Environmental Engineering Reference
In-Depth Information
From the properties of the vector product this is a vector perpendicular to the plane
containing
r
and
p
.
y
p
q
p − q
r
⊥
r
x
Figure 4.1
Particle travelling in the
x
-
y
plane.
Example 4.1.1
A particle moves with momentum
p
and position vector
r
as shown
in Figure 4.1. Calculate the angular momentum.
Solution 4.1.1
=
|
|
=
l
r
×
p
rp
sin
θ
but
sin
θ
=
sin
(π
−
θ) so
l
=
r
⊥
p.
We say that
is the moment of
p
about the origin and the direction of
l
may be
determined as explained in Chapter 1. Turning the fingers of the right-hand from
the direction of
r
to that of
p
causes the thumb to point out of the page, i.e. in the
positive z-direction.
|
l
|
Note that
l
depends on the choice of origin, as the following example demon-
strates.
Example 4.1.2
A particle is travelling with momentum p along the positive y -axis
of a Cartesian co-ordinate system. Calculate the angular momentum relative to: (a)
the origin; (b) the point (a,0,0).
Solution 4.1.2
Relative to the origin the angular momentum is
ijk
0
y
0
0
p
0
l
=
r
×
p
=
=
0
.
Relative to the point (a, 0, 0) however the position vector of the particle becomes
(
−
a, y,
0
) and
i jk
−
l
=
ay
0
0
=−
ap
k
.
p
0