Page 196 - Dynamics of Mechanical Systems

0593_C06_fm Page 177 Monday, May 6, 2002 2:28 PM

Forces and Force Systems 177

and

M = ( M ⋅ )
*
R
O n n =−11 .03 m 1 − 88 .22 m 2 − 88 .22 m ftlb (6.6.21)
R
3
*
F passes through point Q located relative to O by the expression:
OQ = ( R M ) 2
×
*
1
2
O R =−2 .853 n + 3 .742 n − 3 .385 n 3 (6.6.22)
*
Observe that Q is not a geometrically significant point of the box. This means that wrenches,
although physically simple, may not be especially convenient in practical problems.

6.7 Physical Forces: Applied (Active) Forces

The forces acting upon a mechanical system may be divided into two categories: applied
forces and inertia forces, or alternatively into active forces and passive forces, respectively.
Applied (or active) forces arise externally to the system. They are generally composed of
gravity, spring, and/or contact forces. Occasionally, they will arise from electrical, mag-
netic, or radiation fields. Inertia (or passive) forces arise due to motion (that is, acceleration)
of the system.
For ideal mechanical systems with rigid bodies, we may replace the forces by equivalent
force systems. For nonideal systems, but for systems with nearly rigid bodies whose
geometry does not change significantly during the application of the forces, we may still
replace the forces by an equivalent force system if we neglect local effects at the points of
application (see Section 6.5).

6.7.1 Gravitational Forces
For gravitational (or weight) forces, we may replace the forces on bodies by a single force
passing through the center of gravity of the body. For bodies near the surface of the Earth,
having dimensions small compared with the radius of the Earth, the center of gravity may
be identified with the mass center of the body (see Section 6.8). For homogeneous bodies,
the mass center is at the centroid of the geometrical figure occupied by the body. The
magnitude of the single force is simply the weight of the body (the product of the mass
and the gravity acceleration). The line of action of the force passes through the mass center
and the Earth center, and the force is directed toward the Earth center.
For nonhomogeneous bodies, the mass center must be located using definitions and
procedures presented in Section 6.8. For systems remote from the Earth’s surface, if the
gravitational forces are replaced by a single force passing through the mass center, there
must be an accompanying couple. The torque of this couple (called the gravitational
moment) is generally non-zero (although in many cases its magnitude may be insignificant).
For systems with a significant gravitational moment, it is also possible to find a point
through which the gravitational forces may be replaced by an equivalent single force. This
point is then the center of gravity. In the sequel, we will not consider such systems. We
will therefore make no distinction between the mass center and the center of gravity.