Equilibrium and Stability
The Earth exerts an attractive force on the mass of an object; in fact, every
small element of mass in the object is attracted by the Earth. The sum of
these forces is the total weight of the body. This weight can be considered
a force acting through a single point called the center of mass or center of
gravity. As pointed out in Appendix A, a body is in static equilibrium if the
vectorial sum of both the forces and the torques acting on the body is zero. If a
body is unsupported, the force of gravity accelerates it, and the body is not in
equilibrium. In order that a body be in stable equilibrium, it must be properly
supported.
The position of the center of mass with respect to the base of support determines
whether the body is stable or not. A body is in stable equilibrium under
the action of gravity if its center of mass is directly over its base of support
(Fig. 1.1). Under this condition, the reaction force at the base of support cancels
the force of gravity and the torque produced by it. If the center of mass
is outside the base, the torque produced by the weight tends to topple the
body (Fig. 1.1c).
The wider the base on which the body rests, the more stable it is; that is, the
more difficult it is to topple it. If the wide-based body in Fig. 1.1a is displaced
as shown in Fig. 1.2a, the torque produced by its weight tends to restore it to
its original position (Fr shown is the reaction force exerted by the surface on
the body). The same amount of angular displacement of a narrow-based body
results in a torque that will topple it (Fig. 1.2b). Similar considerations show
that a body is more stable if its center of gravity is closer to its base.
Section 1.2 Equilibrium Considerations for the Human Body
FIGURE 1.1 Stability of bodies
FIGURE 1.2 (a) Torque produced by the weight will restore the body to its original
position. (b) Torque produced by the weight will topple the body.
Equilibrium Considerations for the Human Body
The center of gravity (c.g.) of an erect person with arms at the side is at
approximately 56% of the person’s height measured from the soles of the feet
(Fig. 1.3). The center of gravity shifts as the person moves and bends. The
act of balancing requires maintenance of the center of gravity above the feet.
A person falls when his center of gravity is displaced beyond the position of
the feet.
When carrying an uneven load, the body tends to compensate by bending
and extending the limbs so as to shift the center of gravity back over the
feet. For example, when a person carries a weight in one arm, the other arm
FIGURE 1.3 Center of gravity for a person.
swings away from the body and the torso bends away from the load (Fig. 1.4).
This tendency of the body to compensate for uneven weight distribution often
causes problems for people who have lost an arm, as the continuous compensatory
bending of the torso can result in a permanent distortion of the spine. It
is often recommended that amputees wear an artificial arm, even if they cannot
use it, to restore balanced weight distribution.
Stability of the Human Body under the Action of an
External Force
The body may of course be subject to forces other than the downward force
of weight. Let us calculate the magnitude of the force applied to the shoulder
that will topple a person standing at rigid attention. The assumed dimensions
of the person are as shown in Fig. 1.5. In the absence of the force, the person
is in stable equilibrium because his center of mass is above his feet, which are
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