DYNAMICS
Dynamics is composed of 2 major topics:
(1) Kinematics
Mathematical description of motion without regards to Newton’s
Second Law.
(2) Kinetics
Basically Newton’s Second Law or Euler’s First Law.
Basic Definitions of “Particle” Kinematics
The following definitions are made in abstract algebra notation, therefore, they are valid in all
coordinate systems.
Displacement
The displacement of a particle is referred to as the
location of a particle with respect to the origin.
Usually, the “displacement” is represented by the
position vector
~
r
(
t
)
.
Velocity
The average velocity over a time period
Δ
t
is defined
as
~v
avg
=
~
r
(
t
+ Δ
t
)

~
r
(
t
)
Δ
t
.
while the instantaneous velocity at a given time
t
is
defined as
~v
=
lim
Δ
t
→
0
~v
avg
=
d~
r
dt
P
~r
ˆ
e
2
ˆ
e
3
ˆ
e
1
Acceleration
The average acceleration over a time period
Δ
t
is defined as
~a
avg
=
~v
(
t
+ Δ
t
)

~v
(
t
)
Δ
t
.
while the instantaneous acceleration at a given time
t
is defined as
~a
=
lim
Δ
t
→
0
~a
avg
=
d~v
dt
=
d
2
~
r
dt
2
Integral Equations of Motion
From the definitions of instantaneous velocity and acceleration, we can write the integral definitions
as
Z
d~v
=
Z
~a dt
and
Z
d~
r
=
Z
~v dt
.
Clearly, the above integrals preserve the vector properties. There is another integral form which is
commonly used for particles traveling along a path, i.e.,
Z
~a
·
d~
r
=
Z
~v
·
d~v
,
but this formulation retains only the tangential component. The ntcoordinate system is the best for
this particular form.
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Kinematic Formulas in Frequently Used Bases
After defining a basis, the kinematic quantities can be expressed easily in matrix algebra notations.
Cartesian Coordinates
The basis vectors used are
ˆ
ı
,
ˆ
and
ˆ
k
. The direction of
these unit vectors are fixed so their time derivatives
vanish because they have constant magnitudes and no
rotation.
In abstract algebra notation, the vectors written in
component form can be expressed as
~
r
(
t
) =
x
(
t
)ˆ
ı
+
y
(
t
)ˆ
+
z
(
t
)
ˆ
k
~v
(
t
) = ˙
x
(
t
)ˆ
ı
+ ˙
y
(
t
)ˆ
+ ˙
z
(
t
)
ˆ
k
~a
(
t
) = ¨
x
(
t
)ˆ
ı
+ ¨
y
(
t
)ˆ
+ ¨
z
(
t
)
ˆ
k
where
x
(
t
)
,
y
(
t
)
,
z
(
t
)
,
˙
x
(
t
)
,
˙
y
(
t
)
,
˙
z
(
t
)
,
¨
x
(
t
)
,
¨
y
(
t
)
and
¨
z
(
t
)
are scalar functions of time. At each instant of
time, these scalar functions represents the magnitude
of the components.
x
y
z
P
~r
ˆ
ı
ˆ
ˆ
k
The same expressions in matrix algebra notation can be written as
~
r
(
t
) =
x
(
t
)
y
(
t
)
z
(
t
)
,
~v
(
t
) =
˙
x
(
t
)
˙
y
(
t
)
˙
z
(
t
)
and
~a
(
t
) =
¨
x
(
t
)
¨
y
(
t
)
¨
z
(
t
)
Cylindrical (Polar) Coordinates
The basis vectors used are
ˆ
e
r
,
ˆ
e
θ
and
ˆ
e
z
. The direction
of
ˆ
e
r
and
ˆ
e
θ
changes with the angle
θ
but
ˆ
e
z
do not
change.
In abstract algebra notation, the position vector written
in component form is
~
r
(
t
) =
r
(
t
)ˆ
e
r
+
z
(
t
)ˆ
e
z
where
r
(
t
)
and
z
(
t
)
are scalar functions of time. There
is no component in the
ˆ
e
θ
direction for the displacement
vector but do remember that
ˆ
e
r
is a function of
θ
. To
obtain the velocity vector, take time derivative of the
displacement vector
~
r
(
t
)
to yield
~v
(
t
) =
d~
r
dt
= ˙
r
(
t
)ˆ
e
r
+ ˙
z
(
t
)ˆ
e
z
+
r
(
t
)
d
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 Spring '08
 DocWong
 Cartesian Coordinate System, Cos, Tn, Polar coordinate system, Cartesian coordinate systems

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