Environmental Engineering Reference
In-Depth Information
Differentiating Eq. (4.1) with respect to time gives
d
l
d
t
=
d
r
d
t
×
d
p
d
t
p
+
r
×
=
r
×
f
(4.2)
d
r
d
t
since
0
(because the momentum and velocity are in the same direction).
We have used Newton's Second Law to rewrite the rate-of-change of momentum
as the force
f
. As soon as we do so, it is to be understood that we are working in
an inertial frame. The quantity
×
p
=
τ
=
r
×
f
is known as the torque. So, for a particle,
we have
d
l
d
t
.
τ
=
(4.3)
Since this is a vector equation, it is independent of the choice of the origin of our
co-ordinate system, a statement that we can check explicitly. We can trivially write
d
d
t
(
b
b
×
f
=
×
p
),
(4.4)
where
b
is any constant vector. Eq. (4.4) can now be added to Eq. (4.3) to obtain
d
d
t
((
r
(
r
+
b
)
×
f
=
+
b
)
×
p
)
thus
d
l
d
t
,
τ
=
(4.5)
τ
and
l
where
are the torque and angular momentum calculated with respect to
the point
r
b
.Since
b
can be any constant vector we have shown that Eq. (4.3)
is always true in an inertial frame of reference, even though the vectors
l
and
=−
τ
themselves depend on the choice of origin.
4.2 CONSERVATION OF ANGULAR MOMENTUM IN SYSTEMS OF
PARTICLES
So far we have not introduced any new physics into our study of angular
momentum; we have shown that, given the definitions of angular momentum and
torque, Eq. (4.3) is just another way of expressing Newton's Second Law for
the motion of a particle. In particular, the conservation of angular momentum
seems to be nothing more than a trivial consequence of Newton's Second Law,
i.e. Eq. (4.3) tells us that d
l
/
d
t
=
=
0
. The subject really starts to address
new physics when we consider the rotation of systems of particles and of extended
bodies.
0
if
f
