Fundamental Engineering Concepts - Mathematics and Statistics for the Environment - page 233

If a force is applied to an object and no movement occurs, no effective work is done. The energy pos-
sessed by a body determines the amount of work it can do. Newton's third law states that for every
action there is an equal and opposite reaction.

■ EXAMPLE 11.2
Problem: A system of pulleys having a mechanical advantage of 5 is used to move a 1-ton weight
up an inclined plane. The plane has an angle of 45° from horizontal. The weight has to be moved
a vertical distance of 30 ft. Ignoring friction, how much work is required to be done to move the
weight up to the top (vertical lift work)?

Solution:

W = F × s

W = 2000 lb × 30 ft = 60,000 ft-lb

With regard to the safety aspects of environmental engineering, a key relationship between force
F and a body on which it acts is

F = s × A (11.4)

where
s = Force or stress per unit area (e.g., pounds per square inch).
A = Area (square inches, square feet, etc.) over which a force acts.

Frequently, two or more forces act together to produce the effect of a single force, called a resul-
tant . This resolution of forces can be explained by either the triangle law or the parallelogram law.
The triangle law provides that if two concurrent forces are laid out vectorially with the beginning
of the second force at the end of the first, the vector connecting the beginning and the end of the
forces represents the resultant of the two forces (see Figure 11.1A). The parallelogram law provides
that if two concurrent forces are laid out vectorially, with either forces pointing toward or both
away from their point of intersection, a parallelogram represents the resultant of the force. The
concurrent forces must have both direction and magnitude if their resultant is to be determined (see
Figure 11.1B). If the individual forces are known or if one of the individual forces and the resultant
are known, the resultant force may be simply calculated by either the trigonometric method (sines,
cosines, and tangents) or the graphic method (which involves laying out the known force, or forces,
at an exact scale and in the exact directions in either a parallelogram or triangle and then measuring
the unknown to the same scale).

11.3 SLINGS
Slings must be used in accordance with recommendations of the sling manufacturer (PNNL, 2013).
Slings manufactured from conventional three-strand natural or synthetic fiber rope are not recom-
mended for use in lifting service. Natural or synthetic fiber rope slings must be used only if other
sling types are not suitable for the unique application. For natural or synthetic rope slings, the

DID YOU KNOW?

The stress a material can withstand is a function of the material and the type of loading.