Here r and f are the magnitudes of the radial and
force vectors, respectively, and f sin  is the component of f along the line DE drawn perpendicular
to the radial vector and in the plane containing r
and f (Fig. 7.4). Thus, force acting parallel to the
position vector produces no torque and force
acting at right-angles to the position vector will
contribute all of its magnitude to the torque.
The angular momentum, ⌽, is defined in terms
of the position vector of the particle, r, and the
linear momentum as:
(7.9)
Because the linear momentum acts in the same
direction as the force, the angular momentum, ⌽,
acts in the same direction as the torque, (Fig. 7.4).
That is, ⌽ acts along the normal n with a direction
determined by the right-hand convention. The
magnitude of the angular momentum is:
(7.10)
If the position vector and the linear momentum
are parallel to one another, the magnitude of the
angular momentum is zero. If the velocity of
the particle is exactly perpendicular to r, then the
magnitude of the angular momentum is simply
the product of the distance from the origin, the
mass, and the magnitude of the velocity, rmv.
Recall that the time rate of change of the linear
momentum is equivalent to the force acting on a
particle (7.6). There is an analogous relationship
for the time rate of change of the angular momentum and the torque, which is derived from (7.9) as
follows (Resnick and Halliday, 1977, p. 234):
(7.11)
Here the standard form for the derivative of a
product is used with the proviso that, for a cross
product, the order of variables must be preserved.
Also, note that the cross product of two parallel
vectors, such as v and mv, is always zero. The
torque is equal to the time rate of change of the
angular momentum.
It is important to recognize that the physical
quantities we have defined in this section refer to
ϭ r ϫ f ϩ v ϫ mv ϭ r ϫ f ϭ
d
dt
⌽ ϭ
d
dt
(r ϫ p) ϭ r ϫ
dp
dt
ϩ
dr
dt
ϫ p
⌽ ϭ r (p sin ) ϭ r (mv sin ), for 0 Յ  Յ
⌽ ϭ r ϫ mv ϭ r ϫ p
the particular location of the origin of the coordinate system, O, and inertial frame of reference
(Fig. 7.4). That is, we speak of the torque with
respect to the origin, or the angular momentum
with respect to the origin. If we were to move the
location of the origin these quantities would
change in magnitude and direction. Furthermore, the particle is not tied to the origin, so it
will move along a path defined by the line of
action of the applied force, f, rather than spin
about an axis of rotation parallel to n. The line
defined by n is not an axis of rotation for the particle per se, but rather it is the line along which
the torque and angular momentum are directed.
7.2 Rigid-body dynamics and
statics
In this section we generalize the relationships of
particle dynamics so they apply to an aggregate of
particles making up a rigid body, one that does not
change shape or size with time, and does not experience any gain or loss of mass. This rigid body,
like the particle, satisfies conservation of mass by
definition. The first step toward understanding the
dynamics of a rigid body is to define the center of
mass. Then we relate the forces acting on the body
to its linear and angular momentum. Conservation of momentum is the underlying principle for
rigid-body dynamics and provides the necessary
conditions for static equilibrium of such a body.
7.2.1 Center of mass
To define the center of mass no restrictions need
be placed on the size of the body: it may represent
a few cubic meters or many cubic kilometers.
Also, no restrictions are placed on the complexity
of the geological structures within the body. For
example, consider the structural block diagram
(Chapter 7, frontispiece) of a part of the Penninic
Alps (Argand, 1911). The distance across the front
of the block is about 75 km and the distance from
front to back is about 50 km, so this is a crustalscale diagram. Note how the map pattern of
the rocks is reflected in the cross section on the
front of the block. This repetition of patterns
occurs because the structures are plunging to the
248
CONSERVATION OF MASS AND MOMENTUM
force vectors, respectively, and f sin  is the component of f along the line DE drawn perpendicular
to the radial vector and in the plane containing r
and f (Fig. 7.4). Thus, force acting parallel to the
position vector produces no torque and force
acting at right-angles to the position vector will
contribute all of its magnitude to the torque.
The angular momentum, ⌽, is defined in terms
of the position vector of the particle, r, and the
linear momentum as:
(7.9)
Because the linear momentum acts in the same
direction as the force, the angular momentum, ⌽,
acts in the same direction as the torque, (Fig. 7.4).
That is, ⌽ acts along the normal n with a direction
determined by the right-hand convention. The
magnitude of the angular momentum is:
(7.10)
If the position vector and the linear momentum
are parallel to one another, the magnitude of the
angular momentum is zero. If the velocity of
the particle is exactly perpendicular to r, then the
magnitude of the angular momentum is simply
the product of the distance from the origin, the
mass, and the magnitude of the velocity, rmv.
Recall that the time rate of change of the linear
momentum is equivalent to the force acting on a
particle (7.6). There is an analogous relationship
for the time rate of change of the angular momentum and the torque, which is derived from (7.9) as
follows (Resnick and Halliday, 1977, p. 234):
(7.11)
Here the standard form for the derivative of a
product is used with the proviso that, for a cross
product, the order of variables must be preserved.
Also, note that the cross product of two parallel
vectors, such as v and mv, is always zero. The
torque is equal to the time rate of change of the
angular momentum.
It is important to recognize that the physical
quantities we have defined in this section refer to
ϭ r ϫ f ϩ v ϫ mv ϭ r ϫ f ϭ
d
dt
⌽ ϭ
d
dt
(r ϫ p) ϭ r ϫ
dp
dt
ϩ
dr
dt
ϫ p
⌽ ϭ r (p sin ) ϭ r (mv sin ), for 0 Յ  Յ
⌽ ϭ r ϫ mv ϭ r ϫ p
the particular location of the origin of the coordinate system, O, and inertial frame of reference
(Fig. 7.4). That is, we speak of the torque with
respect to the origin, or the angular momentum
with respect to the origin. If we were to move the
location of the origin these quantities would
change in magnitude and direction. Furthermore, the particle is not tied to the origin, so it
will move along a path defined by the line of
action of the applied force, f, rather than spin
about an axis of rotation parallel to n. The line
defined by n is not an axis of rotation for the particle per se, but rather it is the line along which
the torque and angular momentum are directed.
7.2 Rigid-body dynamics and
statics
In this section we generalize the relationships of
particle dynamics so they apply to an aggregate of
particles making up a rigid body, one that does not
change shape or size with time, and does not experience any gain or loss of mass. This rigid body,
like the particle, satisfies conservation of mass by
definition. The first step toward understanding the
dynamics of a rigid body is to define the center of
mass. Then we relate the forces acting on the body
to its linear and angular momentum. Conservation of momentum is the underlying principle for
rigid-body dynamics and provides the necessary
conditions for static equilibrium of such a body.
7.2.1 Center of mass
To define the center of mass no restrictions need
be placed on the size of the body: it may represent
a few cubic meters or many cubic kilometers.
Also, no restrictions are placed on the complexity
of the geological structures within the body. For
example, consider the structural block diagram
(Chapter 7, frontispiece) of a part of the Penninic
Alps (Argand, 1911). The distance across the front
of the block is about 75 km and the distance from
front to back is about 50 km, so this is a crustalscale diagram. Note how the map pattern of
the rocks is reflected in the cross section on the
front of the block. This repetition of patterns
occurs because the structures are plunging to the
248
CONSERVATION OF MASS AND MOMENTUM