v
0
t

dt
ds
v

Let’s find the instantaneous velocity.

x

z

y

P

P

r

r

v

'
v

x

z

y

t
v
a

v

x

z

y

P

P

r

r

v

'
v

x

z

y

x

z

y

t
v
a

v

t
v
lim
a
t

0
dt
v
d

Note that the acceleration is not

necessarily along the direction of

the velocity.

11.10 DERIVATIVES OF VECTOR
FUNCTIONS

u
P
lim
du
P
d
u

0

u
)
u
(
P
)
u
u
(
P
lim
0
u

du
Q
d

du
)
Q
P
(
d

du
P
d

du
P
d
f

P
du
df

du
)
P
f
(
d

du
)
Q
P
(
d

Q
du
P
d

du
Q
d
P

du
Q
d
P

du
)
Q
P
(
d

Q
du
P
d

k
ˆ
du
dP
z

i
ˆ
du
dP
x

j
ˆ
du
dP
y

du
P
d

k
ˆ
P
z

i
ˆ
P
x

j
ˆ
P
y

P

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
OF VELOCITY AND
ACCELERATION

r

k
ˆ
z

j
ˆ
y

i
ˆ
x

j
ˆ
y

v

i
ˆ
x

k
ˆ
z

j
ˆ
y

a

i
x
ˆ

k
ˆ
z

x

z

y

r

j
ˆ
y
k
ˆ
z
i
ˆ
x
x

z

y

P

v

i
ˆ
v
x
j
ˆ
v
y
k
ˆ
v
z
a

x

z

y

j
ˆ
a
y
k
ˆ
a
z
i
ˆ
a
x
a

Velocity Components in Projectile Motion

0
x
a
x

xo
x
v
x
v

t
v
x
xo

0
z
a
z

0
v
z
v
zo
z

0
z

g
y
a
y

gt
v
y
v
yo
y

2
2
1
yo
gt
t
v
y

x

z

y

x’

z’

y’

O

A

B

A
B
A
B
r
r
r
/

11.12 MOTION RELATIVE TO A
FRAME IN TRANSLATION

B
r

A
/
B
r

A
r

A
/
B
A
B
r
r
r

A
/
B
A
B
r
r
r

A
/
B
A
B
v
v
v

A
/
B
A
B
v
v
v

A
/
B
A
B
a
a
a

A
/
B
A
B
r
r
r

A
/
B
A
B
a
a
a

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.

11.13 TANGENTIAL AND NORMAL
COMPONENTS

Plane Motion of a Particle

O

x

y

t
e
ˆ
v
v

t
e
ˆ
'
t
e
ˆ
t
e
ˆ

n
e
ˆ
'
n
e
ˆ
P

P

t
0
e
ˆ
lim

t
0
n
e
ˆ
lim
e
ˆ

2
sin
2
lim
e
ˆ
0
n

d
e
ˆ
d
e
ˆ
t
n

n
e
ˆ

2
2
sin
lim
e
ˆ
0
n

t
e
ˆ
'
t
e
ˆ
t
e
ˆ

dt
v
d
a

d
e
ˆ
d
e
ˆ
t
n

t
e
ˆ
v
v

t
e
ˆ
dt
dv

dt
e
ˆ
d
v
t

n
e
ˆ
v

O

x

y

t
e
ˆ
'
t
e
ˆ
P

P

s

s

d
ds
s
lim
0

t
e
ˆ
dt
dv
a

dt
e
ˆ
d
v
t

dt
ds
ds
d
d
e
ˆ
d
dt
e
ˆ
d
t
t

v
d
e
ˆ
d
t

t
e
ˆ
dt
dv
a

n
2
e
ˆ
v

t
e
ˆ
dt
dv
a

n
2
e
ˆ
v

n
n
t
t
e
ˆ
a
e
ˆ
a
a

dt
dv
a
t

2
n
v
a

Discuss changing radius of curvature for highway curves

Motion of a Particle in Space

The equations are the same.

O

x

y

t
e
ˆ
'
t
e
ˆ
n
e
ˆ
'
n
e
ˆ
P

P

z

COMPONENTS

Plane Motion

x

y

P

r
e
ˆ

e
ˆ
r

r
e
ˆ

e
ˆ

e
ˆ

r
e
ˆ

r
e
ˆ

e
ˆ

e
ˆ
d
e
ˆ
d
r

r
e
ˆ
d
e
ˆ
d

dt
d
d
e
ˆ
d
dt
e
ˆ
d
r
r

e
ˆ

dt
d
d
e
ˆ
d
dt
e
ˆ
d

r
e
ˆ

e
ˆ
v
e
ˆ
v
r
r

r
v
r

r
v

dt
r
d
v

)
e
ˆ
r
(
dt
d
r

r
r
e
ˆ
r
e
ˆ
r

e
ˆ
r
e
ˆ
r
v
r

x

y

r
e
ˆ

e
ˆ
r

sin
j
ˆ
cos
i
ˆ
e
ˆ
r

e
ˆ
cos
j
ˆ
sin
i
ˆ
d
e
ˆ
d
r

r
e
ˆ
sin
j
ˆ
cos
i
ˆ
d
e
ˆ
d

e
ˆ
r
e
ˆ
r
v
r

e
ˆ
r
e
ˆ
r
e
ˆ
r
e
ˆ
r
e
ˆ
r
a
r
r

r
2
r
e
ˆ
r
e
ˆ
r
e
ˆ
r
e
ˆ
r
e
ˆ
r
a

e
ˆ
)
r
2
r
(
e
ˆ
)
r
r
(
a
r
2

dt
dv
a
r
r

dt
dv
a

2
r
r
r
a

r
2
r
a

Note

Extension to the Motion of a Particle in Space:

Cylindrical Coordinates

k
ˆ
z
e
ˆ
R
r
r

k
ˆ
z
e
ˆ
R
e
ˆ
R
v
R

k
ˆ
z
e
ˆ
)
R
2
R
(
e
ˆ
)
R
R
(
a
R
2