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
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
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