Page 244 - Fisika Terapan for Engineers and Scientists

444 CHAPTER 14 Statics and Elasticity
FIGURE 14.20 Block and tackle
used for tensioning power line.
Tension applied to
leg is twice weight
of hanging mass.
FIGURE 14.21 Block and tackle in
traction apparatus for fractured leg.
Ratio of small
displacements equals
ratio of lever arms.
l
x'
x
l'
FIGURE 14.22 Rotation of lever by a
small angle produces displacements
x of the ends.
Block-and-tackle arrangements have
Tension in power
cable is twice weight many practical applications. For instance,
of hanging mass. they are used to provide the proper tension
in overhead power cables for electric trains
power cable
and trams (see Fig. 14.20); without such
an arrangement, the cables would sag on
warm days when thermal expansion
increases their length, and they would
stretch excessively tight and perhaps snap
on cold days, when they contract. One
common cause of power failures on cold
winter nights is the snapping of power lines
Hanging mass maintains lacking such compensating pulleys.
tension when power cable Another practical application of block
expands or contracts.
and tackle is found in the traction devices
used in hospitals to immobilize and align
fractured bones, especially leg bones. A typical arrangement is shown in Fig. 14.21;
here the pull applied to the leg is twice as large as the magnitude of the weight attached
on the lower end to the rope. Also, as in the case of the power line, the tension remains
constant even if the leg moves.
The mechanical advantage provided by levers, arrangements of pulleys, or other
devices can be calculated in a general and elegant way by appealing to the Law of
Conservation of Energy. A lever merely transmits the work we supply at one end to the
load at the other end. We can express this equality of work input and work output by
F ¢x F ¢x (14.16)
where x is the displacement of our hand and x the displacement of the load.
According to this equation, the forces F and F are in the inverse ratio of the dis-
placements,
F ¢x
(14.17)
F ¢x
Consider, now, the rotation of the lever by a small angle (see Fig. 14.22). Since the
two triangles included between the initial and final positions of the lever are similar,
the distances x and x are in the same ratio as the lever arms l and l ; thus, we imme-
diately recognize from Eq. (14.17) that the mechanical advantage of the lever is l l .
Likewise, we immediately recognize from Eq. (14.17) that the mechanical advan-
tage of the arrangement of pulleys shown in Fig. 14.19 is 3, since whenever our hand
pulls a length x of rope out of the upper pulley, the load moves upward by a distance
of only x 3.
✔ Checkup 14.3
x and
QUESTION 1: Figure 14.23 shows two ways of using a lever. Which has the larger
mechanical advantage?
QUESTION 2: Is Eq. (14.15) for the ratio of the forces F and F on a lever valid if one
or both of these forces are not perpendicular to the lever?
QUESTION 3: Suppose that the pulleys in a block and tackle are of different sizes. Does
this affect the mechanical advantage?