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
.
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