Environmental Engineering Reference
In-Depth Information
angular momentum vector is constant, it follows that the the position vector lies
always in the same plane and hence the motion is planar.
For motion in a plane, we can use polar co-ordinates to write (see Section 1.3.4
2
)
r θ
e
θ
˙
r
=˙
r
e
r
+
(9.41)
thus, using Eq. (9.39),
µr
2
θ
e
r
×
L
=
e
θ
.
(9.42)
Example 9.4.1
Show that the radius vector
r
sweeps out area at a constant rate.
Solution 9.4.1
Figure 9.7 illustrates how the radius vector sweeps out area in a
plane. For an infinitesimal displacement
d
r
=
r
d
θ , the area swept out is
1
2
r
2
d
θ
=
d
A
and hence
d
A
d
t
1
2
r
2
θ.
=
θ it follows that
µr
2
Since
|
L
|=
L
=
d
A
d
t
L
2
µ
=
(9.43)
which is a constant of the motion. This result is often known as Kepler's Second
Law.
d
r
r
Figure 9.7 Kepler's Second Law informs us that the radius vector
r
sweeps out area at a
constant rate.
2
Note that we have changed to the notation more commonly used for basis vectors in advanced dynam-
ics, i.e.
e
r
≡
ˆ
θ
r
and
e
θ
≡
.