1
Total Differentiation of a Vector in a
Rotating Frame of Reference
•
Before we can write Newton’s second law of motion for a
reference frame rotating with the earth, we need to
develop a relationship between the total derivative of a
vector in an inertial reference frame and the
corresponding derivative in a rotating system.
k
ˆ
A
j
ˆ
A
i
ˆ
A
A
z
y
x
+
+
=
r
in an inertial frame of reference, and
k
ˆ
A
j
ˆ
A
i
ˆ
A
A
z
y
x
′
′
+
′
′
+
′
′
=
r
in a rotating frame of reference.
Let
be an arbitrary vector with Cartesian components
A
r
k
ˆ
A
j
ˆ
A
i
ˆ
A
A
z
y
x
+
+
=
r
in an inertial frame of reference, then
If
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
=
dt
dA
k
dt
k
d
A
dt
dA
j
dt
j
d
A
dt
dA
i
dt
i
d
A
dt
A
d
z
z
y
y
x
x
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
r
Since the coordinate axes are in an inertial frame of reference,
0
ˆ
ˆ
ˆ
=
=
=
dt
k
d
dt
j
d
dt
i
d
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
=
dt
dA
k
dt
k
d
A
dt
dA
j
dt
j
d
A
dt
dA
i
dt
i
d
A
dt
A
d
z
z
y
y
x
x
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
r
k
dt
dA
j
dt
dA
i
dt
dA
dt
A
d
z
y
x
ˆ
ˆ
ˆ
+
+
=
r
(Eq. 1)

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