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Bonus 08EQUILIBRIUM
10 points for working the 10 problems
Commentary
The conditions acm= 0 and a = 0 can describe a system that is in either
static equilibrium or dynamic equilibrium; for simplicity, we will focus
primarily on cases of static equilibrium, that is, on systems for which vcm= 0
and w= 0 in the inertial frame under consideration.The first condition of
equilibrium alone is sufficient to establish equilibrium for a particle (point
mass); by definition, a particle has no extent in space, so that all forces acting
on it must act at a point and therefore the torqueabout any axis through the
particle is zero. We made use of this simplification in our study of Newton's
laws when we used vector addition to determine the single force that would
balance a set of forces acting on a particle. In effect, a particle has only three
degrees of freedom corresponding to translation along the three coordinate
axes.The first condition of equilibrium is generally not sufficient to
determine the equilibrium of an extended rigid body, since forces may act at
different points on the body and thus exert a net torque on it even though
they add vectorially to zero. This gives rise to the second condition of
equilibrium for a rigid body: that the next external torque on the body must be
zero. The two conditions of equilibrium give rise to six separate equations, all
of which must be satisfied simultaneously. This is not a trivial problem in the
general case, but fortunately it is often possible to simplify the solution since
many problems involve coplanar forces; in such cases, one force-component
equation and two torque-component equations are automatically satisfied if
the coordinate system is properly chosen.Virtually all manmade structures
designed to achieve static equilibrium exist in a uniform gravitational field.
Applying the equilibrium conditions to a rigid body in a gravitational field
would be an exceedingly complex task were it not for the fact that all of the
gravitational force components acting on the uncountable mass elements that
make up the body are equivalent to a single force applied at its center of
gravity. This single force, acting at the center of gravity, can also be used to
calculate the gravitational torque on the body about any point. Note that the
center of gravity always coincides with the center of mass if the gravitational
field is uniform, which includes virtually all cases of practical interest in
engineering design.
8-1: TRANSLATIONAL EQUILIBRIUM
OBJECTIVE: Analyze translational equilibrium problems by identifying all
forces, making a free-body diagram, and applying the first condition of
equilibrium to solve for the unknown parameters. These problems may
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