UNIT 10
COMPONENTS OF ACCELERATION AND CURVATURE
INTRO: In this Unit we will continue our study of the trajectory-aligned coordinate system
ˆ
T,
ˆ
N,
ˆ
B
, in particular how it may be used to decompose acceleration along a trajectory into
its components along and perpendicular to the trajectory. The curvature
κ
of the trajectory
will play a role in all of this.
1. Orthogonal Decomposition of a Vector
In the previous Unit we defined a
complete orthonormal
set of vectors
ˆ
T
,
ˆ
N
and
ˆ
B
for
R
3
,
determined by a trajectory in
R
3
, and a
complete orthonormal
set of vectors
ˆ
T,
ˆ
N
in
R
2
. By
orthonormal we mean that these vectors are mutually orthogonal to each other and that they
are unit vectors: they have norm 1.
By complete we mean there are 3 of them for
R
3
and
two of them for
R
2
. Completeness guarantees that all
directions
in the vector space will be
accounted for. For example, we know
ˆ
i,
ˆ
j,
ˆ
k
also form a complete orthonormal set of vectors in
R
3
and
ˆ
i,
ˆ
j
a complete orthonormal set in
R
2
. The vector
ˆ
T
is the
unit tangent vector
, which
is tangent to the trajectory at each point along the curve. For any particle traveling along the
curve, at any speed,
ˆ
T
will be pointing in the direction of its velocity. The vector
ˆ
N
is the
unit
normal vector
, perpendicular to
ˆ
T
and pointing to the center of the circle which the trajectory
is “instantaneously” tracing, therefore in the direction of its centripetal acceleration. This is
a complete set of orthonormal vectors in
R
2
. In
R
3
the vector
ˆ
B
is the
unit binormal vector
,
which is perpendicular to both
ˆ
T
and
ˆ
N
and defined so as to make a right-handed coordinate
system, thus
ˆ
B
=
ˆ
T
×
ˆ
N
. Since the set is mutually orthogonal, we know
ˆ
T
·
ˆ
N
=
ˆ
T
·
ˆ
B
=
ˆ
N
·
ˆ
B
= 0
and since the vectors are unit vectors,
||
ˆ
T
||
=
||
ˆ
N
||
=
||
ˆ
B
||
= 1
In two dimensions
ˆ
T
and
ˆ
N
form a complete “moving” coordinate system as one travels along
a trajectory. In three dimensions
ˆ
T
,
ˆ
N
and
ˆ
B
form a complete “moving” ccordinate system as
one travels along a trajectory. As long as the travel is along the trajectory,
ˆ
T
will always be
pointing in the direction of the velocity vector
~v
, and the unit normal vector
ˆ
N
will always be
perpendicular to the velocity vector.
We know that any vector
~w
=
< w
x
, w
y
, w
z
>
can be decomposed into its components along
ˆ
i,
ˆ
j,
ˆ
k
, namely,
~w
=
w
x
ˆ
i
+
w
y
ˆ
j
+
w
z
ˆ
k
, where the coefficients are just the
x, y, z
components of
~w
. We also know from Unit 5 that these coefficients are also the scalar projections of
~w
onto
the unit vectors
ˆ
i,
ˆ
j,
ˆ
k
,
w
x
=
~w
·
ˆ
i
w
y
=
~w
·
ˆ
j
w
z
=
~w
·
ˆ
k,
More generally, any vector in
R
3
can be decomposed in the same fashion in terms of any three
orthogonal unit vectors using the scalar projections onto the unit vectors (and any vector in
R
2